Energy-based model reduction of transport-dominated phenomena [original]

Energy-Based Model Reduction of

Transport-Dominated Phenomena

vorgelegt von

M. Sc.

Philipp Schulze

ORCID: 0000-0002-7299-4628

von der Fakultät II - Mathematik und Naturwissenschaften

der Technischen Universität Berlin

zur Erlangung des akademischen Grades

Doktor der Naturwissenschaften

- Dr. rer. nat. -

genehmigte Dissertation

Promotionsausschuss:

Vorsitzender: Prof. Dr. Wilhelm Stannat

Gutachter: Prof. Dr. Volker Mehrmann

Gutachter: Prof. Dr. Matthias Heinkenschloss

Gutachter: Prof. Dr. Benjamin Peherstorfer

Tag der wissenschaftlichen Aussprache: 17. April 2023

Berlin 2023

Abstract

Transport-dominated systems are characterized by the propagation of waves

and occur in many applications such as aerodynamics and chemical engineer-

ing. To predict the dynamics of such systems, mathematical models should

ideally be fast to evaluate and at the same time sufficiently accurate. One

possibility for deriving such models is to start with a complex and accurate

full-order model (FOM) and use model order reduction (MOR) techniques to

obtain a corresponding reduced-order model (ROM). Classical MOR methods

are based on approximating the FOM state by a linear combination of ansatz

functions or modes, but such approaches are often inadequate in the context of

transport-dominated systems. This is one of the reasons why there has been an

increasing research effort in the past years to develop MOR techniques which

are based on nonlinear approximation ansatzes.

As the field of nonlinear MOR is relatively new, there are still many open

research questions to be addressed. These include for instance suitable choices

for the approximation ansatz as well as appropriate ways for the construction

of corresponding ROMs. Furthermore, nonlinear MOR approaches typically

lead to ROMs whose evaluation scales with the dimension of the FOM and thus

may be too expensive. In fact, similar issues may also occur in the context

of linear MOR approaches and, therefore, one uses so-called hyperreduction

techniques to obtain fast ROMs. However, classical hyperreduction methods

suffer from similar difficulties as classical MOR schemes when being applied to

transport-dominated systems. Another challenge is to develop nonlinear MOR

techniques which preserve important system properties such as stability.

In this thesis, we present a new nonlinear model reduction framework which

is based on approximating the state of the FOM by a linear combination of

transformed modes. The transformations may be, e.g., achieved by shift opera-

tors and are parametrized by so-called paths or shift amounts, which constitute

a part of the ROM state. The resulting class of ansatzes is well-suited for ob-

taining low-dimensional and accurate approximations of transport-dominated

systems. For the determination of the modes, we present an optimization ap-

proach based on given snapshot data of the FOM state. Furthermore, the

construction of the ROM is carried out via a residual minimization approach

and we also suggest a new hyperreduction framework to ensure that the ROM

can be efficiently evaluated. In addition, we demonstrate how to preserve sta-

bility via an energy-based formulation using the framework of so-called port-

Hamiltonian systems. Finally, we illustrate the new methodology by means of

numerical experiments for some transport-dominated test cases.

iii

Zusammenfassung

Transportdominierte Systeme sind durch die Ausbreitung von Wellen charak-

terisiert und kommen in vielen Anwendungen vor, z. B. in der Aerodynamik

und Verfahrenstechnik. Um die Dynamik solcher Systeme vorherzusagen, soll-

ten mathematische Modelle schnell auswertbar und dabei hinreichend genau

sein. Hierzu kann man z. B. ausgehend von einem sehr genauen Originalmod-

ell (FOM) mit Verfahren der Modellreduktion (MOR) ein reduziertes Modell

(ROM) herleiten. Klassische MOR-Methoden basieren auf der Approxima-

tion des FOM-Zustandes durch eine Linearkombination von Ansatzfunktionen

bzw. Moden. Solche Ansätze sind jedoch bei transportdominierten Systemen

oft unzureichend. Unter anderem deshalb wird seit ein paar Jahren vermehrt

an MOR-Verfahren geforscht, die auf nichtlinearen Ansätzen basieren.

Da das Gebiet der nichtlinearen Modellreduktion relativ neu ist, gibt es

noch viele offene Forschungsfragen. Diese betreffen z.B. eine adäquate Wahl

des MOR-Ansatzes sowie geeignete Methoden für die Erstellung von ROMs

basierend auf einem konkreten Ansatz. Ferner führen nichtlineare Ansätze

häufig zu ROMs, deren Auswertung mit der Dimension des FOMs skaliert

und dadurch zu aufwendig sein kann. Ähnliche Probleme können auch bei

linearen MOR-Ansätzen auftreten und daher werden sogenannte Hyperreduk-

tionsverfahren verwendet, um effizient auswertbare ROMs zu erhalten. Klas-

sische Hyperreduktionsmethoden sind jedoch bei der Anwendung auf trans-

portdominierte Systeme von ähnlichen Schwierigkeiten betroffen wie klassische

MOR-Verfahren. Eine weitere Herausforderung ist die Entwicklung nichtlin-

earer MOR-Methoden, welche Eigenschaften wie Stabilität erhalten.

In der vorliegenden Arbeit stellen wir eine nichtlineare MOR-Methode vor,

die auf der Approximation des FOM-Zustandes durch eine Linearkombination

von transformierten Moden basiert. Die Transformationen können z. B. durch

Translationen realisiert werden und sind durch sogenannte Pfade bzw. Transla-

tionsstrecken parametrisiert, die einen Teil des ROM-Zustandes bilden. Solche

Ansätze sind gut geeignet, um niedrigdimensionale und genaue Approxima-

tionen transportdominierter Systeme zu erhalten. Für die Modenbestimmung

stellen wir einen Optimierungsansatz vor, der auf Daten des FOM-Zustandes

basiert. Zudem erstellen wir die ROMs durch Residuumsminimierung und

gewährleisten eine effiziente Auswertung durch ein neues Hyperreduktionsver-

fahren. Des Weiteren zeigen wir, wie Stabilität durch eine energiebasierte For-

mulierung erhalten werden kann, indem wir eine sogenannte port-Hamiltonsche

Darstellung verwenden. Schließlich veranschaulichen wir die neue Methodik

anhand numerischer Experimente für transportdominierte Anwendungsfälle.

Acknowledgements

First, I would like to thank Volker Mehrmann for his supervision, proofreading,

valuable feedback, and continuous support over the years. I am also grateful to

Matthias Heinkenschloss and Benjamin Peherstorfer for agreeing to examine

this thesis. In addition, I acknowledge funding by the Deutsche Forschungs-

gemeinschaft via the Collaborative Research Centers 1029 and Transregio 154

as well as by the Berlin Mathematical School. Besides, I am grateful to my

colleagues and friends at TU Berlin for the good working atmosphere, mem-

orable group trips, and joyful summer and Christmas parties. Furthermore,

I thank all my co-authors for the pleasant and fruitful collaboration and in

addition Amelie Binder, Felix Black, Philipp Krah, Riccardo Morandin, Julius

Reiss, and Christoph Zimmer for the inspiring discussions and valuable hints

regarding the subjects of this thesis. Special thanks go to Benjamin Unger for

teaming up with me in many research activities and conferences as well as in

non-mathematical disciplines like the Christmas office chair races. Moreover,

I owe many thanks to Robert Altmann for the nice atmosphere in the office

as well as for his careful proofreading of this thesis and the precious feedback.

Finally, I thank Marine and my family for their unconditional and permanent

support.

vii

Contents

List of Figures xi

List of Algorithms xiii

List of Acronyms xv

1. Introduction 1

1.1. ProblemSetting........................... 3

1.2. Motivation.............................. 6

1.2.1. Transport-Dominated Systems . . . . . . . . . . . . . . 6

1.2.2. Port-Hamiltonian Systems . . . . . . . . . . . . . . . . . 14

1.3. Review of Existing Approaches . . . . . . . . . . . . . . . . . . 15

1.3.1. Model Reduction Techniques for Transport-Dominated

Systems ........................... 15

1.3.2. Structure-Preserving Model Reduction for Port-Hamiltonian

Systems ........................... 24

1.4. Contributions, Outline, and Previously Published Results . . . . 31

2. Preliminaries 37

2.1. Notation............................... 37

2.2. Nonlinear Optimization . . . . . . . . . . . . . . . . . . . . . . . 39

2.3. Functional Analysis . . . . . . . . . . . . . . . . . . . . . . . . 40

2.4. Differential Equations . . . . . . . . . . . . . . . . . . . . . . . 46

2.4.1. Abstract Evolution Equations and Semigroups . . . . . . 46

2.4.2. Finite-Dimensional Systems of Differential Equations . . 48

2.5. Parametric Model Order Reduction . . . . . . . . . . . . . . . . 53

2.5.1. Proper Orthogonal Decomposition . . . . . . . . . . . . 55

2.5.2. POD-Greedy Algorithm . . . . . . . . . . . . . . . . . . 57

2.5.3. Galerkin Projection . . . . . . . . . . . . . . . . . . . . 59

2.5.4. Hyperreduction . . . . . . . . . . . . . . . . . . . . . . . 61

2.6. Port-Hamiltonian Systems . . . . . . . . . . . . . . . . . . . . . 63

2.6.1. Formulations and Basic Properties . . . . . . . . . . . . 63

2.6.2. Structure-Preserving Model Order Reduction . . . . . . . 67

3. Mode Identification 69

3.1. Residual Minimization . . . . . . . . . . . . . . . . . . . . . . . 70

3.1.1. Solving the Full Optimization Problem . . . . . . . . . . 76

3.1.2. Using Variable Projection . . . . . . . . . . . . . . . . . 77

Contents

3.2. Greedy Algorithm based on Transformed Modes . . . . . . . . . 86

3.3. Boundary Treatment . . . . . . . . . . . . . . . . . . . . . . . . 92

3.3.1. Extended Domain . . . . . . . . . . . . . . . . . . . . . . 93

3.3.2. ZeroPadding ........................ 97

3.3.3. Constant Extrapolation . . . . . . . . . . . . . . . . . . 98

3.4. Comparison with Other Approaches . . . . . . . . . . . . . . . . 100

4. Projection-Based Model Order Reduction 105

4.1. Continuously Optimal Reduced-Order Models . . . . . . . . . . 105

4.2. Relation with Symmetry Reduction . . . . . . . . . . . . . . . . 118

4.3. Hyperreduction ...........................122

4.3.1. Case of a Linear Full-Order Model . . . . . . . . . . . . 124

4.3.2. Case of a Nonlinear Full-Order Model . . . . . . . . . . 131

5. Structure-Preserving Model Reduction for Port-Hamiltonian Sys-

tems 141

5.1. Linear Approximation Ansatz . . . . . . . . . . . . . . . . . . . 144

5.2. Nonlinear Separable Approximation Ansatz . . . . . . . . . . . 155

5.3. Nonlinear Factorizable Approximation Ansatz . . . . . . . . . . 167

6. Numerical Examples 175

6.1. Linear Wave Equation . . . . . . . . . . . . . . . . . . . . . . . 176

6.2. Linear Advection–Diffusion Equation . . . . . . . . . . . . . . . 185

6.3. Nonlinear Reaction–Diffusion Equation . . . . . . . . . . . . . . 196

7. Conclusion 209

7.1. Summary ..............................209

7.2. Outlook ...............................211

A. Properties of the Periodic Shift Operator 215

B. Linear Time-Varying Port-Hamiltonian Systems 221

C. Discrete Gradient Schemes 227

C.1. Discrete Gradients for Hamiltonian Systems . . . . . . . . . . . 227

C.2. Discrete Gradients for a Special Class of Port-Hamiltonian Sys-

tems .................................229

D. Technical Details for Chapter 6 237

D.1. Discretized Shift Operators . . . . . . . . . . . . . . . . . . . . . 237

D.1.1. Periodic Shift Operator . . . . . . . . . . . . . . . . . . . 237

D.1.2. Shift Operator used in Section 6.2 . . . . . . . . . . . . . 239

D.1.3. Shift Operator used in Section 6.3 . . . . . . . . . . . . . 242

D.2. Approximation of the Integral in (6.19) . . . . . . . . . . . . . . 245

Bibliography 247

List of Figures

1.1. Example 1.2.1: pseudocolor plot of the analytical solution (left)

and some selected snapshots (right). . . . . . . . . . . . . . . . . 9

1.2. An exemplary mode (left) and its shifted analogue (right) using

a periodic shift operator. . . . . . . . . . . . . . . . . . . . . . . 10

1.3. Pulsed detonation combustion process. . . . . . . . . . . . . . . 13

1.4. Pseudocolor plot of the density ρfor the pulsed detonation com-

bustion (PDC) process based on simulation data from [122]. . . 14

3.1. Example 3.3.1: pseudocolor plots of the analytical solution (left)

and a corresponding approximation with one transformed mode

based on the family of periodic shift operators (right). . . . . . . 93

3.2. An exemplary mode (left) and its shifted analogue (right), where

the gray area indicates a region of undetermined values. . . . . . 94

3.3. An exemplary mode defined on the extended domain Ωe=R

(here depicted on the domain b

Ωe= (−0.25,1), left) and its

shifted analogue on Ω = (0,1) (right)................ 96

3.4. An exemplary mode (left) and its shifted analogue (right) using

a zero padding shift operator. . . . . . . . . . . . . . . . . . . . 98

3.5. An exemplary mode (left) and its shifted analogue (right) using

a constant extrapolation shift operator. . . . . . . . . . . . . . . 99

4.1. Example 4.3.8: initial value for the density. . . . . . . . . . . . . 129

4.2. Example 4.3.8: pseudocolor plots of the analytical solution for

the density (left) and the velocity (right). . . . . . . . . . . . . . 130

4.3. Example 4.3.12: pseudocolor plots of the traveling wave solu-

tion xof the Burgers’ equation (left) and of the corresponding

nonlinear term −x∂ξx(right). ...................133

4.4. Example 4.3.12: singular value decays of the snapshot matrices

depicted in Figure 4.3. . . . . . . . . . . . . . . . . . . . . . . . 133

6.1. Linear wave equation: pseudocolor plots of the FOM solution

for the density (left) and the velocity (right). . . . . . . . . . . . 179

6.2. Linear wave equation: some selected snapshots of the FOM so-

lution for the density (left) and the velocity (right). . . . . . . . 180

6.3. Linear wave equation: singular value decay of the snapshot matrix.180

6.4. Linear wave equation: density (left) and velocity (right) com-

ponents of the determined modes. . . . . . . . . . . . . . . . . . 182

List of Figures

6.5. Linear wave equation: online values of the amplitudes (left) and

thepaths(right). ..........................185

6.6. Linear wave equation: Comparison of the error in conserva-

tion of the ROM Hamiltonian using the implicit midpoint rule

and the midpoint discrete gradient pair method outlined in ap-

pendixC.2. .............................186

6.7. Linear advection–diffusion equation: pseudocolor plot of the

FOM solution (left) and some selected snapshots (right). . . . . 188

6.8. Linear advection–diffusion equation: Comparison of the discrete

time derivative of the ROM Hamiltonian and the corresponding

dissipation and supplied power with time step size ∆t= 0.08

when using the implicit midpoint rule and the midpoint discrete

gradient pair approach from appendix C.2. The inset highlights

an energy inconsistency of the implicit midpoint rule, where the

discrete time derivative of the Hamiltonian is positive despite a

vanishing power supply. Here, t= 0.28 corresponds to the first

midpoint where the input becomes permanently zero. . . . . . 193

6.9. Linear advection–diffusion equation: Convergence of the im-

plicit midpoint rule and the midpoint discrete gradient pair

method from appendix C.2. . . . . . . . . . . . . . . . . . . . . 194

6.10. Linear advection–diffusion equation: pseudocolor plots of the

FOM solution for d= 10−3.5(left) and d= 10−2.5(right). . . . . 195

6.11. Linear advection–diffusion equation: ROM accuracy for differ-

ent values of the diffusion coefficient dwhen using a ROM with

r= 3 transformed modes based on FOM snapshots with d= 10−3.195

6.12. Linear advection–diffusion equation: relative online error for dif-

ferent mode numbers and values of dwhen using the greedy al-

gorithm from section 3.2 for determining the modes. The black

circles highlight the respective worst-case parameter values. . . . 197

6.13. Nonlinear reaction–diffusion equation: pseudocolor plot of the

FOM solution (left) and some selected snapshots (right). . . . . 201

6.14. Nonlinear reaction–diffusion equation: pseudocolor plot of the

FOM nonlinearity (left) and some selected snapshots (right). . . 205

xii

List of Algorithms

2.1. POD-greedy algorithm . . . . . . . . . . . . . . . . . . . . . . . 58

3.1. Evaluation of the cost function and its gradient for (3.9) . . . . 78

3.2. Evaluation of the cost function (3.22) and its gradient . . . . . . 85

3.3. Greedy algorithm based on transformed modes . . . . . . . . . . 86

xiii

List of Acronyms

ANN artificial neural network

DAE differential–algebraic equation

DDT deflagration-to-detonation transition

DEIM discrete empirical interpolation method

EIM empirical interpolation method

FEM finite element method

FOM full-order model

IRKA iterative rational Krylov algorithm

LQG linear quadratic Gaussian

MFEM moving finite element method

MOR model order reduction

ODE ordinary differential equation

PDC pulsed detonation combustion

PDE partial differential equation

pH port-Hamiltonian

POD proper orthogonal decomposition

PSD proper symplectic decomposition

ROM reduced-order model

SOBMOR structured optimization-based model order reduction

SVD singular value decomposition

1. Introduction

Nowadays, in many applications design decisions are made based on numerical

simulations. The increasing demand for accuracy and complexity results in

very high-dimensional systems that need to be simulated. This, however, is in

conflict with multi-query applications like control, optimization, or uncertainty

quantification. The goal of model order reduction (MOR) techniques is to

replace the high-dimensional full-order model (FOM) by a low-dimensional

surrogate model, which is usually called a reduced-order model (ROM). The

main requirement for such a ROM is that it needs to capture the relevant part

of the system accurately enough for a reasonable solution of the task at hand,

while at the same time the computational complexity in evaluating the ROM

should be much smaller in comparison to an evaluation of the FOM. In the

past decades, model reduction methods have experienced an immense research

effort and have been employed in various applications. For an overview, we

refer to the book chapters, books, and survey papers [14, 16, 23, 29, 30, 31,

142, 144, 235, 255].

In short, model reduction aims at constructing a surrogate model which can

be evaluated much faster than the full-order model and which yields a good

approximation of the solution of the problem at hand for a desired range of

parameter or input settings. Usually, the process of constructing the reduced

model is referred to as the offline phase, while the repeated evaluation of the

reduced model, for instance for control or optimization purposes, is called the

online phase. An essential requirement is that the computation time in the

online phase is as small as possible, whereas the computational effort needed

for the offline phase should be at least significantly smaller than the solution

of the optimization, simulation, or control task using the full model. While

this aspect rather concerns the efficiency of the model reduction process, there

are additional goals regarding the approximation quality of the ROM. In the

following we summarize the most relevant requirements for ROMs in terms of

both efficiency and approximation quality:

(i) The approximation error with respect to the quantities of interest should

be smaller than a prescribed tolerance in some specified norm.

(ii) The required time for evaluating the ROM in the online phase should be

significantly smaller than the time needed for solving the corresponding

FOM.

(iii) When considering optimization tasks, the computation time needed for

1. Introduction

the offline phase should be considerably smaller than solving the corre-

sponding full-order problem.

(iv) The availability of cheap-to-compute, reliable, and sharp error estimators

is desirable, e.g., for supporting the fulfillment of (i) or for facilitating

techniques which allow to adaptively build the ROM.

(v) In some applications it may be necessary that the reduced model reflects

the physics in the sense that certain properties of the original model

such as conservation laws are preserved. Similarly, if the full model is

given by an interconnected system of several submodels, then it is usually

important that the reduced model maintains this modular structure as

well.

In this thesis, we develop new model reduction techniques with special em-

phasis on two different aspects. First, we aim for MOR methods which are

suitable for the effective reduction of transport-dominated systems, i.e., sys-

tems whose dynamics is dominated by the propagation of waves or similar

structures. Especially, in cases where the corresponding wave profiles exhibit

sharp fronts, classical MOR techniques are usually observed to fail in provid-

ing low-dimensional and accurate approximations. Consequently, we consider

a new model reduction framework which overcomes such difficulties by explic-

itly accounting for the wave propagation in the model reduction ansatz.

The second central aspect of this thesis is the preservation of certain sys-

tem properties such as stability and passivity. Passivity corresponds, roughly

speaking, to the inability of the system to internally generate energy out of

nowhere. In this context, we consider port-Hamiltonian (pH) systems which

generalize classical Hamiltonian systems such that they are not only useful for

describing conservative systems, but they also account for energy dissipation

and energy exchange with the environment. Such systems can be shown to

be inherently passive and in many cases even stable. Thus, this structure is

especially well-suited for preserving such system properties when performing

model reduction. More precisely, there are model reduction methods which

preserve the pH structure and, thus, also the associated properties encoded

in this structure. In this thesis, we are especially interested in investigating

how the new model reduction scheme mentioned in the last paragraph can be

applied in a way which allows to preserve the pH structure.

The remainder of the introductory chapter is structured as follows. The

subsequent section provides a brief introduction to the mathematical problem

setting which is considered in most parts of this thesis. In section 1.2, we pro-

vide a motivation for considering transport-dominated systems and structure-

preserving methods for pH systems. A literature review on these two research

areas is outlined in section 1.3, whereas in section 1.4 we list the main contri-

butions of this thesis and provide a brief overview on the subsequent chapters.

1.1. Problem Setting

The FOMs we consider in this thesis are of the general form

˙x(t) = F(t, x(t)) for all t∈I:= [t0, tend], x(t0) = x0,(1.1)

with state x:I→W, initial time t0∈R≥0, final time tend ∈R>t0, initial value

x0∈W, right-hand side operator F:R≥0×W→X, a real Hilbert space

X, and a subspace W⊆X. Classical model reduction schemes are based on

first determining a suitable r-dimensional subspace of Wfor approximating

the state x, where ris much smaller than the dimension of W. This subspace

is usually represented by a corresponding basis of so-called ansatz functions or

modes φ1, . . . , φr∈W. The corresponding approximation ansatz then reads

x(t)≈ˆx(t):=

i=1

αi(t)φifor all t∈I,(1.2)

where α1, . . . , αr:I→Rare the so-called coefficients or amplitudes corre-

sponding to the respective modes φ1, . . . , φr. A common way for determin-

ing suitable ansatz functions is the proper orthogonal decomposition (POD)

method which yields an optimal approximation of given snapshot data in the

sense that the error in (1.2) is minimized, cf. section 2.5.1. Once the modes

have been determined, a usual way of constructing a corresponding ROM is a

Galerkin projection of (1.1) onto the subspace spanned by the modes, cf. sec-

tion 2.5.3. The state of the resulting reduced-order model corresponds to the

amplitudes αiin the approximation ansatz (1.2). Thus, since the latter one is

linear in the reduced-order state, (1.2) is called a linear approximation ansatz.

As mentioned in the introduction of this chapter, the power of linear ap-

proximation ansatzes as in (1.2) is often observed to be rather limited when

facing FOM dynamics which involve the propagation of waves with sharp wave

fronts. This shortcoming of linear ansatzes is subject to the discussion in sec-

tion 1.2.1. Roughly speaking, the major reason is that (1.2) allows the modes

only to change their amplitude as time evolves, but the modes themselves are

fixed and cannot follow the propagation of the wave profiles. To overcome

this issue, we add a time-dependent coordinate transformation to the approx-

imation ansatz which allows to explicitly account for the propagation. More

precisely, we extend (1.2) and consider the approximation ansatz

x(t)≈ˆx(t):=

i=1

αi(t)Ti(pi(t)) φifor all t∈I(1.3)

with suitable families of transformation operators Ti:b

Pi→ L(V,X), so-called

paths pi:I→b

Piwhich parametrize the transformation, finite-dimensional real

Banach spaces b

Pi, modes φi∈Vfor i= 1, . . . , r, and a real Banach space V.

Here, L(V,X)denotes the space of linear and bounded operators from Vto

1. Introduction

X, cf. section 2.3. In all examples of this thesis, the considered transformation

operators are given by translation or shift operators and the paths are real-

valued functions determining by which amount the modes are shifted. In

contrast to the classical ansatz (1.2), in (1.3) the modes are allowed to be in

a general Banach space V, which does not need to be a subspace of X. This

flexibility allows us, for instance, to define the modes on an extended spatial

domain as discussed in section 3.3.1.

To ensure that the approximation ˆxlies within the domain of the right-hand

side operator Fof the FOM, it is sufficient to require that the modes satisfy

φi∈\

η∈b

(Ti(η))−1(W)

for i= 1, . . . , r. For the remainder of this thesis, we restrict ourselves for

simplicity to the case that only one transformation family T1=. . . =Tr=:T

is used and that the corresponding path space b

P1=. . . =b

Pr=:b

Pis given by

P=R. These restrictions lead to the more special ansatz

x(t)≈ˆx(t):=

i=1

αi(t)T(pi(t)) φifor all t∈I.(1.4)

Here, the modes are required to satisfy

φ1, . . . , φr∈Y:=\

η∈R

(T(η))−1(W).(1.5)

A different setting allowing for higher-dimensional path spaces and for the

simultaneous use of different types of transformation operators is discussed in

[37], where some of the topics of this thesis have been already addressed. In

particular, the nonlinear projection framework outlined in section 4.1 has been

presented first in [37], however, in a slightly different setting with W-invariant

transformation operators Ti(η)∈ L(X):=L(X,X)for η∈b

Pi,i= 1, . . . , r.

A more extensive discussion of results, which are presented in this thesis and

have been published before, is provided in section 1.4.

While model reduction based on a linear approximation ansatz of the form

(1.2) has been studied for decades and is fairly well-understood, the use of the

new ansatz (1.4) poses new questions to be addressed, for instance:

1. What are suitable families of transformation operators T?

2. For a given transformation family Tand for given snapshot data of the

FOM state, how can we determine rand αi,φi,pifor i= 1, . . . , r such

that the approximation error in (1.4) is small?

3. For a given transformation family Tand for given modes φ1, . . . , φr,

how can we construct a ROM for determining the time evolution of the

1.1. Problem Setting

amplitudes α1, . . . , αrand the paths p1, . . . , pr, while at the same time

ensuring that the ROM can be evaluated in an efficient way?

4. If the FOM possesses desirable properties like stability, does an approxi-

mation ansatz of the form (1.4) allow for a model reduction scheme which

preserves these properties?

All of these questions are at least partially addressed in this thesis. While a

general answer to the first question of suitable choices of transformation opera-

tors is out of the scope of this thesis, we introduce several families of translation

operators and illustrate their usefulness via various examples, cf. sections 1.2.1

and 3.3 and chapter 6. The second question of approximating given snap-

shot data via an ansatz of the form (1.4) is addressed in chapter 3. Here

we especially focus on determining the modes and the amplitudes, whereas

ris assumed to be given and the paths may be for instance determined in

a pre-processing step. The ROM construction, which is subject of the third

question, is discussed in detail in chapter 4 and, especially, the question of how

to obtain a ROM which may be evaluated in an efficient way is addressed in

section 4.3. Finally, the fourth and last question is so general that it cannot

be completely answered in this thesis. Instead, we focus on preserving port-

Hamiltonian structures in chapter 5 and demonstrate that this often also leads

to stable ROMs.

Remark 1.1.1 (Parameter dependency).In many situations, the right-hand

side and the initial value in (1.1) may depend additionally on a parameter

vector µ∈M⊆Rnpand, thus, so does the solution x, i.e., the FOM takes

the form

˙x(t;µ) = F(t, x(t;µ); µ)for all (t, µ)∈I×M,

x(t0;µ) = x0(µ)for all µ∈M(1.6)

with F:R≥0×W×M→X,x0:M→W, and x:I×M→W. In this

case, we consider instead of (1.4) the slightly modified ansatz

x(t;µ)≈ˆx(t;µ):=

i=1

αi(t;µ)T(pi(t;µ)) φifor all (t, µ)∈I×M,(1.7)

i.e., the amplitudes αiand the paths pimay also depend on the parameter

vector µ. On the other hand, the modes φiare assumed to be chosen such that

they are suitable for approximating the solution xover the complete parameter

domain M. Alternatively, one could also determine modes separately for dif-

ferent parameter values and use interpolation afterwards for constructing the

reduced-order model, see for instance [30, sec. 4] for a detailed comparison of

these two different approaches in the context of finite-dimensional linear time-

invariant control systems. In the remainder of this thesis, we usually drop the

parameter dependency for simplicity, but we discuss in several remarks how to

incorporate parameters, cf. Remarks 3.1.4, 4.1.7, 4.3.10, and 4.3.14. ¨

1. Introduction

Remark 1.1.2 (Ansatz based on clustered modes).In practice, it is often use-

ful to not have one transformation operator per mode, but instead to allow

for groups or clusters of modes to be transformed by the same operator. An

example, where this may be helpful, is when the dynamics exhibit multiple

traveling waves whose profiles change due to diffusion as time evolves. In such

a scenario, time-dependent translation operators may be used for describing

the propagation of the waves, whereas the diffusion can be taken into account

by considering a time-dependent linear combination of multiple modes for each

traveling wave. Such a clustering approach may be represented by an approx-

imation ansatz of the form

x(·)≈ˆx(·):=

i=1 T(pi(·))

j=1

αi,j(·)φi,j,(1.8)

where the total number of modes is r:=Pnt

i=1 riand nt∈Ndenotes the

number of transformations. We emphasize that the mentioned example of

traveling and diffusing waves may be likewise described by the approximation

ansatz (1.4). However, the ansatz (1.8) yields more robustness for the ROM

since the modes corresponding to one traveling wave are forced to share the

same path, see for instance [37, sec. 7.1] for a numerical comparison of the

ansatzes (1.4) and (1.8). Moreover, (1.8) is often also advantageous for the

theory in terms of solvability of the reduced-order model, cf. section 4.1 or [37,

Rem. 5.8]. ¨

1.2. Motivation

In this section, we first provide a motivation for considering special MOR

approaches for transport-dominated systems in section 1.2.1. Afterwards, in

section 1.2.2 we mention several reasons for preserving a port-Hamiltonian

structure when performing model reduction.

1.2.1. Transport-Dominated Systems

In this thesis, we use the term transport-dominated for dynamical systems

whose solutions are mainly characterized by the propagation of one or several

waves or similar structures through the physical domain. Such phenomena

may be observed, for instance, in the context of hyperbolic conservation laws

like the wave equation or in combustion processes with traveling shock waves

and reaction fronts.

By means of a simple example, we illustrate why classical model reduction

methods which are based on linear subspace approximations are often ineffec-

tive when applied to transport-dominated systems.

Example 1.2.1 (Linear advection equation with periodic boundary condi-

tions).We consider the linear advection equation with periodic boundary con-

1.2. Motivation

ditions











∂tx(t, ξ) = −c∂ξx(t, ξ),for all (t, ξ)∈I×Ω,

x(0, ξ) = x0(ξ),for all ξ∈Ω,

x(t, a) = x(t, b),for all t∈I,

(1.9a)

(1.9b)

(1.9c)

on a one-dimensional domain Ω = (a, b)with a∈R,b∈R>a and time interval

I= [0, tend]with tend ∈R>0. Furthermore, c∈Rdenotes the advection speed.

As initial value we consider x0:Ω→Rdefined via

x0(ξ) = 









3(ξ−a)3−7

2(ξ−a)2+2

(ξ−a)+1,if ξ∈[a, a +],

2(ξ−b)2+2

(ξ−b)+1,if ξ∈[b−, b],

0,otherwise,

(1.10)

with small parameter ∈(0,b−a

2). This initial value is zero everywhere except

for regions of length adjacent to the boundaries. Moreover, the initial con-

dition is constructed such that it is continuously differentiable and that the

values of x0and of its first derivative coincide at the boundaries, i.e.,

x0(a) = x0(b) = 1 and x0

0(a) = x0

0(b) = 2

.

The analytical solution of (1.9) is given by x(t, ξ) = ˇx0(ξ−ct)for all (t, ξ)∈

I×Ω, where ˇx0:R→Rdenotes the (b−a)-periodic continuation of the

initial value x0. This solution is unique, which follows from the fact that all

continuously differentiable solutions of the linear advection equation (1.9a)

have to be constant along lines which are parallel to {(t, ξ)∈R2|ξ=ct}.

These lines are usually referred to as the characteristic ground curves, cf. [198,

sec. 1.2.1]. For c= 1, we consider fully discrete snapshots of the analytical

solution stored in the snapshot matrix X∈Rn,q, whose entries are given by

[X]i,j =x(tj, ξi)(1.11)

with discrete points in space ξi= (i−1)∆ξand discrete points in time tj=

(j−1)∆tfor i= 1, . . . , n and j= 1, . . . , q. Here, we choose n=q∈N>1, the

mesh width ∆ξ= (b−a)/n > , and the time step size ∆t= ∆ξ. Thus, the

solution is considered in a time horizon where the advected profile is in the

end just one grid point apart from its initial position.

Next, we consider the approximation of the solution by means of a linear

approximation ansatz of the form (1.2), which corresponds (in the fully discrete

case) to a low rank factorization of the snapshot matrix X. For the considered

example, the special choice of the initial value (1.10), the transport behavior

of the advection equation, and the relatively coarse sampling of the analytical

solution cause the snapshot matrix to be equal to the n×nidentity matrix.

Consequently, the singular values of Xdo not decay at all and, hence, there

1. Introduction

exists no low-rank factorization of Xwith a small approximation error. For

similar examples we refer to [1, 242]. We emphasize that the singular values

would decay if the spatial and the temporal sampling resolution was increased,

but nevertheless the singular value decay would be very slow, in particular for

small values of .

Considering the singular value decay of a snapshot matrix is one possibility

for judging how well-suited a linear approximation ansatz of the form (1.2)

is for a certain example. However, there are also other ways for seeing that

the considered advection equation problem is challenging for linear approxi-

mations. To this end, we set a= 0,b= 1,n= 100,= 0.005 and depict

the corresponding snapshots in a pseudocolor plot in Figure 1.1, left. Here,

the snapshot matrix contains values of the analytical solution within the range

[0,0.99] ×[0,0.99] ⊂I×Ω, cf. (1.11). In accordance with the considerations

of the last paragraph, the traveling wave corresponds to a diagonal line in the

space time diagram. On the contrary, the classical linear approximation ansatz

(1.2) is based on a sum of dyadic products, which, roughly speaking, is well-

suited for approximating horizontal and vertical structures in the space time

diagram, but not for approximating diagonal structures as in Figure 1.1, left.

Furthermore, Figure 1.1, right, depicts some selected snapshots of the solution.

If we were aiming for approximating the snapshots within a low-dimensional

subspace, we would ideally desire them to be at least almost linearly depen-

dent, i.e., that the angles between them are as small as possible. However,

the depicted snapshots in Figure 1.1, right, have compact non-overlapping

supports and, thus, the snapshots are pairwise orthogonal. Thus, the angles

between them are maximal and this is the worst possible situation in terms

of approximability by a subspace whose dimension is smaller than the num-

ber of snapshots. Of course, if the temporal resolution were increased and we

would add more and more snapshots, then there would be also overlapping

snapshots, which are not orthogonal to each other. Nevertheless, even in that

case most snapshot pairs would be orthogonal to each other, which makes the

set of snapshots hard to approximate within a low-dimensional subspace.

Finally, let us mention that the approximability of a solution manifold by

means of a linear approximation ansatz may be also studied by considering

the so-called Kolmogorov n-widths. These are measures for the worst-case ap-

proximation error obtained by the best linear subspace of a certain dimension,

cf. [72, 167, 274]. Thus, if the Kolmogorov n-widths decay slowly, we can in

general not expect to obtain accurate approximations by a linear approxima-

tion ansatz of the form (1.2) with just a few ansatz functions. In [220, sec. 5.1]

the authors demonstrate for a linear advection test case with jump disconti-

nuity that the Kolmogorov n-widths decay slowly and they conclude that any

MOR scheme which is based on a linear approximation ansatz is condemned

to failure. l

We emphasize that Example 1.2.1 is an academic example which has been

constructed to illustrate the limitations of linear approximation ansatzes and

1.2. Motivation

0 0.5

0.5

x(t, ξ)

0 0.5

t= 0.1t= 0.3t= 0.5t= 0.7t= 0.9

Figure 1.1.: Example 1.2.1: pseudocolor plot of the analytical solution (left) and some

selected snapshots (right).

to motivate considering more general approximation ansatzes which go beyond

the linear one in (1.2). For instance, drawing inspiration from Example 1.2.1,

where the analytical solution is given by applying a time-dependent shift to

the initial value, a reasonable modification of the linear approximation ansatz

(1.2) is given by

x(t)≈

i=1

αi(t)T(p(t)) φifor all t∈I,(1.12)

which constitutes a special case of (1.4) with p1=. . . =pr=:p, i.e., all modes

are transformed using the same time-dependent transformation operator T(p).

For Example 1.2.1, a natural choice for Tis given by the family T=Tper of

periodic shift operators as defined in Definition 1.2.2, see also Figure 1.2 for an

illustration and appendix A for a summary of some mathematical properties

of Tper.

Definition 1.2.2 (Periodic shift operator).For given Ω=(a, b)with a∈

Rand b∈R>a, we define the family of periodic shift operators Tper :R→

L(L2(Ω)) via Tper(η)f:=g, where, for given η∈Rand f∈L2(Ω),gis the

unique element in L2(Ω) satisfying

a+ζ

Za|g(ξ)−f(ξ+b−a−ζ)|dξ+

a+ζ|g(ξ)−f(ξ−ζ)|dξ= 0

with ζ:=ηmod (b−a).K

Note that despite the different notation used in Example 1.2.1, we may

apply the approximation ansatz (1.12) to Example 1.2.1 by considering xas a

function mapping from Ito a suitable subspace of L2(Ω). In fact, the analytical

1. Introduction

0 0.5 1

−3

−2

−1

φ(ξ)

0 0.5 1

−3

−2

−1

(Tper(0.25)φ)(ξ)

Figure 1.2.: An exemplary mode (left) and its shifted analogue (right) using a periodic

shift operator.

solution of Example 1.2.1 can be expressed via the ansatz (1.12) with just one

mode, e.g., by setting

r= 1,T=Tper, φ1=x0, p(t) = ct, α1(t)=1

for all t∈I. The ansatz (1.12) also allows for low-dimensional representa-

tions of more complex systems including non-constant transport velocities and

moving structures with varying shape, see, for example, [54].

While the advection equation considered in Example 1.2.1 can be well-

described by an ansatz of the form (1.12) with just one transformation param-

etrized by a single path p(t) = ct, in practice one often encounters multiple

wave profiles with different wave speeds. This motivates using a more gen-

eral ansatz with multiple paths as in (1.4). One of the simplest examples for

the occurrence of multiple waves is given by the dynamics of the linear wave

equation.

Example 1.2.3 (Linear wave equation with periodic boundary conditions).

We consider the linear acoustic wave equation with periodic boundary condi-

tions 









∂tρ(t, ξ) = −ρref ∂ξv(t, ξ)for all (t, ξ)∈I×Ω,

∂tv(t, ξ) = −c2

ρref

∂ξρ(t, ξ)for all (t, ξ)∈I×Ω,

ρ(0, ξ) = ρ0(ξ)for all ξ∈Ω,

v(0, ξ) = v0(ξ)for all ξ∈Ω,

ρ(t, a) = ρ(t, b)for all t∈I,

v(t, a) = v(t, b)for all t∈I

(1.13)

on a one-dimensional domain Ω=(a, b)with a∈R, b ∈R>a and time interval

I= [0, tend]with tend ∈R>0. Here, the unknowns are the density variation

ρ:I×Ω→Rand the velocity variation v:I×Ω→R, whereas the reference

density ρref ∈R>0and the velocity of sound c∈R>0are assumed to be given,

cf. [177, § 64]. Moreover, the initial values ρ0, v0∈C1(Ω) are assumed to

1.2. Motivation

satisfy the boundary conditions

ρ0(a) = ρ0(b), ρ0

0(a) = ρ0

0(b), v0(a) = v0(b), v0

0(a) = v0

0(b).(1.14)

By following a similar approach as for instance presented in [198, p. 23 f.], we

may derive the analytical solution

"ρ(t, ξ)

v(t, ξ)#="ρref

c#ϑr(ξ−ct) + "ρref

−c#ϑl(ξ+ct),(1.15)

where the so-called Riemann invariants ϑr:R→Rand ϑl:R→Rare given

by the (b−a)-periodic continuations of

2 1

ρref

ρ0+1

cv0!and 1

2 1

ρref

ρ0−1

cv0!,(1.16)

respectively. Especially, the analytical solution (1.15) involves time-dependent

shifts of the Riemann invariants and, thus, we may expect similar difficulties

in approximating the solution by a linear approximation ansatz of the form

(1.2) as for the linear advection equation considered in Example 1.2.1. A

corresponding theoretical analysis of the decay of the Kolmogorov n-widths

for the linear wave equation has been carried out in [123] and confirms this

intuition. However, if we instead consider an approximation based on the

ansatz (1.4) with

r= 2, φ1="ρref

c#ϑr|Ω, φ2="ρref

−c#ϑl|Ω,T="Tper 0

0Tper#,

α1(t) = α2(t)=1, p1(t) = −p2(t) = ct for all t∈I,

(1.17)

then the analytical solution coincides with the approximation. This equality

may be established by considering the analytical solution at each time instance

as element of (L2(Ω))2. Thus, the solution allows for a description using only

two modes, which are transformed by different time-dependent transformations

T(ct)and T(−ct). This motivates using an approximation ansatz as in (1.4),

which allows to incorporate more than just one time-dependent transformation

operator. l

The difficulty of standard model reduction schemes with treating transport-

dominated phenomena does not only apply to the simple academic Exam-

ples 1.2.1 and 1.2.3, but has also been observed in many applications where

the dynamics involve the propagation of wave profiles with locally large first

derivative. For instance, in [184, ch. III] the author considers a one-dimensional

nozzle flow, which involves a moving shock, and observes that many POD

modes are necessary to obtain a suitable approximation in that part of the

spatial domain which is affected by the moving shock. A different application

is addressed in [32], where the authors consider among others a batch chro-

1. Introduction

matography model and compare a POD-based ROM with a surrogate model

obtained via a purely data-driven approach. They observe a rather slow decay

of the singular values of the snapshot matrix and attribute this to the trans-

port within the system. The same also applies to [163], where a mathematical

model of a solidification process is considered, which leads to dynamics with

propagating fronts. In [213], the authors consider a convection-diffusion test

case and observe that the smaller the diffusion coefficient is chosen, the slower

is the corresponding singular value decay. Similarly, in [180] the authors study

the performance of the POD method for the flow around an airfoil for different

Mach numbers and note that the singular value decay is significantly slower for

higher Mach numbers, i.e., higher flow velocities. Another application where

a slow singular value decay results from some kind of transport is given by

seismic data, as for instance discussed in [174]. Here, the transport is not

explicitly depending on time, but rather results from different positions of the

seismic sources and receivers.

Another example is a pulsed detonation engine which has, for instance, been

a research subject within the Collaborative Research Center 1029 Substantial

efficiency increase in gas turbines through direct use of coupled unsteady com-

bustion and flow dynamics. A schematic depiction of the pulsed detonation

combustion (PDC) cycle is given in Figure 1.3, cf. [256]. Here, a combus-

tion tube is considered whose geometry features a convergent-divergent nozzle,

cf. [122]. In the first stage, the combustion tube is filled with a fuel-air mixture

before the mixture is ignited by a spark plug. As a result, a deflagration flame,

i.e., a subsonic diffusion-driven flame, cf. [51], is propagating through the com-

bustion tube. While the gray scale in Figure 1.3 indicates where the gas is

already burnt and where not, pressure waves are not visible in Figure 1.3. In

fact, there is a fast leading shock traveling in front of the deflagration flame,

which is for instance visible when considering corresponding snapshots of the

density, cf. Figure 1.4. As the shock approaches the nozzle, it gets partially

reflected and one observes a huge pressure increase in the focus point of the re-

flecting pressure waves. As a consequence, a small explosion is observed which

leads to a so-called deflagration-to-detonation transition (DDT). Afterwards,

a detonation wave, which is a supersonic flame coupled to a shock wave, con-

tinues to propagate through the pipe, cf. [51, 122]. In the end, the combustion

products are exhausted, the tube is purged, and afterwards the next cycle of

the PDC process begins.

From a thermodynamic perspective, a detonation wave means a higher ther-

modynamic efficiency than a deflagration flame, since it is closer to an ideal

constant-volume combustion. Thus, it is desirable to reduce the DDT length

as much as possible while still achieving a DDT in order to exploit the more

efficient detonation combustion. However, the DDT process depends on many

different parameters and extensive parameter studies are very expensive or

even infeasible, both numerically and experimentally. This motivates for us-

ing parametric model order reduction techniques to significantly reduce the

1.2. Motivation

Filling Stage

fuel

air

Spark Ignition

Propagation of Deflagration Flame

Deflagration-to-Detonation Transition (DDT)

Propagation of Detonation Wave

Exhaust

Figure 1.3.: Pulsed detonation combustion process.

1. Introduction

0 0.5 1

0.5

ρ(t, ξ)

leading shock

reaction front

reflected wave

re-reflected wave

detonation wave

Figure 1.4.: Pseudocolor plot of the density ρfor the PDC process based on simulation

data from [122].

computational effort for carrying out parameter studies. However, the PDC

process exhibits several transports within the system including a reaction front

and multiple shock waves. This is for instance visible in Figure 1.4, where

simulated density data from [122] are depicted in a pseudocolor plot over the

spatial coordinate ξand the time t. Similarly as for the academic examples

addressed in Examples 1.2.1 and 1.2.3, the transport of reaction fronts and

shock waves causes standard MOR techniques to be ineffective in constructing

low-dimensional accurate surrogate models. Thus, for these kinds of applica-

tions new methods are needed for decreasing the computation time by several

orders of magnitude to allow to perform extensive parameter studies numer-

ically. Since the performance of model reduction methods based on linear

approximation ansatzes is very limited for such transport-dominated systems,

model reduction schemes based on nonlinear approximation ansatzes as the

ones considered in this thesis provide a promising research direction to be able

to tackle such challenging problems as the PDC process. For the specific PDC

simulation data from [122], approximations of the snapshot data based on a

decomposition of the form (1.8) have been presented in [240, 259]. In partic-

ular, it has been demonstrated that this can be done by taking significantly

fewer modes than when using a linear approximation ansatz as in (1.2).

1.2.2. Port-Hamiltonian Systems

The classical model reduction objectives are to obtain low-dimensional and

accurate reduced-order models, ideally in combination with an estimate for

quantifying the approximation error a priori or a posteriori. Depending on

the application, also other objectives may be of importance, as preserving

certain qualitative properties of the original system, such as algebraic con-

straints [29, 248, 268], network or other structures [67, 108, 176, 181], passiv-

ity [15, 44, 98, 239, 267], or stability [62, 206, 234]. In this thesis, we aim

1.3. Review of Existing Approaches

for preserving port-Hamiltonian structures, since these come with many de-

sirable properties, see for instance [278] for a general overview, [237] for a

recent survey article on infinite-dimensional pH systems, [168] for a mono-

graph about structure-preserving discretization schemes, and [197] for a re-

cent survey article on port-Hamiltonian descriptor systems. In particular, a

port-Hamiltonian structure implies passivity and often also stability of the dy-

namical system. Moreover, pH structures are closed under power-preserving

interconnection, which makes them especially attractive for control purposes

[85, 187, 189, 222, 276, 277, 254] and for modeling networks [8, 102, 139, 275].

Furthermore, since the energy balance plays the central role within the port-

Hamiltonian framework, it may be applied to many physical systems and is

especially well-suited for the coupling of different physical domains, see for in-

stance [58, 95, 112, 173, 207, 281, 282, 289]. The flexibility of the pH modeling

framework becomes especially noticeable when looking at the wide range of

applications including acoustics [273], chemistry [146, 236, 283], electromag-

netism [69, 116, 225], fluid dynamics [7, 19, 208, 238], structural dynamics

[47, 48, 186, 284], thermodynamics [87, 182], and even economics [185].

Important properties of linear time-invariant port-Hamiltonian systems are,

for instance, their robustness with respect to perturbations in the coefficient

matrices [193, 195], the existence of efficient solvers for associated linear equa-

tion systems [128], and special algebraic properties of the associated matrix

pencil [194].

1.3. Review of Existing Approaches

As mentioned at the beginning of this chapter, the major focus of this the-

sis is on model reduction for transport-dominated systems and on structure-

preserving MOR schemes for port-Hamiltonian systems. In the following two

subsections we give an overview of the most relevant approaches proposed in

the past years within these two subtopics of model reduction.

1.3.1. Model Reduction Techniques for

Transport-Dominated Systems

In recent years, there has been an increasing effort in the model reduction com-

munity to develop new methods which are suitable for transport-dominated

systems. In this subsection, we give an overview of some of these approaches,

which we subdivide into three classes, see also [37, sec. 2], where parts of this

overview have been originally presented. The first class mainly uses time-

dependent coordinate transformations to account for the transport and these

methods are hence referred to as transformation-based methods. For instance,

in the case of a simple advection problem, a natural coordinate transformation

is given by a time-dependent translation or shift that describes the advective

1. Introduction

behavior, cf. section 1.2.1. These transformations are either used for trans-

forming the FOM such that it allows for a more effective reduction by stan-

dard MOR techniques or they are incorporated into the MOR approximation

ansatz. In the latter case, the approximation ansatz is typically nonlinear, for

instance if the time dependency of the coordinate transformations is implicit

via some time-dependent parameters and if these are considered as unknowns

of the ROM. This is especially true for the ansatz (1.4), where the transfor-

mations are parametrized by the paths pi, which correspond in the case of a

shift operator to the time-dependent shift amounts.

The second class summarizes methods which include a time-dependent on-

line update of the MOR basis functions, which is not based on a coordinate

transformation. We refer to these methods as adaptive basis methods. Most

of these approaches are based on a time-discrete update of the basis functions

and lead to switched reduced-order models.

As mentioned before, transformation-based methods often involve specific

nonlinear approximation ansatzes, where the nonlinearity originates from a

suitable coordinate transformation, which is typically motivated by physical

insights. The third class consists of methods employing a more generic non-

linear approximation ansatz which is not explicitly targeting the transport

within the dynamics by a suitable coordinate transformation. Therefore, we

refer to them as generic nonlinear methods. Such generic nonlinear approx-

imation ansatzes may be for instance characterized by a quadratic function

or by an artificial neural network (ANN) architecture. Especially, ANN-based

methods have received much attention in the past years and they are often

based on autoencoder architectures, which naturally lead to a low-dimensional

state space for the ROM, see for instance [135, 157, 164, 179, 244].

Remark 1.3.1 (Approximation of transport-dominated problems by time-delay

systems).Most of the methods mentioned in the remainder of this subsection

are based on some kind of projection of the FOM. A completely different

approach is based on describing the input-output dynamics of a transport-

dominated system by a time-delay system. Some first results in this direction

are obtained in [103, 233, 253, 257, 258]. ¨

Transformation-Based Methods

A common approach among the methods using coordinate transformations

consists of formulating the FOM dynamics in a new coordinate system, which

we refer to as the reference frame. The first developments in this direction are

presented within the symmetry reduction framework, cf. [36, 245, 246]. The

main idea is to approximate the solution by a composition of a time-dependent

group action and a so-called frozen solution, cf. section 4.2. Ideally, the group

action and the time-dependent group element are chosen such that the frozen

solution is almost constant over time, which supports a low-dimensional ap-

proximation. The group action can, for example, be chosen based on physical

1.3. Review of Existing Approaches

considerations or from snapshot data of the full-order solution. Related model

reduction techniques may be roughly divided into two categories: The first one

is based on first transforming the FOM and then applying standard MOR tech-

niques, while the second one is based on augmenting the approximation ansatz

by a suitable transformation. While both of these categories usually involve

a projection of the FOM, in [154] the authors present a purely data-driven

approach which is based on a modified version of the operator inference frame-

work introduced in [227]. The classical operator inference method constructs

a ROM directly from given snapshot data, whereas the approach introduced

in [154] involves a preceding coordinate transformation applied to the snap-

shot data in order to obtain a low-dimensional and accurate ROM even for

advection-dominated problems.

An example for the first category, where the FOM is transformed before ap-

plying classical MOR techniques, is presented in [219] and relies on Lie group

actions for transforming the original partial differential equation (PDE). In

this context it is assumed that the right-hand side of the FOM is equivariant

under the group action. The resulting transformed PDE is closed by alge-

braic equations, so-called phase conditions, which determine the group com-

ponent of the solution and which are usually chosen such that the temporal

change of the state is minimized, see also [36]. The resulting transformed

system is then reduced by a classical MOR approach based on projecting the

FOM onto a low-dimensional subspace. Moreover, the authors achieve an

efficient offline/online decomposition by employing the empirical operator in-

terpolation method, which is a hyperreduction technique originally proposed

in [84]. In [205], the authors present a model reduction framework which is

based on applying standard MOR techniques, such as the POD method, to a

one-dimensional nonlinear scalar convection-diffusion equation formulated in

a Lagrangian, i.e., co-moving, coordinate system. They also present numeri-

cal experiments for a one-dimensional Euler equations test case and observe

that considering the system in a Lagrangian coordinate system significantly

improves the performance of classical MOR methods. The combination of

their framework with hyperreduction techniques, which would be necessary to

obtain fast ROMs, is not considered in [205]. Instead, in [74] the authors ex-

tend the approach from [205] in several aspects including hyperreduction and

a time-windowing technique. The methods presented in [74, 205, 219] have in

common that they either have restrictive assumptions on the right-hand side of

the FOM or they assume that the transformed problem is already at hand. By

contrast, in [270] the author presents a general method for constructing a bijec-

tive parameter-dependent transformation of the spatial domain such that the

transformed PDE may be effectively reduced by standard MOR approaches.

The major focus is on parameter-dependent PDEs, whereas time-dependent

evolution problems are not explicitly considered, but only indirectly, for in-

stance by discretizing in time and considering time as another parameter. The

parameter-dependent transformation map is constructed in the offline phase

1. Introduction

based on solving a nonconvex and nonlinear optimization problem, which aims

for minimizing the distance between the transformed state and a suitably cho-

sen template function. Once this map is constructed, a ROM may be obtained

by first transforming the spatial domain of the PDE and afterwards applying a

standard MOR scheme such as projection onto POD modes. In the follow-up

work [271], the authors extend the approach from [270] by constructing trans-

formation maps acting on space and time and by combining the method with a

hyperreduction scheme to achieve considerable speed-ups in the online phase.

The application of the approach from [270] to transport-dominated problems

on two-dimensional spatial domains is discussed in [101, 272]. In [125], the

authors propose a model reduction technique for nonlinear scalar hyperbolic

PDEs by first introducing a relaxation of the original PDE. This relaxation

leads to a system of two hyperbolic partial differential equations with con-

stant wave speeds and nonlinear right-hand side. For the semi-discretization

in space, they approximate the state by a linear combination of shifted finite el-

ement basis functions, which takes care of the transport within the system and,

thus, allows the successful application of classical model reduction techniques

based on linear subspaces to the semi-discretized system. Other approaches

based on the idea to first transform the FOM and afterwards apply standard

model reduction schemes are, e.g., discussed in [183, 204, 212]. Further meth-

ods which only consider the approximation of snapshot data but not the ROM

construction are for instance presented in [152, 264, 266].

Instead of first transforming the full-order model and then reducing the

transformed system, the authors in [119] directly reduce the untransformed

FOM using an approximation ansatz that includes a translation as coordi-

nate transformation. They construct a corresponding ROM for a linear one-

dimensional advection equation by enforcing the residual to be orthogonal to

the shifted modes, whereas the update of the shift is determined based on the

advection speed of the advection equation. The question of how to achieve an

efficient offline/online decomposition has not been addressed in [119]. A simi-

lar approach has been considered in [245], but in contrast to [119] the method

from [245] may be applied to a more general class of nonlinear problems with

equivariant right-hand side. Especially, they present numerical results for the

one-dimensional Kuramoto–Sivashinsky equation with periodic boundary con-

ditions. Furthermore, the authors propose to compute the time-dependent

shift based on a reconstruction equation, which may for instance be obtained

via template fitting. The latter is based on defining a template function, for

example the initial value, and choosing the shift amount such that the dis-

tance between the shifted state and the template function is minimized. This

results in an algebraic constraint, but instead of considering the corresponding

differential–algebraic equation (DAE), the authors differentiate the algebraic

constraint with respect to time and obtain an ordinary differential equation

(ODE) system as ROM. Similarly as in [119], the question of efficiently eval-

uating the ROM has not been addressed in [245]. In [54] the authors also

1.3. Review of Existing Approaches

use an approximation ansatz that includes a coordinate transformation and

for the application of the method they also consider a simple translation as

transformation. For constructing the ROM, the authors first discretize in time

and then substitute the approximation ansatz into the semi-discrete full-order

model. They present an algorithm for updating the time-discrete states of

the ROM by minimizing the time-discrete residual. The evaluation of the

ROM still scales with the dimension of the FOM, but for a Burgers’ equation

test case they present an additional approximation of the ROM which allows

achieving an efficient offline/online decomposition. This hyperreduction ap-

proach is based on constructing interpolation-based approximants of the ROM

coefficient functions in the offline phase and makes use of the periodic bound-

ary conditions of the considered test case and of the quadratic nature of the

nonlinearity of the Burgers’ equation. Another online-efficient approach is the

manifold approximation via transported subspaces method which has been in-

troduced in [243] for time- and parameter-dependent scalar conservation laws.

As approximation ansatz, the authors use a linear combination of transformed

modes, where all modes are affected by the same coordinate transformation,

which is itself described by a linear combination of so-called transport modes.

Both kinds of modes are determined in the offline phase based on snapshot

data and, moreover, the authors present a time-discrete online update of the

time-dependent coefficients occurring in the approximation ansatz. To achieve

an efficient offline/online decomposition, they use the empirical interpolation

method with collocation points which are also updated as time evolves. In

[171], the authors consider one mode with fixed amplitude one for the shape of

the wave profile and a rather general coordinate transformation, which is sim-

ilarly as in [243] described by a linear combination of a few space-dependent

ansatz functions. They discuss two different methods for determining these

ansatz functions in the offline phase, one of which is based on the observation

that their approximation ansatz may be interpreted as a special autoencoder

structure, which allows to determine the ansatz functions by training a neural

network. For the ROM construction, they consider an intrusive and a non-

intrusive scheme, where the former one is the manifold Galerkin method, which

is also addressed below in the passage addressing the generic nonlinear meth-

ods. They also propose a hyperreduction method for a rather general class of

FOM nonlinearities, but they also emphasize that this method requires to eval-

uate the FOM nonlinearity at a number of sample points which is significantly

higher than the dimension of the ROM. Furthermore, the time-dependent up-

date of the sample points involves a matrix-vector product which scales with

the dimension of the FOM. Nevertheless, their method achieves speed-ups for

the considered two-dimensional test cases.

The methods mentioned in the last paragraphs have in common that they

use a single coordinate transformation to obtain low-dimensional and accurate

ROMs for transport-dominated systems. However, in some applications it may

be advantageous to apply multiple coordinate transformations, for instance

1. Introduction

when there are several traveling waves with different propagation speeds. A

simple example for the occurrence of more than one traveling wave with differ-

ent advection speeds is provided by the dynamics of the linear wave equation

as discussed in Example 1.2.3. To be able to also achieve low-dimensional ap-

proximations of such phenomena, recently some methods have been proposed

which use multiple transformations, see also section 3.4 for a more detailed

summary of some of these methods. For instance, the shifted POD method,

which has been introduced in [241] and enhanced in [240, 259], is based on an

approximation ansatz involving shift operators with different shift amounts for

different sets of modes. The goal of this method is to obtain a low-dimensional

approximation of a given snapshot matrix. The first version of the shifted

POD algorithm in [241] is based on transforming the snapshots into the differ-

ent reference frames and compressing the transformed data via singular value

decompositions (SVDs). To separate the dynamics corresponding to the dif-

ferent reference frames, a heuristic iterative procedure is introduced and its

performance is illustrated by various numerical examples. In contrast to this,

the shifted POD version presented in [240] is based on an optimization problem

which aims for maximizing the largest singular values in each reference frame.

Another optimization-based approach is introduced in [259] and it directly tar-

gets the minimization of the difference between the original snapshot data and

their approximation. While the methods presented in [240] and [259] consider

the coordinate transformations to be given, an extension of the method in [259]

which also optimizes the paths is provided in [40]. An alternative approach is

considered in [242], where the authors also present a method for approximating

snapshots by a linear combination of shifted modes using multiple coordinate

transformations. In contrast to the shifted POD method, the technique intro-

duced in [242] is based on a greedy-type algorithm. As a consequence, this

approach is not able to yield the two traveling wave profiles when applied

to a linear wave equation test case where the analytical solution is known.

Nevertheless, they also present some interesting extensions which have not yet

been addressed within the shifted POD framework, such as advection velocities

which may vary within the spatial domain. In [199], the authors propose an

optimization scheme for determining the advection speeds of multiple traveling

waves and afterwards compute low-rank approximations within each reference

frame. Since this method is lacking an iteration as for instance in [241], it is

similarly as the approach in [242] not able to obtain optimal decompositions

for examples like the linear wave equation. An application of the approach

in [199] in the context of a rotating detonation engine has been presented in

[200] and an extension to two-dimensional wave phenomena in [201]. A special

class of coordinate transformations where the modes are composed with lin-

ear affine mappings is considered in [124]. In contrast to the other approaches

mentioned in this paragraph, the method presented in [124] assumes the modes

to be given and focuses on the optimization of the amplitudes and of the path

variables, i.e., the quantities which parametrize the transformations.

1.3. Review of Existing Approaches

All of the methods mentioned in the previous paragraph have in common

that they focus on a low-dimensional decomposition of snapshot data, whereas

no ROMs are constructed based on the identified modes. On the contrary, in

[37] the authors introduce a framework that allows constructing ROMs based

on transformed modes, which can, for instance, be computed by one of the

contributions mentioned in the previous paragraph. Notably, their framework

is not restricted to the case that all modes are transformed by the same type

of coordinate transformation, but they consider the general ansatz (1.3) which

allows incorporating different families of transformation operators.

Another class of methods which is based on multiple coordinate transforma-

tions is described in [214, 251]. Especially, these methods consider transports

with respect to parameter changes instead of with respect to time increments

and they use snapshots from close-by parameter values as ansatz functions.

While the authors in [214] consider stationary problems, the authors in [251]

consider time- and parameter-dependent problems in a time-discrete setting.

Even though it has not been proposed in the context of model order re-

duction, we also mention the moving finite element method (MFEM), which

has been proposed in [202, 203] and which is also based on the idea of refer-

ence frames. The MFEM extends the classical finite element method (FEM)

by allowing the FEM basis functions to move in space as time evolves. This

method is especially suitable for problems with moving shocks and the idea is

that the basis functions follow the movements of the shocks such that a fine

resolution is only achieved where it is needed. The unknowns of the result-

ing semi-discretized system are the amplitudes and the positions of the basis

functions. The corresponding approximation ansatz is nonlinear in these un-

known state variables, which means that the approximation of the PDE state

is not restricted to a linear subspace but to a nonlinear manifold. For the time

evolution of the state variables, the authors in [203] propose to minimize the

residual with respect to the time derivative of the state variables. The resulting

semi-discretized system is inherently nonlinear and features a state-dependent

mass matrix. To ensure that this matrix is always nonsingular, the authors

propose a regularization technique which punishes the relative movement of

different basis functions. The major difference between the MFEM and the

model reduction framework presented in [37], see also section 4.1, is that the

MFEM uses generic basis functions with compact support, whereas [37] em-

ploys problem-specific basis functions which are usually non-zero on the entire

computational domain.

Adaptive Basis Methods

While transformation-based methods involve a coordinate transformation of

the FOM or of the basis functions, the adaptive basis methods considered

in the following are based on time-dependent updates of the ROM which do

not originate from a coordinate transformation. One example for such an ap-

proach is presented in [81], where the authors divide the total time interval

1. Introduction

into several subintervals and determine different reduced bases for the differ-

ent subintervals. The segmentation of the total time interval is carried out in

an adaptive way which ensures that a given error tolerance is met and that

the number of basis functions in a subinterval cannot exceed a certain pre-

defined value. In the online phase, the ROM is simulated step by step on

each subinterval specified in the offline phase. When an interface between two

subintervals is reached, the initial value of the ROM corresponding to the new

subinterval is determined based on projecting the current approximation of

the full-order state onto the span of the basis functions of the new subinterval.

Consequently, the ROM may in total be regarded as a switched system with

a switching condition which depends only on time.

Differently from [81], the authors in [93] propose a scheme that adapts the

reduced basis only in the online phase and only if the error estimator returns

values which are very high or very low. In the offline phase, they define a

tree structure that represents hierarchical orthogonal decompositions of the

underlying vector space of the FOM. Based on this tree structure, they are

able to adaptively increase the number of basis functions if the error of the

ROM is very high or to reduce the number of basis functions if the error is

smaller than a prescribed error threshold. As in [81], the obtained ROM is

a switched system, but in contrast to [81], the switching condition is state-

dependent, since the error estimators are based on the current state of the

ROM. Thus, the switching times are a priori not known. The work in [93] is

based on the ideas presented in [59] and extends them towards more general

refinement trees and a more general and efficient basis compression scheme.

Another online-adaptive scheme is proposed in [226], where the basis functions

are regularly modified via a low-rank update, but the number of basis functions

remains constant in contrast to [93].

In [115], the authors present an approach that also involves a reduced basis

which is adapted as time evolves. However, in contrast to the works men-

tioned in the previous two paragraphs, the rules for updating the basis are

more problem-specific. Concretely, they propose to use the eigenfunctions of

a linear Schrödinger operator associated with the initial value of the FOM as

basis functions. Then, their time evolution is performed in such a way that

the basis functions remain eigenfunctions of a linear Schrödinger operator as-

sociated with the time-dependent FOM state. Consequently, they obtain an

additional evolution equation for the basis functions. In contrast to the works

mentioned in the previous two paragraphs, the ROM is not a switched system

with time-discrete changes in the basis functions, but instead they obtain a

time-continuous equation for their evolution.

An approach which combines the idea of using time-dependent basis func-

tions with machine learning techniques is provided in [229]. To this end, the

authors use a linear combination of space-, time-, and parameter-dependent

basis functions with time- and parameter-dependent coefficients as approxima-

tion ansatz. In the offline phase, they use FOM snapshots to train two neural

1.3. Review of Existing Approaches

networks: one for the coefficients and one for the basis functions. Afterwards,

they project the time-discrete FOM onto the span of the basis functions to

obtain a reduced-order model. Thus, each time step of the online phase in-

volves the evaluation of the neural network for the basis functions and the

construction of the projected ROM based on the current reduced basis. Con-

sequently, the online phase still scales with the FOM dimension. Nevertheless,

they observe a computational speed-up, at least for cases where a high spatial

resolution is used for the FOM.

Generic Nonlinear Methods

The idea of the third class of methods is to approximate the solution via a

generic nonlinear approximation ansatz. In [179], the authors use an autoen-

coder, which is a type of artificial neural network, to obtain a low-dimensional

description of the FOM solution. Based on the snapshot data, a decoder and

an encoder mapping are learned, where the decoder is a mapping from the

reduced state space to the full state space and the encoder vice versa. Es-

pecially, the lifting of the reduced state to an approximation of the full-order

state is performed by the decoder mapping, which thus describes the approx-

imation ansatz. The projection of the FOM is carried out by substituting

the approximation ansatz into the FOM and then constructing the ROM via

minimization of the residual. They propose two different approaches: The

manifold Galerkin method which is based on minimizing the residual for the

time-continuous FOM and the manifold least-squares Petrov–Galerkin method

which considers the FOM in a time-discrete setting. The idea to use autoen-

coders for the purpose of model order reduction has been previously presented

in [135] and [157]. The mentioned approaches have in common that the eval-

uation of the ROM usually still scales with the dimension of the FOM, which

prevents an efficient online phase. Therefore, in [164] the authors combine

the approach from [179] with a classical hyperreduction approach for linear

subspace methods. Furthermore, in contrast to [179], the method in [164]

is based on shallow de- and encoders and, moreover, they propose to use a

sparse decoder with a sparsity structure which reflects for instance spatial lo-

cality properties in the solver used for simulating the FOM. As a consequence,

they are able to show that the evaluation of the hyperreduced ROMs does in

general not scale with the FOM dimension. Nevertheless, in their numerical

experiments for one- and two-dimensional Burgers’ equation test cases their

approach only achieves moderate speed-ups. A possible explanation could be

that the sample points used for the hyperreduction scheme are fixed and do

not move along the transport within the system. As a consequence, the re-

ported number of hyperreduction sample points is significantly larger than the

corresponding ROM dimension. A similar hyperreduction approach has been

presented in [244]. Instead of obtaining the ROM via projection onto a non-

linear manifold, in [106] the authors introduce a method which approximates

the FOM state by means of an autoencoder and describes the ROM dynamics

1. Introduction

by means of a deep feedforward neural network. This approach only requires

solution snapshots of the FOM state, but not access to the FOM itself. Fur-

thermore, since no projection is involved, no hyperreduction is needed and

the only computational cost in the online phase is the evaluation of the neu-

ral networks trained in the offline phase. This framework has been further

developed and applied to challenging test cases in [104, 105, 107]. Further

MOR approaches based on artificial neural networks are, for instance, given in

[50, 86, 191].

Instead of using an approximation ansatz based on an ANN architecture, in

[20] the authors propose a model reduction framework which is based on ap-

proximating the FOM state by a quadratic approximation ansatz. For the

offline phase, a two-step procedure is introduced where the snapshots are

first compressed using the classical POD method and, afterwards, a purely

quadratic ansatz is used for approximating the part of the dynamics which is

not captured by the POD modes. The second step involves solving ninde-

pendent linear least squares problems, where ndenotes the FOM dimension.

To obtain a ROM, the authors use a special case of the manifold least-squares

Petrov–Galerkin method for quadratic manifolds. Furthermore, they achieve

an efficient offline/online decomposition by suitably extending the energy-

conserving sampling and weighting method, which is a hyperreduction tech-

nique originally proposed in [97] and is based on a quadrature approximation

of the ROM nonlinearity. The numerical experiments presented in [20] reveal

that this new method is able to achieve considerable speed-ups in comparison

to classical POD-based ROMs. Also in [113] the authors use a quadratic ap-

proximation ansatz. However, in contrast to [20], their approach is not based

on projection, but instead corresponds to an extension of the non-intrusive

operator inference framework to quadratic approximation ansatzes.

Further model reduction techniques which are based on nonlinear approxi-

mation ansatzes are for instance presented in [13, 56, 89, 190].

1.3.2. Structure-Preserving Model Reduction for

Port-Hamiltonian Systems

Most standard model reduction schemes in general do not preserve stability

or passivity of the FOM and this shortcoming may lead to unphysical ROMs

and unbounded errors. One possibility to guarantee preservation of stability

and passivity during the model reduction is achieved by making use of a pH

representation of the governing equations. Further properties which motivate

for preserving the pH structure are mentioned in section 1.2.2.

Structure-preserving model reduction techniques for pH systems may be

roughly divided into two classes: The first class of methods aims for approxi-

mating the input-output map of a port-Hamiltonian system, whereas the sec-

ond class targets a good approximation of the state, for instance by using

a POD-based approach. As port-Hamiltonian systems arose from the control

1.3. Review of Existing Approaches

systems community, the former class has received considerably more attention.

In the following we provide a summary of structure-preserving MOR techniques

and begin with those ones which are based on control system strategies and

afterwards we discuss methods which aim for a good state approximation. We

also refer to [197, sec. 8] for a detailed overview on structure-preserving model

reduction schemes with particular emphasis on methods for DAE systems.

Remark 1.3.2 (Passivity-preserving MOR techniques).In the special case of

controllable and observable linear time-invariant ODE systems, it is for in-

stance shown in [26] that the subclasses of port-Hamiltonian systems and pas-

sive systems coincide, see also [68] for corresponding statements in the DAE

case. Consequently, in this case we may regard passivity-preserving MOR

schemes as structure-preserving schemes for port-Hamiltonian systems, see for

instance [44] for a more detailed discussion. ¨

Balancing-Based Methods

We start by summarizing structure-preserving MOR methods which are based

on some kind of balancing technique and a subsequent truncation. For in-

stance, in [109] the author presents a modification of the balanced truncation

approach for nonlinear control systems originally introduced in [110] for gen-

eral unstructured systems. While the original work is based on balancing

suitably defined observability and controllability functions, the approach in

[109] is based on either replacing the observability or the controllability func-

tion by an appropriately weighted version. By this modified balancing, the

method ensures that the corresponding ROM preserves the pH structure of

the FOM. The author also investigates the existence of the required weighting

matrices and derives equivalent conditions based on the column spans of the

state-dependent dissipation and input matrices. Consequently, the approach

is restricted to a certain subclass of nonlinear pH systems and for this subclass

it is shown that the weighted version of the controllability or observability

function coincides with the Hamiltonian. Furthermore, in the nonlinear case

the approach presented in [109] requires solving a Hamilton–Jacobi equation

and, thus, may be only applied to nonlinear FOMs of small size.

In [137] the authors consider linear time-invariant port-Hamiltonian FOMs

with even state space dimension and invertible structure matrix. They present

an approach which is based on balanced truncation and singular perturbation

arguments to obtain a port-Hamiltonian ROM. Furthermore, they present con-

ditions for the FOM and the ROM to be asymptotically stable and in that case

they obtain an a priori error bound as in the classical unstructured balanced

truncation method.

Similarly as in [109], in [160] the authors consider a modification of bal-

anced truncation, where they balance the controllability function and the

Hamiltonian. In contrast to [109], their approach is presented for linear time-

invariant pH systems with positive definite dissipation matrix. Furthermore,

1. Introduction

they present a bound for the error in the supplied energy for a given input

signal and this bound involves the sum of the truncated singular values. An

extension of this approach to nonlinear pH systems is provided in [161] and

it involves similarly as in [109] computing a solution of a Hamilton–Jacobi

equation.

Instead of considering classical balanced truncation, in [42] the authors pro-

pose structure-preserving variants of the generalized balanced truncation and

extended balanced truncation methods. The former one has been originally

introduced in [145] and is based on balancing solutions of Lyapunov inequal-

ities instead of Lyapunov equations as in the classical balanced truncation

method. Similarly, also the extended balanced truncation method introduced

in [250] relies on balancing solutions of linear matrix inequalities, which may

be regarded as extended versions of the ones considered in the generalized

balanced truncation method and provide additional degrees of freedom. The

structure-preserving variants of both methods presented in [42] are based on

the idea to simultaneously diagonalize the energy matrix of the pH system

and the solutions of the respective matrix inequalities by a suitable congru-

ence transformation. To this end, they derive for each of the two methods

sufficient conditions to ensure that such a congruence transformation exists

and demonstrate how such a transformation may be constructed. Moreover,

both methods provide a priori error bounds similarly as in the classical bal-

anced truncation method and both the theory and the numerical experiments

show that the error bound obtained via extended balanced truncation is in

general smaller than the one obtained via generalized balanced truncation.

While the balancing approaches discussed so far mainly focus on the open-

loop behavior, in [45] the authors propose a structure-preserving variant of the

linear quadratic Gaussian (LQG) balanced truncation method, which is espe-

cially suitable for constructing low-dimensional controllers. Their approach is

based on appropriately choosing the weighting matrices of a pair of algebraic

Riccati equations to ensure that the resulting system obtained after balancing

and truncation is port-Hamiltonian. Moreover, they derive an error bound in

the gap metric and demonstrate that replacing the FOM Hamiltonian by one

which is based on a maximal solution of a Kalman-Yakubovich-Popov inequal-

ity may lead to a significantly lower error bound and error. In addition, they

also propose a similar approach for classical balanced truncation and also ob-

tain an H∞error bound provided that a certain condition on the dissipation

and input ports is satisfied. We note that the same condition also appears

in [42] in the context of generalized balanced truncation. An extension of

the structure-preserving LQG balanced truncation method from [45] to port-

Hamiltonian DAE systems is presented in [43]. Since the techniques presented

in [43, 45] require the solutions of quadratic matrix equations whose size scales

with the FOM dimension, the application of these methods is limited to FOMs

of moderate size.

1.3. Review of Existing Approaches

Methods Based on Interpolating the Transfer Function

In [230] the authors present a structure-preserving model reduction scheme

for linear time-invariant pH systems via moment matching at infinity, i.e.,

via matching the Markov parameters of the transfer function. They present

structure-preserving variants of the Arnoldi and the Lanczos method, cf. [14,

sec. 10.4], by exploiting that the considered linear time-invariant pH struc-

tures are preserved by Galerkin or suitable Petrov–Galerkin projections. For

the structure-preserving variant of the Lanczos method, they require an ad-

ditional assumption on the port-Hamiltonian system matrices to accomplish

the matching of 2rMarkov parameters with a ROM of state space dimension

r. This additional assumption is for instance satisfied if the FOM is purely

damped or purely undamped. Further structure-preserving techniques, which

achieve moment matching at other expansion points than at infinity, are pre-

sented in [231, 286].

All interpolation-based methods mentioned so far are based on matching

the first moments of the FOM transfer function at a fixed expansion point.

On the other hand, in [130] the authors present a structure-preserving MOR

scheme for linear time-invariant multiple-input/multiple-output pH systems

via tangential interpolation of the transfer function. This approach naturally

allows for interpolation at various points and, moreover, the authors propose

an algorithm for selecting the interpolation points and directions, which is

inspired by the iterative rational Krylov algorithm (IRKA) presented in [129]

for general unstructured systems. However, while the original IRKA method

yields ROMs which are optimal with respect to the H2norm, the algorithm

introduced in [130] only enforces parts of the necessary optimality conditions

and does hence in general not guarantee H2-optimal ROMs. Another structure-

preserving method which achieves moment matching at multiple expansion

points is presented in [153].

In [117], the authors present a Krylov-based structure-preserving MOR

scheme for parameter-dependent pH systems. The structure preservation is

achieved by a suitable modification of the matrix interpolation method, which

has been originally presented in [224] for the unstructured case.

While the interpolation-based methods mentioned so far consider model re-

duction for pH systems of ODEs, in [88] the authors use Krylov methods for

reducing a port-Hamiltonian DAE system, which arises from the modeling of

a transport network. The structure of the FOM allows to preserve both the

port-Hamiltonian structure and the algebraic constraints by a Galerkin pro-

jection which only acts on the differential equations and the corresponding

variables. They also achieve the preservation of further properties like mass

conservation by ensuring that the projection matrices satisfy some algebraic

compatibility conditions.

While the approach in [88] is rather specific for the considered special class

of port-Hamiltonian DAE systems, in [27] the authors present a more general

structure-preserving model reduction framework for port-Hamiltonian DAEs.

1. Introduction

In particular, they discuss for different classes of semi-explicit index-one and

index-two systems how to construct ROMs which satisfy given tangential in-

terpolation conditions and match the polynomial part of the FOM transfer

function, while preserving the pH structure. Furthermore, for determining

suitable interpolation points and directions, they propose an extension of the

pH-IRKA method introduced in [130] to the DAE case.

Effort-Constraint and Flow-Constraint Method

A general procedure for structure-preserving MOR of port-Hamiltonian sys-

tems is presented in [232] and it is based on the representation of a pH system

in terms of a Dirac structure. The method assumes that the FOM is already

balanced in the sense that a part of the state variables has little contribu-

tion to the input-output map of the system. Based on this assumption, the

authors propose the idea to obtain energy-consistent ROMs by cutting the en-

ergy flow related to the less relevant state variables. To this end, they present

two methods, which both remove the less relevant state variables and yield

port-Hamiltonian ROMs: the so-called effort-constraint and flow-constraint

methods. Both techniques are presented in a general setting which also in-

cludes nonlinear pH systems, but the authors also demonstrate the two ap-

proaches for the special case of a linear time-invariant pH system. While the

effort-constraint method yields a ROM which has the same structure as the

FOM, the flow-constraint method leads to an additional feedthrough term in

the ROM. Furthermore, the application of the effort-constraint method re-

quires a certain submatrix of the energy matrix to be invertible, whereas the

flow-constraint method requires a certain submatrix of the structure matrix

to be invertible. Even though it is not explicitly mentioned in [232], we note

that comparing the ROMs obtained via the effort-constraint method and those

derived in [137] via balanced truncation and singular perturbation arguments

appear to coincide. This suggests a relation between these two approaches in

the special case where the FOM is balanced via classical Lyapunov balancing.

In [138], the authors extend the effort-constraint and flow-constraint meth-

ods to linear time-invariant DAEs. For this purpose, they first transform the

port-Hamiltonian DAE such that the differential and algebraic equations are

separated and such that the differential equations do not depend on the al-

gebraic variables. Afterwards, they perform the model reduction only on the

differential part and, hence, preserve not only the pH structure but also the

algebraic constraints. In addition, they present a structure-preserving MOR

technique for port-Hamiltonian DAEs which is based on moment matching.

Optimization-Based Methods

Most of the methods discussed so far have in common that they are based

on projection-based MOR techniques. As an alternative, in [252] the author

proposes to directly determine the coefficients of a linear time-invariant asymp-

1.3. Review of Existing Approaches

totically stable ROM by solving an optimization problem. More precisely, it

is proposed to choose the ROM coefficients such that the error in the trans-

fer function is minimized with respect to the H2norm. To ensure that the

ROM is pH and asymptotically stable, the optimal ROM matrices are sought

within a product manifold which involves the spaces of skew-symmetric and

of symmetric positive definite matrices. To solve this optimization problem,

a Riemannian trust-region algorithm is presented, which involves solving lin-

ear matrix equations scaling with the FOM dimension in every iteration and

is, thus, only applicable for problems of moderate size. As starting value, it

is proposed to use a ROM obtained via the pH-IRKA approach from [130]

and the numerical results indicate a clear improvement in comparison to this

starting value, especially when the dimension of the ROM is very small. How-

ever, this advantage becomes smaller as the ROM dimension increases and the

ROM obtained via pH-IRKA is even partially better when considering the H∞

instead of the H2error.

In [209], the authors propose a more efficient way for solving the optimiza-

tion problem considered in [252] based on pole-residue formulations of the

transfer functions of the FOM and of the ROM. This avoids having to solve

high-dimensional linear matrix equations in each step and, thus, extends the

applicability to large-scale FOMs.

While [209] and [252] are based on optimization with respect to the H2

distance, in [261] the authors propose the structured optimization-based model

order reduction (SOBMOR) method using a cost function which is motivated

by H∞minimization, but only requires transfer function evaluations at certain

sampling points on the imaginary axis. In their numerical experiments, they

demonstrate that minimizing this cost function is advantageous in terms of

computational cost and accuracy compared to a direct minimization of the

H∞error. Furthermore, their numerical experiments reveal that this new

method outperforms other structure-preserving MOR techniques as the effort-

constraint balanced truncation method introduced in [232] and the pH-IRKA

method proposed in [130]. In addition, an adaptive sampling strategy for

the SOBMOR approach is presented in [260], whereas an extension of the

approaches from [209] and [261] to DAE systems is provided in [210, 262].

An Approach Based on POD

The methods mentioned so far are based on considering port-Hamiltonian sys-

tems as control systems and they build ROMs by approximating the input-

output map. Due to the focus on the input-output behavior, the ROMs ob-

tained from these methods do typically not lead to a good approximation of

the FOM state variables. In contrast to this, in [65] the authors present a

structure-preserving model reduction technique which is based on the POD

method for generating the projection matrices and, thus, this approach may

be used for applications where a good approximation of the FOM state is

desirable. The considered class of FOMs consists of nonlinear time-invariant

1. Introduction

port-Hamiltonian systems of ordinary differential equations. To ensure struc-

ture preservation, the authors propose to use a Petrov–Galerkin projection,

where the trial and test spaces are chosen such that they provide good ansatz

spaces for the FOM state and the gradient of the FOM Hamiltonian, respec-

tively, see also section 2.6.2 for more details. Those spaces may be obtained

by computing SVDs of matrices containing snapshots of the state or of the

gradient of the Hamiltonian, respectively.

Since the evaluation of the ROM involves in general the evaluation of the

FOM nonlinearity and is thus not efficient, they also present a structure-

preserving hyperreduction technique, which is based on an additional approx-

imation of the nonlinearity and thereby renders the ROM evaluation efficient.

The hyperreduction method is based on a modification of the discrete empir-

ical interpolation method (DEIM) and involves first writing the Hamiltonian

as the sum of a quadratic function and a remainder term. Afterwards, they

introduce a DEIM-inspired approximation of the remainder term such that the

gradient of the approximate Hamiltonian may be evaluated in an efficient way.

Finally, this leads to a ROM which is port-Hamiltonian and whose evaluation

does not scale with the FOM dimension.

Next to the POD-based approach, the authors also present another method,

where the linear subspace used for approximating the FOM state is determined

based on applying the pH-IRKA method from [130] to a linearized version of

the FOM.

Methods for Hamiltonian Systems

While this literature overview mainly focuses on general pH systems, we em-

phasize that there has been also much research effort to develop structure-

preserving model reduction techniques for Hamiltonian systems. In this con-

text, researchers are typically not only interested in preserving the energy

conservation properties, but also in preserving the symplectic structure. For

instance, in [228] the authors present a structure-preserving model reduction

scheme based on the proper symplectic decomposition (PSD), which is a mod-

ification of the POD method to determine a symplectic projection matrix, and

a subsequent symplectic Galerkin projection. As an alternative to the PSD, in

[3] a greedy procedure for parameter-dependent Hamiltonian systems is intro-

duced, which gradually constructs a symplectic projection matrix by adding

two basis vectors per iteration. This approach has been also applied to dissipa-

tive Hamiltonian systems in [4] by first coupling the FOM to a heat bath, which

collects the dissipated energy such that the coupled system may be regarded as

a conservative system. In [143, 223], the authors present a structure-preserving

MOR framework using time-dependent basis functions and thereby not only

achieve preservation of the Hamiltonian structure, but this approach is also

suitable for achieving an effective reduction of transport-dominated systems.

This applies also to the symplectic manifold Galerkin method, which is intro-

duced in [50] and involves a structure-preserving projection of a Hamiltonian

1.4. Contributions, Outline, and Previously Published Results

system onto a nonlinear manifold, which may be for instance constructed based

on an ANN.

1.4. Contributions, Outline, and Previously

Published Results

In this section, we provide an overview of the major contributions of this thesis

and briefly summarize the structure of its remainder. Moreover, since some of

the results of this thesis originate from joint work with other researchers and

have already been published, we address these previously published works at

the end of this section.

We begin by listing the main contributions of this thesis in the following.

(i) Given snapshot data of a finite-dimensional FOM of the form (1.1), we

present a new method for computing a decomposition of the form (1.4).

To this end, we assume the transformation operators and paths to be

given and propose to solve an optimization problem which aims for min-

imizing the error of the approximation (1.4). Moreover, we propose two

different optimization strategies: the optimization parameters of the first

one consist of the amplitudes and the modes, whereas the second one is

based on a reduced optimization problem, which has only the modes as

optimization parameters, cf. section 3.1.

(ii) For the case of parameter-dependent FOMs, we introduce a greedy al-

gorithm for combining the parameter sampling with a gradual determi-

nation of the modes in the offline phase. This algorithm extends the

classical POD-greedy algorithm to nonlinear approximation ansatzes of

the form (1.8) and involves the repeated application of the optimization

procedure from (i), cf. section 3.2.

(iii) While the framework in this thesis is mostly presented for a rather general

class of transformation operators, we propose various families of trans-

lation operators for one-dimensional spatial domains. Especially, these

translation operators differ in their boundary treatment and we illus-

trate their usefulness by means of some exemplary PDEs with different

boundary conditions, cf. section 3.3 and chapter 6.

(iv) Based on given modes, we present a nonlinear projection framework for

constructing dynamic reduced-order models for determining the ampli-

tudes and the paths in the online phase. In particular, the ROMs are

constructed in such a way that the residual is minimized at each time

instant and, due to the nonlinearity of the approximation ansatzes (1.4)

and (1.8), the ROMs are in general nonlinear even in the case of a linear

FOM. We also present a residual-based error bound and discuss existence

1. Introduction

and uniqueness of solutions of the ROM. In addition, we also discuss the

case where the state-dependent mass matrix of the ROM may become

singular and propose a regularization strategy for ensuring invertibility

of the mass matrix, see section 4.1.

(v) We show that in a special case the ROMs constructed as in (iv) are equiv-

alent to ROMs obtained via a symmetry reduction technique, provided

that the same paths are used in both frameworks. Moreover, we compare

the residual minimization strategy for fixing the paths as considered in

this thesis with two different strategies within the symmetry reduction

framework and show a relation between the corresponding minimization

problems, cf. section 4.2.

(vi) To obtain ROMs whose evaluation does not scale with the dimension

of the FOM, we introduce a hyperreduction framework, which is based

on additional approximations of the ROM nonlinearities. One source

of nonlinearity of the ROM is the nonlinearity of the approximation

ansatz and it leads to path-dependent coefficient matrices whose eval-

uation involves large-scale matrix-vector products. To avoid evaluating

these path-dependent matrices in the online phase, we propose to de-

termine corresponding interpolation- or regression-based approximants,

which may be evaluated in the online phase in an efficient way. Fur-

thermore, we present an extension of the empirical interpolation method

(EIM)/DEIM which is based on approximating the FOM nonlinearity by

an ansatz of the form (1.8), cf. section 4.3.

(vii) We discuss in a general finite-dimensional setting how a port-Hamiltonian

representation of the FOM may be exploited in the context of projection-

based model reduction to obtain port-Hamiltonian ROMs. To this end,

we consider different linear and nonlinear as well as time-invariant and

time-varying classes of port-Hamiltonian FOMs and present structure-

preserving schemes based on different linear and nonlinear as well as

time-invariant and time-varying approximation ansatzes. The consid-

ered nonlinear approximation ansatzes include in the finite-dimensional

setting (1.4) as a special case, which allows us to obtain port-Hamiltonian

ROMs which are based on the ansatz (1.4). Moreover, we provide ad-

ditional conditions which ensure that the port-Hamiltonian ROMs are

stable and we demonstrate that in many cases we are able to obtain

ROMs which are at the same time pH and optimal in the sense of resid-

ual minimization, cf. chapter 5.

(viii) We illustrate the numerical performance of the new MOR framework by

means of several transport-dominated test cases including an example

with more than one traveling wave, an example with non-periodic and

inhomogeneous boundary conditions, and a nonlinear example. In this

context, we also provide details on suitable discretizations of the involved

1.4. Contributions, Outline, and Previously Published Results

translation operators and on structure-preserving discretization schemes.

All in all, the numerical results illustrate that the new method not only

yields accurate ROMs of very low dimension, but it also allows for a great

flexibility in increasing the time step size compared to the corresponding

FOMs and to corresponding POD-based ROMs, cf. chapter 6.

The remainder of this thesis is subdivided into six chapters and four ap-

pendices. In chapter 2, we introduce some notation used throughout this

thesis and present a couple of preliminary results from different relevant areas

such as nonlinear optimization, model reduction, and port-Hamiltonian sys-

tems. We proceed in chapter 3 by addressing the problem of identifying modes

based on given snapshot data, whereas chapter 4 is devoted to the construc-

tion of ROMs, once the modes have been determined. Afterwards, we discuss

structure-preserving, projection-based MOR for different classes of linear and

nonlinear port-Hamiltonian systems and using different linear and nonlinear

approximation ansatzes in chapter 5. In chapter 6, we demonstrate the appli-

cation of some of the methods presented in chapters 3 to 5 by means of three

transport-dominated numerical examples. Finally, a summary of the whole

thesis and an outlook to future work are presented in chapter 7.

For the sake of a better readability of the main chapters, some of the results

are presented only at the end of this thesis in one of the appendices. In

appendix A we discuss some properties of the periodic shift operator since

this transformation operator is used in many examples of this thesis and some

of them require the properties formally proven in appendix A. Furthermore,

we analyze some properties of a class of linear time-varying port-Hamiltonian

systems in appendix B, since we use this class in chapter 5 and since it differs

from the classical definition introduced in [25]. In appendix C, we discuss time

integration schemes based on discrete gradients and propose a new scheme

for nonlinear time-invariant port-Hamiltonian systems with a state-dependent

mass matrix. The latter one is used in chapter 6 to ensure a dissipation

inequality for the ROMs after time discretization. At the end, we provide in

appendix D some details about the discretization of the shift operators used

in chapter 6 and about the spatial discretization of the nonlinear term of the

reaction–diffusion test case considered in section 6.3.

We close this section by stating which of the results of this thesis have

been obtained in joint work with other researchers and are already published

elsewhere.

•The nonlinear projection framework considered in section 4.1 has been

originally developed in joint work with F. Black and B. Unger and was

first published in [37]. As mentioned in section 1.1, the setting con-

sidered in this thesis is slightly different, but nevertheless large parts

of section 4.1 are directly based on the results from [37], whereas the

regularization method presented at the end of section 4.1 is new. In

1. Introduction

section 4.2, we consider even the same setting as in [37] and in par-

ticular the content of section 4.2 essentially coincides with the one of

[37, sec. 6], even though section 4.2 is partially more detailed. Also the

infinite-dimensional formulation of the optimization problem considered

in section 3.1 is based on the one presented in [37] with only slight mod-

ifications. In addition, the extended domain shift operator presented in

section 3.3.1 has been originally introduced in [37]. Finally, the numer-

ical test cases considered in sections 6.1 and 6.2 are very similar to the

ones considered in sections 7.2 and 7.3 from [37]. However, the numerical

experiments are not identical, as we use for instance different initial and

boundary values as well as different discretization methods in this thesis.

•A preliminary version of the optimization framework considered in sec-

tion 3.1 has been originally presented in [259] in joint work with J. Reiss

and V. Mehrmann. However, section 3.1 provides a significantly more

profound analysis and, especially, discusses in detail the connection to

the variable projection framework as introduced in [120].

•The hyperreduction framework which is subject of section 4.3 has been

originally presented in a very similar fashion in [39] in joint work with

F. Black and B. Unger. A notable difference to [39] is that the method

presented in section 4.3.2 is formulated for the case where the FOM is a

PDE.

•The original idea of approximating the FOM state by an augmented

approximation ansatz including multiple transformation operators has

been presented in [241] in joint work with J. Reiss, V. Mehrmann, and

J. Sesterhenn. While the results from [241] are not presented in detail in

this thesis, we emphasize that the constant extrapolation shift operator

presented in section 3.3.3 has been already used in [241]. Moreover, in

section 3.4 we compare the approach from [241] to the one outlined in

section 3.1.

In addition to the listed papers, there are also other previously published

works, which are at least mentioned in this thesis. For instance, the optimiza-

tion framework presented in [40] (joint work with F. Black and B. Unger) is

compared in section 3.4 to the approach from section 3.1 and in addition men-

tioned several times throughout this thesis. Also the study [38] (joint work

with F. Black and B. Unger), which addresses the application of the projection

framework from [37] to the linear wave equation, is mentioned a few times and

its findings are also used for the numerical test case presented in section 6.1.

Further previously published papers which are mentioned in this thesis include

contributions on port-Hamiltonian modeling as in [7] (with R. Altmann), [19]

(with H. Bansal, M. H. Abbasi, H. Zwart, L. Iapichino, W. H. A. Schilders, and

N. van de Wouw), and [69] (with K. Cherifi and V. Mehrmann), on structure-

preserving model reduction as in [45] (with T. Breiten and R. Morandin) and

1.4. Contributions, Outline, and Previously Published Results

[43] (with T. Breiten), and on structured data-driven modeling as in [103] (with

E. Fosong and B. Unger), [257] (with B. Unger), and [258] (with B. Unger,

C. Beattie, and S. Gugercin).

2. Preliminaries

In this chapter, we introduce the notation which is used throughout this thesis,

cf. section 2.1, and provide some standard results from different areas which

are relevant for the content following in the next chapters. Since we encounter

various optimization problems throughout this thesis, the most relevant re-

sults of nonlinear optimization are summarized in section 2.2. Furthermore, in

section 2.3 we recap the relevant concepts from functional analysis, whereas in

section 2.4 we state standard existence, uniqueness, and stability results for dif-

ferential equations. In section 2.5, we summarize the general idea of parametric

model order reduction and present some common techniques based on linear

approximation ansatzes. Finally, some basic properties of port-Hamiltonian

systems and a corresponding structure-preserving model reduction scheme are

addressed in section 2.6.

2.1. Notation

The sets of natural, real, and integer numbers are denoted with N,R, and Z,

respectively. Here, we use the convention that Ndoes not contain 0. Further-

more, for a general field F, we use Fm,n for the set of m×nmatrices. For

column vectors, we abbreviate Fm,1as Fmand for the special case F=R, we

write k·k for the Euclidean norm on Rm. Moreover, given a symmetric and

positive definite matrix A∈Rm,m, we use the notation k·kAfor the associated

weighted norm defined by kxkA:=√x>Ax for all x∈Rm. In addition, the

Frobenius norm of a matrix A∈Rm,n is denoted with kAkFand the Frobenius

inner product of two matrices A, B ∈Rm,n with hA, BiF. Besides, we write

[A]jto refer to the jth column vector of a matrix Aand [A]i,j for the ith

entry of [A]j. Conversely, the definition of an m×nmatrix Aby its entries

a1,1, a1,2, . . . , am,n ∈Fis denoted as A:= [ai,j]ij ∈Fm,n. Similarly, we define a

column vector vof length nby its entries via v:= [vi]i∈Fn. The n×niden-

tity matrix is denoted by Inand its entries can be described by the Kronecker

delta, i.e., [In]i,j =δij for i, j = 1, . . . , n or, equivalently, In= [δij]ij ∈Fn,n.

Moreover, for a column vector in Fnwhose entries are all equal to one, we

use the symbol 1n, while 0m×ndenotes an m×nmatrix whose entries are all

equal to zero. Furthermore, we introduce the following short-hand notation

2. Preliminaries

for (block-)diagonal and tridiagonal matrices:

diag (A1, . . . , An):=





A10··· 0

0A2....

.......0

0··· 0An







tridiagn(a, b, c):=







b c 0··· 0

a b c ....

0a b ...0

..........c

0··· 0a b







∈Rn,n.

Here, A1, . . . , Ancan be matrices of any size, whereas a, b, c are scalars. The

Kronecker product of two matrices A∈Fm,n,B∈Fp,q is denoted as A⊗B,

i.e.,

A⊗B=





a1,1B··· a1,nB

.....

am,1B··· am,nB





∈Rmp,nq.

For a matrix A∈Fm,n, we use the notations ker(A)and im(A)for the kernel

and the image of the associated linear mapping x7→ Ax from Fnto Fm, respec-

tively. Furthermore, the transpose of Ais denoted with A>and, for the special

case F=R, the Moore–Penrose pseudoinverse of Ais denoted with A+, cf. [57,

ch. 1]. Moreover, to indicate that a matrix A∈Rm,m is positive (semi-)definite,

we write A > 0(A≥0). Besides, we use the notation σmax(A)and σmin(A)

for the largest and the smallest singular values of A∈Rm,n, respectively. If

Ais invertible, we denote its condition number σmax(A)σmin(A)−1with κ(A).

In the special case of a symmetric matrix A, we use λmax(A)and λmin(A)for

the largest and the smallest eigenvalues of A, respectively. If A∈Rn,n is in

addition positive semi-definite, we call the uniquely determined symmetric and

positive semi-definite matrix B∈Rn,n satisfying B2=Athe square root of A

and denote it with A1

2, see for instance [150, Thm. 7.2.6].

Given x∈Rand y∈R\{0}, we define the corresponding modulo operation

via

xmod y:=x−y$x

y%,

where b·c:R→Zdenotes the floor function. Besides, for real scalars x1, . . . , xn

with n∈N, we define max(x1, . . . , xn):=xkwhere xksatisfies xk≥xifor all

i= 1, . . . , n and similarly we use min(x1, . . . , xn)for the minimal value.

For metric spaces Xand Y, we use the notation C(X, Y )for the set of

continuous mappings from Xto Yand, in particular, C(X):=C(X, R)for

the special case Y=R. In addition, the closure of a subset Uof a metric space

Xis denoted by U. Besides, we use the notations [a, b],[a, b),(a, b], and (a, b)

2.2. Nonlinear Optimization

for intervals between the real numbers aand b>a, where a square bracket is

used if the respective boundary belongs to the interval and a round bracket if

not.

Throughout this thesis, we use Lagrange’s notation f0for the total derivative

of a function for alternatively Newton’s notation ˙

for Leibniz’s notation df

in the special case where the independent variable corresponds to the time t,

cf. [55, ch. II]. Furthermore, the partial derivatives of a function depending on

several variables, e.g. f(x1, . . . , xn), are denoted by ∂x1f:=∂f

∂x1etc. Moreover,

if one of the independent variables, e.g. x1, is an element of Rn, we use the

notation ∇x1ffor the partial gradient of fwith respect to x1.

For a general set M, we use the notation |M|for its cardinality and P(M)

for its power set, i.e., for the set of all subsets of M. Moreover, the set of all

finite subsets of Mis denoted as Pf(M). Finally, the identity operator which

maps every element of Mto itself is denoted as IdM.

2.2. Nonlinear Optimization

In this section we recall some basic definitions and theorems from uncon-

strained nonlinear optimization, as these are relevant for most of the chapters

of this thesis. To this end, we consider an optimization problem of the form

min

x∈RnJ(x)(2.1)

with a cost function J:Rn→R. Before stating necessary and sufficient

conditions for solutions of this optimization problem, we introduce the notion

of local and global minimum points, see for instance [24, sec. 4.1].

Definition 2.2.1 (Local and global minimum point).Let n∈Nand J:Rn→

Rbe given. A point ˆx∈Rnis called a local minimum point of Jif there exists

∈R>0such that J(ˆx)≤J(x)holds for all x∈Rnwith kx−ˆxk< . If there

even exists ∈R>0such that J(ˆx)< J(x)is satisfied for all x∈Rn\ {ˆx}

with kx−ˆxk< , then ˆxis called a strict local minimum point of J. Similarly,

we define the terms global minimum point and strict global minimum point by

omitting the restriction kx−ˆxk< .K

The following theorem provides necessary first-order and second-order opti-

mality conditions for a local minimum point. In particular, we call each point

x∈Rnsatisfying (2.2) a critical point of Jor of the associated minimization

problem (2.1). A sufficient condition for a critical point to be a strict local

minimum point is provided in Theorem 2.2.3. The proofs of both theorems

may for example be found in [24, sec. 4.1].

Theorem 2.2.2 (Necessary optimality conditions).For given n∈Nand

J:Rn→R, let ˆx∈Rnbe a local minimum point of (2.1) and let Jbe

2. Preliminaries

continuously differentiable at ˆx. Then, the derivative of Jvanishes at ˆx, i.e.,

J0(ˆx)=0.(2.2)

Furthermore, if Jis even twice continuously differentiable at ˆx, then J00(ˆx)is

positive semi-definite.

Theorem 2.2.3 (Sufficient second-order optimality condition).For given n∈

Nand J:Rn→R, let ˆx∈Rnbe a critical point of (2.1) and let Jbe twice

continuously differentiable at ˆx. If J00(ˆx)is positive definite, then ˆxis a strict

local minimum point of J.

In the special case where Jis continuously differentiable and convex, cf. Def-

inition 2.2.4, it turns out that a simple sufficient condition may be formulated

even for global minima, see Theorem 2.2.5 and [216, p. 53] for a corresponding

proof. Furthermore, a sufficient condition for convexity is provided in Theo-

rem 2.2.6, cf. [216, p. 55].

Definition 2.2.4 (Convex continuously differentiable functions).For n∈N,

we call f∈C1(Rn)convex on Rnif it satisfies

f(y)≥f(x) + f0(x)(y−x)for all (x, y)∈Rn.K

Theorem 2.2.5 (Sufficient first-order optimality condition for a convex func-

tion).Let J∈C1(Rn)with n∈Nbe convex and let J0(ˆx) = 0 hold for some

ˆx∈Rn. Then, ˆxis a global minimum point of J.

Theorem 2.2.6 (Sufficient condition for convexity).Let f∈C2(Rn)with

n∈Nbe given and let f00(x)be positive semi-definite for all x∈Rn. Then, f

is convex.

2.3. Functional Analysis

In this section, we recall some standard notions and results from functional

analysis. In particular, we commence by introducing Lebesgue and Sobolev

spaces and afterwards address linear bounded operators as well as differentiable

mappings between Banach spaces.

Lebesgue and Sobolev Spaces

We start by introducing Lebesgue spaces following [12, ch. X]. To this end,

we consider a complete σ-finite measure space (Ω,A, µ), cf. [12, ch. IX], and

first introduce the notions of µ-simple and µ-measurable functions from Ωto

a Banach space B.

Definition 2.3.1 (µ-simple and µ-measurable functions).Let (Ω,A, µ)be a

complete σ-finite measure space and (B,k·kB)a Banach space. Then, we call

a function f: Ω → B µ-simple if each of the following conditions is satisfied:

2.3. Functional Analysis

(i) f(Ω) is a finite set,

(ii) for every b∈ B, the fiber of bunder fis an element of A,

(iii) µ(f−1(B\{0})) <∞.

Besides, we call f: Ω → B µ-measurable if there exists a sequence of µ-simple

functions converging µ-almost everywhere to f. Moreover, the set of all µ-

measurable functions from Ωto Bis denoted with L0(Ω, µ, B).K

If f: Ω → B is µ-measurable, then also kfkp

B: Ω →R≥0is a µ-measurable

function for any p∈R≥1, cf. Remark X.1.2(d) and Theorem X.1.7(i) in [12].

In particular, this allows to define the integral RΩkfkp

Bdµas in [12, p. 98] and

based on this we introduce the Lebesgue spaces.

Definition 2.3.2 (Lebesgue spaces).Let (Ω,A, µ)be a complete σ-finite mea-

sure space and (B,k·kB)a Banach space. For each p∈R≥1, the set of all

µ-measurable functions f: Ω → B satisfying

Ωkfkp

Bdµ < ∞

is denoted with Lp(Ω, µ, B). Furthermore, we use the notation L∞(Ω, µ, B)for

the set of all µ-measurable functions f: Ω → B for which there exists r∈R≥0

with µ({s∈Ω| kf(s)kB> r})=0.K

In [12, sec. X.4] it is shown that Lp(Ω, µ, B)as in Definition 2.3.2 is a vector

space for all p∈[1,∞]. Moreover, it is proven in [12, Rem. X.4.4(b)] that the

set

N:={f∈ L0(Ω, µ, B)|f= 0 µ-almost everywhere}

is a subspace of Lp(Ω, µ, B)for all p∈[1,∞]. In particular, this allows to

introduce the corresponding quotient spaces

Lp(Ω, µ, B):=Lp(Ω, µ, B)/N

for all p∈[1,∞]. While the elements of Lp(Ω, µ, B)are functions from Ωto B,

Lp(Ω, µ, B)consists of equivalent classes of functions which coincide µ-almost

everywhere. As is common in the literature and by abuse of notation, we do

usually not distinguish between the equivalent class [f]∈Lp(Ω, µ, B)and its

representatives.

If Bis a real Hilbert space with inner product h·,·iB, then also L2(Ω, µ, B)

is a real Hilbert space with inner product

hf, giL2(Ω,µ,B):=Z

Ωhf, giBdµ,

cf. [12, Thm. X.4.10(ii)]. Moreover, in most examples of this thesis, we consider

the case of Ωbeing an open or a closed subset of Rnwith λn(Ω) >0, where λn

2. Preliminaries

denotes the n-dimensional Lebesgue measure, cf. [12, sec. IX.4]. In this case,

we use the abbreviations Lp(Ω,B):=Lp(Ω, λn,B)and Lp(Ω) :=Lp(Ω,R)for

all p∈[1,∞]. In particular, for the case that Ωis open and pin R≥1, it is

stated in [2, Thm. 2.21] that Lp(Ω) is separable, i.e., it has a countable dense

subset. Sometimes, we also consider the special case where Ωis a non-empty

finite subset of Rand then we use the counting measure H0, cf. [12, sec. 9.2],

and the notation Lp(Ω) :=Lp(Ω,H0,R).

In the following, we present a second important class of function spaces,

namely the Sobolev spaces, cf. [2]. To this end, we consider a non-empty open

set Ω⊆Rnand we start by introducing the notion of weak partial derivatives

of a locally integrable function. A function fwhich is almost everywhere

defined on Ωis called locally integrable in Ω, if it satisfies f∈L1(M)for all

non-empty open sets Mwhich are bounded and satisfy M⊆Ω. Furthermore,

the set of locally integrable functions in Ωis denoted with L1

loc(Ω). Especially,

Lp(Ω) is contained in L1

loc(Ω) for all p∈[1,∞], cf. [2, Cor. 2.15].

We proceed with the definition of weak partial derivatives. For this purpose,

we call a vector αin (Z≥0)namultiindex of order |α|:=Pn

i=1 αi. Given such

a multiindex αand a function u: Ω →R, we define

Dαu:=∂α1

ξ1···∂αn

ξnu, (2.3)

where ∂ξidenotes the partial derivative with respect to the ith component of

Ωfor i= 1, . . . , n. If there exists for a given u∈L1

loc(Ω) and a multiindex

α∈(Z≥0)na function v∈L1

loc(Ω) satisfying

Ω

ϕ(ξ)v(ξ) dξ= (−1)|α|Z

Ω

u(ξ)Dαϕ(ξ) dξfor all ϕ∈C∞

c(Ω),

then we call vthe α-th weak partial derivative of uand write Dαu=v. Here,

C∞

c(Ω) denotes the space of infinitely differentiable functions with compact

support in Ω, cf. [2, ch. 1]. In fact, it can be shown that the weak derivative, if

it exists, is uniquely determined up to a set of measure zero, cf. [94, sec. 5.2.1].

Based on the notion of weak derivatives, we introduce the Sobolev spaces.

Definition 2.3.3 (Sobolev space).For n∈N, a non-empty open set Ω⊆Rn,

p∈[1,∞], and k∈Z≥0, we define the Sobolev space Wk,p(Ω) via

Wk,p(Ω) :={u∈Lp(Ω) |Dαu∈Lp(Ω) for all α∈(Z≥0)nwith |α| ≤ k},

where Dαudenotes the α-th weak partial derivative of u.K

In the proof of [2, Thm. 3.3] it is shown that the Sobolev spaces equipped

with suitable norms are Banach spaces. For instance, for p∈R≥1and k

and Ωas in Definition 2.3.3, a norm for Wk,p(Ω) is given by the mapping

2.3. Functional Analysis

k·kWk,p(Ω) :Wk,p(Ω) →Rdefined via

kukp

Wk,p(Ω) =X

|α|≤kZ

Ω|Dαu(ξ)|pdξ.

Moreover, the Sobolev space Wk,2(Ω) is a Hilbert space with inner product

h·,·iWk,2(Ω) :Wk,2(Ω) ×Wk,2(Ω) →Rdefined as

hu, viWk,2(Ω) :=X

|α|≤khDαu, DαviL2(Ω)

for all k∈Z≥0, cf. [2, Thm. 3.6]. Accordingly, we use in the following the

abbreviation Hk(Ω) :=Wk,2(Ω).

In this thesis, we are especially interested in H1(Ω) and often consider the

case where Ωis given by an open interval in R. If this is the case, then every

equivalence class in H1(Ω) has a continuous representative in Ωas the following

theorem specifies, cf. [46, Thm. 8.2].

Theorem 2.3.4 (Continuous representatives in H1(Ω)).Let f∈H1(Ω) with

Ω=(a, b),a∈R,b∈R>a be given. Then, there exists a function ˜

f∈C(Ω)

such that fand ˜

fcoincide almost everywhere on Ωand

f(ξ2)−˜

f(ξ1) =

ξ2

ξ1

D1f(ξ) dξ

holds for all ξ1, ξ2∈Ω.

By Theorem 2.3.4, we may define pointwise evaluations for elements of

H1((a, b)) in [a, b]via the respective continuous representatives. In particu-

lar, this allows to define the subspaces

0(Ω) :={f∈H1(Ω) |f(a) = f(b)=0},

per (Ω) :={f∈H1(Ω) |f(a) = f(b)},

which correspond to vanishing and periodic boundary values, respectively.

Linear Operators and Differentiable Mappings

As mentioned in section 1.1, the approximation ansatz (1.4) involves a trans-

formation family T, which is pointwise a linear bounded operator. In Defini-

tion 2.3.5, we provide a formal definition of such operators, see also for instance

[11, sec. VI.2]. Especially, one may show that L(X, Y )is a vector space and

that k·kL(X,Y )is indeed a norm for non-trivial X, cf. [11, p. 13f.].

Definition 2.3.5 (Linear bounded operator).Let (X, k·kX),(Y, k·kY)be two

normed spaces. We call T:X→Yalinear bounded operator from Xto Y,

2. Preliminaries

if Tis linear and there exists c∈R≥0such that

kTxkY≤ckxkXfor all x∈X.

Moreover, the set of linear bounded operators from Xto Yis denoted with

L(X, Y )and in the special case X=Ywe write L(X):=L(X, X). In

addition, for the case that Xis non-trivial, i.e., X6={0}, we introduce the

operator norm k·kL(X,Y ):L(X, Y )→R≥0via

kTkL(X,Y ):= sup

x∈X\{0}

kTxkY

kxkX

While Definition 2.3.5 is formulated for general normed spaces, we often

consider the special case of Hilbert spaces. In this context, the notions of

adjoint and unitary operators are provided in the following.

Definition 2.3.6 (Adjoint operator).Let (H1,h·,·iH1)and (H2,h·,·iH2)be

real Hilbert spaces and T:H1→H2a linear bounded operator. Then, we call

the unique operator T∗∈ L(H2, H1)satisfying

hTx, yiH2=hx, T∗yiH1for all x∈H1, y ∈H2

the adjoint operator of T.K

Definition 2.3.7 (Unitary operator).For given real Hilbert spaces H1and H2,

we call T∈ L(H1, H2)unitary if it satisfies TT∗= IdH2and T∗T= IdH1.K

The fact that there exists a uniquely determined adjoint for each operator

between two Hilbert spaces H1and H2is for instance shown in the proof

of [155, Thm. 2.4.2]. Moreover, a direct consequence of Definition 2.3.7 is

that a unitary operator is isometric, i.e., it satisfies kTxkH2=kxkH1and

hTx, TyiH2=hx, yiH1for all x, y ∈H1, cf. [158, p. 257f.].

In the remainder of this section, we introduce the differentiability notions

used in this thesis and start with the differentiability of mappings between

Banach spaces, cf. [11, p. 149].

Definition 2.3.8 (Differentiability of functions between Banach spaces).Let

(X, k·kX)and (Y, k·kY)be real Banach spaces and let Ube an open subset of

X. Then, we call a mapping f:U→Ydifferentiable at u0∈Uif there exists

L∈ L(X, Y )satisfying

lim

u→u0

f(u)−f(u0)−L(u−u0)

ku−u0kX

= 0.

Furthermore, if fis differentiable at every u0∈U, then we say that fis

differentiable.K

2.3. Functional Analysis

We emphasize that the mapping Lfrom Definition 2.3.8 depends on u0and

in particular that it is uniquely determined for given u0∈U, provided that fis

differentiable at u0, cf. [11, Prop. VII.2.1(iii)]. If fis differentiable, then we call

the function which maps u0∈Uto the corresponding linear bounded operator

L, the derivative of fand denote it by f0. If additionally f0is continuous, then

we call fcontinuously differentiable and write f∈C1(U, Y ). Furthermore, if

a function f:U→Yis differentiable at a point u0∈Uas in Definition 2.3.8,

then fis also continuous at u0, see [11, Prop. VII.2.1(ii)]. Besides, in this

thesis we often make use of the chain rule as it is formalized in Theorem 2.3.9,

cf. [11, Thm. VII.3.3]. Another important property is the generalized product

rule stated in Theorem 2.3.11, which is a special case of [11, Cor. VII.4.7].

It uses the notion of a bounded or, equivalently, continuous bilinear form as

defined in Definition 2.3.10, cf. [11, sec. VII.4].

Theorem 2.3.9 (Chain rule).Let X,Y, and Zbe real Banach spaces and

U⊆X, V ⊆Ybe open subsets. If f:U→Ywith f(U)⊆Vis differentiable

at u∈Uand g:V→Zis differentiable at f(u), then g◦fis differentiable at

uwith derivative

(g◦f)0(u) = g0(f(u)) ◦(f0(u)).

Definition 2.3.10 (Bounded bilinear form).Let (X, k·kX)and (Y, k·kY)be

real Banach spaces. Then, we call a bilinear form a:X×Y→Rbounded, if

there exists a constant β∈R≥0with

|a(x, y)| ≤ βkxkXkykYfor all x∈X, y ∈Y. K

Theorem 2.3.11 (Generalized product rule).Let Xbe a real Banach space,

U⊆Rbe an open set, a:X×X→Rbe a bounded bilinear form, and

f1, f2:U→Xbe continuously differentiable. Then, the map g:U→Rdefined

via g(u):=a(f1(u), f2(u)) is continuously differentiable with derivative

g0=a(f0

1, f2) + a(f1, f0

2).

The differentiability definition provided in Definition 2.3.8 considers a rather

general setting with Banach spaces, but is restricted to functions defined on

an open set. In this thesis, we often also consider functions which are defined

for instance on a closed interval and for such functions we consider the dif-

ferentiability notion presented in Definition 2.3.12, cf. [10, sec. IV.1]. If fis

differentiable in the sense of Definition 2.3.12 with continuous derivative f0,

then we call fcontinuously differentiable and write f∈C1(U, X). Similarly as

for the differentiability notion from Definition 2.3.8, also Definition 2.3.12 leads

to a chain rule for the derivative of a composition of differentiable functions,

see [10, Thm. IV.1.7]. For the special case U= [a, b]with a∈R,b∈R>a, we

introduce the notation C1

per([a, b], X)for the subset

per([a, b], X):={f∈C1([a, b], X)|f(a) = f(b), f0(a) = f0(b)}.

2. Preliminaries

Definition 2.3.12 (Differentiability at a limit point).Let Ube a subset of R,

u0∈Ube a limit point of U, and Xbe a real normed vector space. Then, we

call a function f:U→Xdifferentiable at u0if the limit

f0(u0):= lim

u→u0

f(u)−f(u0)

u−u0∈X

exists. In that case we call f0(u0)the derivative of fat u0. Furthermore, if

Uis dense-in-itself, i.e., every element of Uis a limit point of U, and if fis

differentiable at every u0∈U, then we say that fis differentiable.K

So far, we have only addressed differentiable functions which are defined on

an open set or on a dense-in-itself subset of R. In addition, we consider also

the differentiability of functions which map from U⊆Rnto Rmwith m, n ∈N,

U=J1×J2×···×Jn, and J1, . . . , Jn⊆Rbeing proper intervals. Especially,

every element of Uis a limit point of Uand one can introduce the notion of

differentiability analogously to Definition 2.3.8. Also most of the important

properties like uniqueness of the derivative, the fact that differentiability im-

plies continuity, and the chain rule may be proven following the lines of the

corresponding proofs for differentiable functions defined on open sets. Simi-

larly as for the other differentiability notions, we denote the set of continuously

differentiable functions from U⊆Rnto Rmwith C1(U, Rm).

2.4. Differential Equations

In section 1.1, we have considered a differential equation as full-order model

which describes the time evolution of the X-valued state x. Such equations

are also referred to as abstract evolution equations. In section 2.4.1, we spec-

ify how classical solutions of the associated initial value problems are defined

and discuss the existence and uniqueness of such solutions for a special class

of right-hand sides. Afterwards, we address finite-dimensional systems of dif-

ferential equations in section 2.4.2, where we also introduce the notions of

equilibrium points and stability.

2.4.1. Abstract Evolution Equations and Semigroups

In the following, we consider an abstract evolution equation of the form

˙x(t) = F(t, x(t)) for all t∈I:= [0, tend],(2.4a)

x(0) = x0(2.4b)

with tend ∈R>0,F:R≥0×W→X, and x0∈W, where Xis a real Banach

space and W⊆Xa subspace. We call x:I→Xaclassical solution of the

initial value problem (2.4) if xis continuously differentiable, x(t)∈Wfor all

t∈I, and (2.4) holds.

2.4. Differential Equations

Before we address the question of existence and uniqueness of classical solu-

tions for a special class of right-hand sides F, we introduce strongly continuous

semigroups and their generators in Definitions 2.4.1 and 2.4.2, see also Defini-

tions I.5.1 and II.1.2 in [92].

Definition 2.4.1 (Strongly continuous semigroup).Let Bbe a Banach space.

Then, we call T:R≥0→ L(B)astrongly continuous semigroup if it satisfies

the following properties:

(i) T(t+s) = T(t)T(s)for all s, t ∈R≥0,

(ii) T(0) = IdB,

(iii) for every x∈ B, the orbit map ξx:R≥0→ B defined via ξx(t):=T(t)xis

continuous. K

Definition 2.4.2 (Generator of a strongly continuous semigroup).Let Bbe

a Banach space and T:R≥0→ L(B)a strongly continuous semigroup. Then,

based on the domain

D(A):=x∈ B | lim

h&0

h(T(h)x−x)exists,

we define the generator A:D(A)→ B of Tvia

Ax := lim

h&0

h(T(h)x−x).K

From Definition 2.4.2 it follows that the domain of a generator is a sub-

space and the generator itself is linear, cf. [92, Lemma II.1.3]. Furthermore,

a generator is bounded if and only if its domain coincides with the complete

underlying Banach space B, cf. [92, Cor. II.1.5].

In the following, we consider the initial value problem (2.4) for the special

case where the right-hand side Fsatisfies

F(t, x):=Ax +f(t)for all (t, x)∈R≥0×W,(2.5)

where Ais the generator of a strongly continuous semigroup and fis contin-

uously differentiable. A sufficient condition for the existence and uniqueness

of classical solutions is provided in Theorem 2.4.3, see [99, sec. I.5] for a cor-

responding proof.

Theorem 2.4.3 (Unique solution of (2.4) for special F).Let Xbe a Banach

space and Abe the generator of a strongly continuous semigroup T:R≥0→

L(X)with domain D(A) =:W. Furthermore, let x0be an element of Wand

f:R≥0→Xbe continuously differentiable on I= [0, tend]with tend ∈R>0.

Besides, let F:R≥0×W→Xbe given by (2.5). Then, there exists a unique

2. Preliminaries

solution to the initial value problem (2.4), which is given by

x(t) = T(t)x0+

T(t−s)f(s) dsfor all t∈I.

Theorem 2.4.4 (Growth bound for strongly continuous semigroups).Let B

be a Banach space and T:R≥0→ L(B)be a strongly continuous semigroup.

Then, there exist ω∈Rand M∈R≥1satisfying

kT(t)kL(B)≤Meωt for all t∈R≥0.

Finally, we note that the growth bound presented in Theorem 2.4.4, cf. [92,

Prop. I.5.5], allows to bound the unique solution mentioned in Theorem 2.4.3

via

kx(t)kX≤ kT(t)x0kX+

T(t−s)f(s) dsX

≤ kT(t)x0kX+

Z0kT(t−s)f(s)kXds

≤ kT(t)kL(X)kx0kX+

Z0kT(t−s)kL(X)kf(s)kXds

≤Meωt 

kx0kX+

e−ωs kf(s)kXds



(2.6)

for all t∈I, where the second inequality follows from [12, Thm. X.2.11(i)].

2.4.2. Finite-Dimensional Systems of Differential Equations

In this subsection, we consider evolution equations on a finite-dimensional

space X=W=Rngiven by

E(t, x(t)) ˙x(t) = F(t, x(t)) for all t∈I,(2.7a)

x(t0) = x0(2.7b)

with time interval I= [t0,∞)or I= [t0, tend]with t0∈R≥0and tend ∈R>t0,

a mass matrix E:R≥0×Rn→Rn,n, a right-hand side F:R≥0×Rn→Rn,

and an initial value x0∈Rn. In the following we only consider the case where

Eis at least invertible at (t0, x0). For the more general case of a singular

mass matrix, we refer to the differential-algebraic equations literature, see for

instance [175] and the references therein.

We call x∈C(I,Rn)asolution of (2.7a) if xis differentiable in Iand satisfies

(2.7a). If additionally (2.7b) holds, then we call xasolution of the initial value

2.4. Differential Equations

problem (2.7). Theorem 2.4.5 provides sufficient conditions for the initial value

problem (2.7) to have locally a unique solution and is based on a corresponding

result for standard ODE systems with E=In, cf. [136, Thm. 1.1]. Based on

additional assumptions on Eand F, one may also provide statements about

the maximal interval of existence as detailed in Theorem 2.4.6.

Theorem 2.4.5 (Local existence and uniqueness of solutions of (2.7)).Con-

sider the initial value problem (2.7) with I= [t0, tend],t0= 0,tend ∈R>0, and

x0∈Rn, and let E:R≥0×Rn→Rn,n and F:R≥0×Rn→Rnbe continu-

ous. Furthermore, let E(t0, x0)be invertible and let there exist a neighborhood

of (t0, x0)in R≥0×Rnwhere ∂xEand ∂xFexist and are continuous. Then,

the initial value problem (2.7) has exactly one solution, provided that tend is

sufficiently small.

Proof. Since the set of invertible matrices is open, cf. [96, Prop. 1.2.1], and

due to the assumptions on Eand F, we infer that there exist 1, 2∈R>0such

that ∂xEand ∂xFexist and are continuous on the compact set

R:={(t, x)∈[0, 1]×Rn| kx−x0k ≤ 2}

and such that E(t, x)is invertible for all (t, x)∈R. Moreover, since Eand

∂xEare continuous on R, this is also true for the pointwise inverse E−1and

its partial derivative with respect to x, cf. [57, Prop. 10.5.1] and Theorem 2

in [178, p. 124]. Consequently, also the function ˜

F:R→Rndefined via

F(t, x):=E−1(t, x)F(t, x)is continuous as well as its partial derivative ∂x˜

Especially, we infer that ˜

Fis uniformly Lipschitz continuous with respect to

x, cf. [11, Rem. VII.3.11(b)]. Thus, by means of classical ODE theory, see for

instance [136, Thm. 1.1], we obtain that the initial value problem

˙x(t) = ˜

F(t, x)for all t∈˜

I,(2.8a)

x(0) = x0(2.8b)

has a unique solution on the time interval ˜

I= [0, 3]with

3:= min 1,2

M,

where M∈R>0is an upper found for k˜

Fkon R. In particular, this solution

also solves the original initial value problem (2.7) on the time interval [0, 3],

which proves the existence. To also show the uniqueness, let ˜xbe a solution of

the initial value problem (2.7) on the time interval [0, 3]. If k˜x(t)−x0k< 2

holds for all t∈[0, 3], then E(t, ˜x(t)) is invertible for all t∈[0, 3]and ˜xhas

to coincide with the unique solution of (2.8). Otherwise, the set A:={t∈

[0, 3]| k˜x(t)−x0k ≥ 2}is non-empty. Especially, we observe that Ais the

preimage of the closed set B:={z∈Rn| kz−x0k ≥ 2}under ˜x. Thus, since

˜xis differentiable and hence continuous, the set Ais closed as well. Since A

2. Preliminaries

is also bounded, it has a minimum value t1∈(0, 3]. Moreover, we must have

k˜x(t1)−x0k=2, since otherwise the openness of {t∈[0, 3]| k˜x(t)−x0k>

2}would imply the existence of t2∈(0, t1)with k˜x(t2)−x0k> 2, which

would contradict the fact that t1is the minimum value in A. In particular, we

infer that E(t, ˜x(t)) is invertible for all t∈[0, t1]and hence ˙

˜xcoincides with

E−1(·,˜x(·))F(·,˜x(·)) on [0, t1]. As a consequence, ˙

˜xis continuous on [0, t1]and

we obtain the bound

2=k˜x(t1)−x0k=

˜x(t) dt=

E−1(t, ˜x(t))F(t, ˜x(t)) dt≤Mt1,

which yields t1≥2

M≥3≥t1and thus t1=3. Hence, we conclude that

k˜x(t)−x0k ≤ 2must hold for all t∈[0, 3]and, consequently, ˜xcoincides

with the unique solution of (2.8), which proves the uniqueness.

Theorem 2.4.6 (Maximal solution).Consider the ODE system (2.7a) with

F∈C1(R≥0×Rn,Rn)and pointwise invertible E∈C1(R≥0×Rn,Rn,n). Then,

for each (t0, x0)∈R≥0×Rnexactly one of the following two statements is true.

(i) There exists a unique δmax ∈R>t0such that for any tend ∈(t0, δmax)the

initial value problem (2.7) with I= [t0, tend]and x(t0) = x0has a unique

solution, but not for any larger end time tend ≥δmax. Furthermore, we

have lim

t%δmax kx(t)k=∞.

(ii) For any tend ∈R>t0, the initial value problem (2.7) with I= [t0, tend]and

x(t0) = x0has a unique solution.

Proof. For given (t0, x0)∈R≥0×Rn, we observe that for any tend ∈R>t0

the function x:I→Rnwith I= [t0, tend]is a solution of (2.7) if and only if

˜x: [0, tend −t0]→Rnwith ˜x(t):=x(t+t0)solves the initial value problem

˜x(t)=(E(t+t0,˜x(t)))−1F(t+t0,˜x(t)) for all t∈[0, tend −t0

|{z }

=:˜

tend

˜x(0) = x0.

(2.9)

Furthermore, using similar arguments as in the proof of Theorem 2.4.5, we

infer that ˜

F:R≥0×Rn→Rndefined via ˜

F(t, ˜x):= (E(t+t0,˜x))−1F(t+t0,˜x)

is continuously differentiable. Then, Corollary 23 in [280, p. 38] implies that

either (2.9) has a unique solution for any ˜

tend ∈R>0or that there exists

a unique ˜

δmax ∈R>0such that (2.9) has a unique solution for any ˜

tend ∈

(0,˜

δmax), but not for any ˜

tend ≥˜

δmax. Furthermore, in the latter case we

have lim

t%˜

δmax k˜x(t)k=∞. The claim then follows from transferring these state-

ments from the shifted initial value problem (2.9) to the original one (2.7) via

introducing δmax :=˜

δmax +t0.

2.4. Differential Equations

We emphasize that the right endpoint δmax of the maximal existence interval

in Theorem 2.4.6 depends in general on the initial value x0and the initial time

point t0. Moreover, we note that also Theorem 2.4.6 is based on standard ODE

theory, cf. [280, sec. 2.4].

Stability

In the following, we introduce some common stability notions for the system

(2.7a) and in particular introduce a sufficient stability criterion in terms of the

existence of a Lyapunov function. First, we introduce the term equilibrium

point in Definition 2.4.7 and associated stability concepts in Definition 2.4.8,

cf. [280, sec. 5.1].

Definition 2.4.7 (Equilibrium point).For given F:R≥0×Rn→Rnand

pointwise invertible E:R≥0×Rn→Rn,n, we call x∗∈Rnan equilibrium point

of (2.7a) if F(t, x∗)=0holds for all t∈R≥0.K

Definition 2.4.8 (Stability).Consider the system (2.7a) with F∈C(R≥0×

Rn,Rn), pointwise invertible E∈C(R≥0×Rn,Rn,n), and equilibrium point

0∈Rn. Furthermore, assume that for each initial condition x(t0) = x0with

t0∈R≥0and x0∈Rn, the initial value problem (2.7) has a unique solution on

[t0,∞). We denote the evaluation of this solution at t∈R≥t0by s(t, t0, x0).

(i) We call the equilibrium point 0stable, if for each ∈R>0and each

t0∈R≥0there exists a δ∈R>0such that

ks(t, t0, x0)k<  (2.10)

holds for all (t, x0)∈R≥t0×Rnwith kx0k< δ.

(ii) We call the equilibrium point 0uniformly stable, if for each ∈R>0there

exists a δ∈R>0such that (2.10) holds for all t0∈R≥0,t∈R≥t0, and

x0∈Rnwith kx0k< δ.

(iii) We call the equilibrium point 0globally exponentially stable, if there exist

constants a, b ∈R>0such that

ks(t, t0, x0)k ≤ akx0ke−b(t−t0)

holds for all t0∈R≥0,t∈R≥t0, and x0∈Rn.K

The following definition of a Lyapunov function is inspired by standard Lya-

punov theory for ODE systems with E=In, see for instance [162, Thm. 4.10],

and by the port-Hamiltonian formulation introduced in [196]. Moreover, The-

orem 2.4.10 gives a relation between the existence of a Lyapunov function as

defined in Definition 2.4.9 and stability of the equilibrium point 0.

2. Preliminaries

Definition 2.4.9 (Globally quadratic Lyapunov function).We consider the

system (2.7a) with Eand Fas in Definition 2.4.7 and call V:R≥0×Rn→R

aglobally quadratic Lyapunov function of (2.7a) if the following conditions are

satisfied.

(i) The function Vis continuously differentiable. Moreover, there exist a

function z:R≥0×Rn→Rnand a constant c1∈R≥0such that for all

(t, x)∈R≥0×Rnwe have

∇xV(t, x) = E(t, x)>z(t, x),(2.11)

and ∂tV(t, x) + z(t, x)>F(t, x)≤ −c1kxk2.(2.12)

(ii) There exist constants c2, c3∈R>0with

c2kxk2≤V(t, x)≤c3kxk2for all (t, x)∈R≥0×Rn.

In addition, we call a globally quadratic Lyapunov function which satisfies (i)

for some c1∈R>0astrong globally quadratic Lyapunov function.K

Theorem 2.4.10 (Lyapunov’s theorem for (2.7a)).We consider the system

(2.7a) with Eand Fas in Definition 2.4.8. Then, the following assertions

hold.

(i) If there exists a globally quadratic Lyapunov function of (2.7a), then the

equilibrium point 0is uniformly stable.

(ii) If there exists a strong globally quadratic Lyapunov function of (2.7a),

then the equilibrium point 0is globally exponentially stable.

Proof. (i) Let t0∈R≥0and x0∈Rnbe arbitrary and let s(t, t0, x0)denote

the evaluation of the solution of the corresponding initial value problem

(2.7) at t∈R≥t0. In particular, since Fis continuous and Eis continuous

and pointwise invertible, we infer from (2.7a) that the solution s(t, t0, x0)

is continuously differentiable with respect to t. Using this solution as well

as a globally quadratic Lyapunov function Vof (2.7a), we introduce the

mapping ˜

V:R≥t0→Rdefined via ˜

V(t):=V(t, s(t, t0, x0)). Condition

(i) in Definition 2.4.9 implies that ˜

Vis continuously differentiable and

that its derivative satisfies

V(t) = ∂tV(t, s(t, t0, x0)) + ∇xV(t, s(t, t0, x0))>∂ts(t, t0, x0)

=∂tV(t, s(t, t0, x0)) + z(t, s(t, t0, x0))>E(t, s(t, t0, x0))∂ts(t, t0, x0)

=∂tV(t, s(t, t0, x0)) + z(t, s(t, t0, x0))>F(t, s(t, t0, x0))

≤0

for all t∈R≥t0. Thus, we infer

V(t, s(t, t0, x0)) = ˜

V(t)≤˜

V(t0) = V(t0, s(t0, t0, x0)) = V(t0, x0)

2.5. Parametric Model Order Reduction

for all t∈R≥t0. Using condition (ii) from Definition 2.4.9, we further

obtain that there exist constants c2, c3∈R>0such that for all t∈R≥t0

we have

ks(t, t0, x0)k2≤1

V(t, s(t, t0, x0)) ≤1

V(t0, x0)≤c3

c2kx0k2.

This implies that (2.10) holds for instance when choosing δ=qc2

c3.

(ii) The proof follows similar arguments as the proof of (i). Since (2.12)

is satisfied with c1∈R>0, we may replace ˙

V(t)≤0by the stronger

statement

V(t)≤ −c1ks(t, t0, x0)k2≤ −c1

V(t, s(t, t0, x0)) = −c1

V(t)(2.13)

for all t∈R≥t0. Moreover, applying standard theory for differential

inequalities to (2.13), see for instance [9, Lem. (16.4)], we obtain

V(t)≤˜

V(t0) exp −c1

(t−t0)

for all t∈R≥t0. This in turn yields

ks(t, t0, x0)k2≤1

V(t, s(t, t0, x0)) = 1

V(t)

≤1

V(t0) exp −c1

(t−t0)

V(0, x0) exp −c1

(t−t0)

≤c3

c2kx0k2exp −c1

(t−t0)

and, consequently,

ks(t, t0, x0)k ≤ sc3

c2kx0kexp −c1

2c3

(t−t0)

for all t∈R≥t0.

2.5. Parametric Model Order Reduction

In this section, we discuss model order reduction techniques for parametric

full-order models of the form

˙x(t;µ) = F(t, x(t;µ); µ)for all (t, µ)∈I×M, x(0; µ) = x0(µ),(2.14)

2. Preliminaries

with I= [0, tend],M⊆Rnp,F:R≥0×W×M→X,x0:M→W,x:I×

M→W, real Hilbert space X, and subspace W⊆X, cf. Remark 1.1.1.

The task of parametric model order reduction is to determine a reduced-order

model of state space dimension rdim(X), which is usually of the form

˜x(t;µ) = ˜

F(t, ˜x(t;µ); µ)for all (t, µ)∈I×M,˜x(0; µ) = ˜x0(µ),

with ROM state ˜x:I×M→Rr, right-hand side ˜

F:R≥0×Rr×M→Rr, and

initial value ˜x0:M→Rr. An approximation of the full-order state is then

given by a mapping Vr:R≥0×Rr×M→W, i.e., x(t;µ)≈Vr(t, ˜x(t;µ); µ)for

all (t, µ)∈I×M. Often, Vrdoes not explicitly depend on tor µand is linear

in the reduced state ˜x. This important special case is in particular subject to

the discussion in section 2.5.3.

Remark 2.5.1 (Goal-oriented model reduction).We emphasize that in many

applications one aims for an accurate approximation of some quantities of

interest or outputs of the form

y(t;µ) = C(t, x(t;µ); µ)for all (t, µ)∈I×M,

with output y:I×M→Rp, output mapping C:R≥0×W×M→Rp, and

output dimension p∈N. Depending on the application, one is interested in

approximating both the FOM state and the output or only in approximating

the output. For instance, in model reduction techniques for control systems

it is commonly exploited that not the complete FOM state needs to be ap-

proximated well, but typically only a few derived quantities. In this context,

one usually targets a good approximation of the input-output map, cf. [14]

and the references therein. Also when considering systems without control

input, so-called goal-oriented model reduction methods make use of the quan-

tities of interest and construct ansatz functions which are tailored for a good

approximation of these quantities, see for instance [41, 52, 265]. ¨

The whole process of parametric model order reduction is usually separated

into two phases: the offline and the online phase, see also the beginning of

chapter 1. The offline phase summarizes all steps required for constructing

a reduced-order model. Especially in the case of nonlinear full-order models,

this often involves simulations of the FOM and a subsequent determination of

modes or ansatz functions based on the simulated FOM data, cf. section 2.5.1.

The parameter samples which are used for simulating the FOM in the offline

phase may be selected via a greedy algorithm as presented in section 2.5.2.

Also the actual construction of the ROM belongs to the offline phase and may

for instance be achieved by means of a Galerkin projection, see section 2.5.3.

On the other hand, the online phase describes the actual use of the reduced-

order model, i.e., its evaluation for many different parameter configurations.

These multi-query evaluations of the ROM are, among others, often needed in

control or optimization tasks.

Since the offline phase is ideally only performed once, the requirements on its

2.5. Parametric Model Order Reduction

computational effort are usually less severe than for the evaluation of the ROM

in the online phase. In this context, the term efficient offline/online decompo-

sition is used if the computation of the online phase does not scale with the

dimension of the full-order model. Then, the evaluation of the reduced-order

model is typically much cheaper than the evaluation of the FOM. However, es-

pecially when the FOM is nonlinear, another approximation of the nonlinearity

is often required for achieving such an efficient offline/online decomposition,

cf. section 2.5.4.

2.5.1. Proper Orthogonal Decomposition

A common technique for determining modes as in the approximation ansatz

(1.2) is the POD method, see for instance [127, 148]. To this end, we assume

in this subsection that Xis a real separable Hilbert space, that W⊆Xis

a dense subspace, and that we have access to a solution trajectory xof the

FOM (1.1), for instance, by means of a numerical simulation. Furthermore,

we present the POD method for a solution trajectory xwhich depends only

on time but not on a parameter vector. Instead, in section 2.5.2 we present a

POD-greedy algorithm to handle the parameter-dependent case, see also [156]

for an alternative POD-based approach for parametric problems.

The POD method is based on a minimization of the approximation error by

considering the optimization problem











min 1

tend

x(t)−

j=1

αj(t)φj

s.t. φj∈Wand hφi, φjiX=δij, i, j = 1, . . . , r

(2.15)

with given rdim(X). Since the best approximation within a subspace

is given by the orthogonal projection onto this subspace, we can replace the

coefficients αjby hx(t), φjiXfor j= 1, . . . , r. This results in the optimization

problem 









min 1

tend

x(t)−

j=1 hx(t), φjiXφj

s.t. φj∈Wand hφi, φjiX=δij, i, j = 1, . . . , r.

(2.16)

According to [127], this optimization problem may be solved by solving an

eigenvalue problem associated with the operator R:X→Xdefined via

Rφ:=

tend

Z0hx(t), φiXx(t) dt. (2.17)

We observe that if the solution xis in L2((0, tend),W), then Rφis in Wfor

all φ∈X, see also the proof of [127, Lemma 1.24]. In that case, we obtain a

2. Preliminaries

solution of (2.16) as follows, cf. [127, Theorem 1.15].

Theorem 2.5.2 (Solution of the POD minimization problem).Let Xbe a

real separable Hilbert space, xbe in L2((0, tend),X)with tend ∈R>0, and R

be as defined in (2.17). Then, there exist eigenvalues λi∈Rand associated

orthonormal eigenfunctions φi∈Xfor i∈ I with

Rφi=λiφifor all i∈ I :={1,...,dim(X)},

λi≥λi+1 ≥0for all i∈ I \{dim(X)},(2.18)

if Xis finite-dimensional, or

Rφi=λiφifor all i∈ I :=N,

λi≥λi+1 ≥0for all i∈ I,(2.19)

otherwise. If xis additionally in L2((0, tend),W)with Wbeing a dense sub-

space of X, then for any r∈ I with λr>0, the rleading eigenfunctions

φ1, . . . , φrform a solution of (2.16).

Remark 2.5.3 (Uniqueness of the POD modes).Since eigenfunctions are in

general not unique, but only the corresponding eigenspaces, we infer from

Theorem 2.5.2 that the POD modes and the solution of (2.16) are in general

not unique. However, as long as the eigenvalues are distinct, the only source

of non-uniqueness is given by the fact that we can replace φiby −φifor any

i∈ {1, . . . , r}, see also for instance [131, sec. 2.4.5]. ¨

For the actual computation of the POD modes, one usually considers a

finite-dimensional full-order model (2.14) with X=W=Rn, which is, for

example, obtained via semi-discretization of a PDE in space. This finite-

dimensional FOM is then simulated and the collected data of the state are

stored in a matrix X∈Rn,q, i.e.,

X:=hx(t1)x(t2)··· x(tq)i.(2.20)

The columns are referred to as snapshots of the solution and the matrix Xis

called a snapshot matrix. The discrete counterpart of the minimization prob-

lem (2.16) may then be solved by computing a singular value decomposition

(SVD) of X, cf. [127, Rem. 1.10]. The leading rleft singular vectors constitute

the matrix Vrwhich can be used for obtaining a reduced-order model based on

a Galerkin projection, see section 2.5.3. Reasonable choices for the dimension

ressentially depend on the singular value decay. In particular, if the singular

values decay very fast, then a few modes are usually sufficient to obtain a

suitable approximation of the snapshot matrix. On the other hand, if the sin-

gular values decay rather slowly, then the POD method requires many modes

to achieve a reasonable approximation and, consequently, the evaluation of a

corresponding reduced-order model may be rather slow.

2.5. Parametric Model Order Reduction

2.5.2. POD-Greedy Algorithm

A common approach for sampling the parameter domain during the offline

phase is to use a POD-greedy algorithm as discussed in the following, see also

[131, sec. 2.4.5]. To this end, we consider a parametric FOM of the form (2.14)

with X=W=Rn. For the POD-greedy algorithm, cf. Algorithm 2.1, we

also consider a grid based on discrete time points t1, . . . , tq∈I, which allows

to compute a POD via an SVD as outlined at the end of section 2.5.1. Fur-

thermore, we require a function ε:Mtrain ×Pf(Rn)→R≥0which can be used

as an error estimator and which usually involves constructing and evaluat-

ing a ROM based on the modes specified by the second input argument of ε,

see section 2.5.3 for more details on the ROM construction. Neglecting the

computational effort, an ideal choice would be the true squared online error

εtrue :Mtrain ×Pf(Rn)→R≥0, which is defined via

εtrue (µ, {φ1, . . . , φr}):=

`=1 x(t`;µ)−

i=1

˜xi(t`;µ)φi

where xand ˜xdenote the solutions of the FOM and of the ROM, respectively.

Here, we assume that the FOM and the ROM are uniquely solvable. Another

important error measure is the true squared projection error εopt :Mtrain ×

Pf(Rn)→R≥0, which is defined as

εopt (µ, {φ1, . . . , φr}):=

`=1 x(t`;µ)−Pspan{φ1,...,φr}x(t`;µ)2

and provides a lower bound for εtrue. Here, Pspan{φ1,...,φr}denotes the orthogonal

projection onto the span of φ1,. . .,φr. We note that both error measures εtrue

and εopt involve in general solving the FOM and, thus, their evaluation is

typically expensive. An alternative approach is based on residual-based error

bounds, which may be often computed in an efficient way, see [131] and the

references therein for more details.

During the iterations of the POD-greedy algorithm, the error estimator is

evaluated on a pre-defined parameter training set Mtrain ⊂Mand the algo-

rithm terminates once the estimated maximum error does not exceed the error

tolerance tol or the maximum number of iterations imax is reached. Besides,

the POD-greedy algorithm may be initialized with an empty basis or with a

given orthonormal basis Φ0.

Algorithm 2.1 is an iterative procedure and at the beginning of each itera-

tion the error estimator is evaluated on the training set Mtrain. When using

a residual-based error estimator, this step usually involves the construction

of a parametric ROM and multiple ROM evaluations. Here, an efficient of-

fline/online decomposition for the ROM and for the error estimator itself is

crucial for a reasonable computation time, especially if the training set Mtrain

is large. Afterwards, the worst-case value of the error estimator is compared to

2. Preliminaries

the tolerance and the algorithm terminates if the tolerance is met. Otherwise,

the FOM is solved for the worst-case parameter value to generate snapshots of

the FOM state. Here, we note that there may be some arbitrariness in line 5

of Algorithm 2.1, since the worst-case parameter value µmax is not necessarily

uniquely determined. After the FOM snapshots have been computed, they are

projected onto the orthogonal complement of the span of the current basis.

This ensures that the basis vector added in the last step of Algorithm 2.1 is

orthogonal to the other basis vectors. Consequently, the output basis is guar-

anteed to be orthonormal as well. We remark that the mode to be added in the

last step is in general not uniquely determined, especially if the multiplicity

of the largest singular value of the projected snapshot matrix is greater than

one, cf. Remark 2.5.3.

Algorithm 2.1 POD-greedy algorithm

Input:

•FOM as in (2.14) with X=W=Rn,M⊆Rnp,n, np∈N

•discrete time points t1, . . . , tq∈Iwith 0 = t1< t2< . . . < tq=tend

and q∈N≥2

•parameter training set Mtrain ⊂Mwith |Mtrain|<∞

•initial set of ansatz vectors Φ0={φ1, . . . , φr0} ⊂ Rnwith r0∈

{0, . . . , n}and hφi, φjiX=δij for i, j = 1, . . . , r0

•error estimator ε:Mtrain ×Pf(Rn)→R≥0

•error tolerance tol ∈R>0

•maximum number of iterations imax ∈N

Output:

•set of ansatz vectors Φk={φ1, . . . , φrk} ⊂ Rnwith k∈N∩{0}and

rk∈N≥r0, satisfying max

µ∈Mtrain

ε(µ, Φk)≤tol or k=imax

1: for i←1to imax do

2: Evaluate the error estimator ε(·,Φi−1)on Mtrain

3: if maxµ∈Mtrain ε(µ, Φi−1)≤tol then return Φi−1

4: end if

5: Solve the FOM for a parameter value µmax ∈arg max

µ∈Mtrain

ε(µ, Φi−1)

6: Project the FOM snapshots x(t1;µmax), . . . , x(tq;µmax)onto the orthog-

onal complement of span{Φi−1}

7: Compute an SVD of the projected snapshot matrix and add the leading

left singular vector to Φi−1and thereby obtain Φi

8: end for

Even without specifying a maximum number of iterations, the POD-greedy

algorithm terminates in theory after a finite number of steps, provided that

2.5. Parametric Model Order Reduction

the error estimator is zero whenever the true projection error is zero. This

follows from the fact that, as long as the maximum error is not small enough,

the POD-greedy algorithm continues extending the orthonormal basis until

the error criterion is met. Thus, since the FOM state is assumed to take val-

ues in a finite-dimensional space, the reduced basis is guaranteed to span this

complete state space after a finite number of steps. However, this theoreti-

cal property is in practice often of minor importance, since we usually desire

ROMs whose state space dimension is orders of magnitude smaller than the

one of the FOM. Thus, we are rather interested in the convergence speed of

the POD-greedy algorithm. In fact, under certain assumptions on the param-

eter training set Mtrain and on the error estimator ε, one can show that an

algebraic or exponential decay of the Kolmogorov n-widths, cf. Example 1.2.1,

leads to an algebraic or exponential convergence of the POD-greedy algorithm,

respectively, see for instance [131, Prop. 2.94].

2.5.3. Galerkin Projection

In sections 2.5.1 and 2.5.2, we have discussed approaches for determining suit-

able modes φibased on snapshot data of the FOM. The focus of this section

is to derive a reduced-order model which describes the time and parameter

dependency of the amplitudes αifor given modes φi. For this purpose, we as-

sume that the modes are linearly independent, as is for instance the case when

using one of the methods from sections 2.5.1 and 2.5.2. A common approach

for obtaining a reduced-order model is to perform a Galerkin projection. To

this end, we substitute the linear approximation ansatz

x(t;µ)≈ˆx(t;µ):=

i=1

αi(t;µ)φi

into the full-order model (2.14) to obtain the residual

i=1

˙αi(t;µ)φi−F t,

i=1

αi(t;µ)φi;µ!(2.21)

at (t, µ)∈I×M. An evolution equation for the coefficients α1, . . . , αris

then obtained by enforcing the residual to be orthogonal to the span of the

modes φ1, . . . , φr. Moreover, we derive initial values for the coefficients via an

orthogonal projection of the FOM initial value x0onto the span of φ1, . . . , φr.

The resulting ROM reads

M˙

˜x(t;µ) = ˜

F(t, ˜x(t;µ); µ),˜x(0; µ) = ˜x0(µ)for all (t, µ)∈I×M(2.22)

2. Preliminaries

with mass matrix M∈Rr,r, ROM state ˜x:I×M→Rr, right-hand side

F:R≥0×Rr×M→Rr, and initial value ˜x0:M→Rrdefined as

M:=hhφi, φjiXiij ,(2.23a)

˜x(t;µ):= [αi(t;µ)]i,(2.23b)

F(t, α;µ):=

*φi, F 

t,

j=1

αjφj;µ

+X



i

,(2.23c)

˜x0(µ):=M−1[hφi, x0(µ)iX]i.(2.23d)

An important property is that for given (t, µ)∈I×Mand for given values

α1(t;µ),. . .,αr(t;µ)∈R, the corresponding values of the time derivatives

˙α1(t;µ),. . .,˙αr(t;µ)∈Rdetermined by the Galerkin ROM (2.22) are optimal

in the sense that they minimize the norm of the residual (2.21). Since the

continuous-time residual is minimized, this property is called continuous opti-

mality in [61] to distinguish it from an alternative approach which minimizes

the residual after time discretization.

The first equation in (2.22) is a system of rdifferential equations and r

unknowns and, thus, the number of ROM equations and unknowns is reduced

in comparison to the FOM. However, the evaluation of the ROM right-hand

side ˜

Fstill involves the evaluation of the high-dimensional FOM right-hand

side Fand hence the online phase may still be expensive. Nevertheless, in

some situations we may transfer those computations which scale with the FOM

dimension from the online to the offline phase. As a consequence, the ROM

can then be evaluated with a computational cost which only scales with the

reduced dimension r. This is for instance possible if Fis linear with respect

to its second argument and if its time and parameter dependency allows for a

separation of the form

F(t, x;µ) =

k=1

θk(t;µ)Fkx(2.24)

with Kn, coefficient functions θ1, . . . , θK:R≥0×M→R, and linear

mappings F1, . . . , FK:W→X. In this case, we have

F(t, α;µ) =

k=1

θk(t;µ)





hφ1, Fkφ1iX. . . hφ1, FkφriX

hφr, Fkφ1iX. . . hφr, FkφriX







|{z }

=:˜







α1

αr







and, thus, ˜

Fmay be efficiently evaluated in the online phase, once the r×r

matrices ˜

F1,..., ˜

FKhave been precomputed in the offline phase. In a similar

way, an efficient offline/online decomposition can be achieved if Fis a poly-

nomial function with respect to its second argument, see for instance [172]

2.5. Parametric Model Order Reduction

and the references therein. Furthermore, the treatment of piecewise polyno-

mial nonlinearities is discussed in [82] and a method for treating nonlinearities

which are linear combinations of elementary functions such as sin(·),cos(·), or

exp(·)is presented in [126].

In the case that Finvolves a more general nonlinearity, it may not be possi-

ble to sufficiently reduce the computational complexity of the ROM evaluations

just by offline precomputations. To this end, several hyperreduction methods

have been proposed in the literature and they usually involve an approximation

of ˜

Fin order to significantly decrease the computational cost for evaluating

the ROM. For instance, a commonly applied hyperreduction method is the

(discrete) empirical interpolation method, which is briefly summarized in sec-

tion 2.5.4.

We note that the Galerkin method presented at the beginning of this sub-

section is a special case of a Petrov–Galerkin scheme. In general, the Petrov–

Galerkin method is based on enforcing the residual to be orthogonal to the

span of some linearly independent test functions ζ1, . . . , ζr∈Xwhich satisfy

the compatibility condition

span{ζ1, . . . , ζr}⊥∩span{φ1, . . . , φr}={0}.(2.25)

This condition is in particular satisfied if the ansatz functions φiand test

functions ζicoincide for i= 1, . . . , r, respectively, and this special case cor-

responds to the Galerkin method. In general, a Petrov–Galerkin projection

yields a ROM of the form (2.22) where the mass matrix, right-hand side, and

initial value are given by

M:=hhζi, φjiXiij ,˜

F(t, α;µ):=

*ζi, F 

t,

j=1

αjφj;µ

+X



i

˜x0(µ):=M−1[hζi, x0(µ)iX]i.

In particular, we note that the mass matrix Mis invertible due to the compat-

ibility condition (2.25). In the special case of a finite-dimensional FOM with

X=W=Rn, the ROM is obtained via matrix multiplications using the ma-

trices Vr:= [φ1··· φr]∈Rn,r and Wr:= [ζ1··· ζr]∈Rn,r. More precisely,

the ROM mass matrix is then given by M=W>

rVr, the right-hand side by

F(t, α;µ) = W>

rF(t, Vrα;µ), and the initial value by ˜x0(µ) = M−1W>x0(µ).

2.5.4. Hyperreduction

As mentioned in the previous subsection, the evaluation of the ROM (2.22)

may still scale with the FOM dimension, especially if the FOM (2.14) is non-

linear. To this end, various hyperreduction techniques have been introduced

for achieving an efficient offline/online decomposition by an additional approx-

imation of the ROM, cf. [6, 17, 21, 60, 64, 97, 134, 140, 166, 217, 247]. In the

2. Preliminaries

following we only discuss the empirical interpolation method (EIM) and the

discrete empirical interpolation method (DEIM), since these methods are the

most relevant ones for the new approach presented in section 4.3.2.

The empirical interpolation method is introduced in [21] for approximating a

general space- and parameter-dependent function g: Ω ×M→Rwith spatial

domain Ω⊂R2and parameter domain M⊂Rnp. Especially, the authors

assume that g(·, µ)is an element of L∞(Ω) for all µ∈Mand present a

procedure for approximating gby a function ˜g: Ω ×M→Rwhich is affine

in the second argument, i.e., there exist K∈N,γ1, . . . , γK:M→R, and

g1, . . . , gK: Ω →Rsatisfying

˜g(ξ, µ) =

k=1

γk(µ)gk(ξ)for all (ξ, µ)∈Ω×M.

Furthermore, the authors also discuss the usefulness of such an affine approx-

imation in the context of model order reduction, see also (2.24) and the corre-

sponding discussion in section 2.5.3.

The DEIM is introduced in [64] and applies the ideas of the EIM in the con-

text of general nonlinear finite-dimensional ODE systems, which usually arise

from a semi-discretization of a PDE in space. Accordingly, in the following

we consider a ROM of the form (2.22)–(2.23) with X=Rnand mode matrix

Φ:= [φ1··· φr]∈Rn,r. In addition, we assume for simplicity that the right-

hand side Fdoes not explicitly depend on time, which can be achieved by mak-

ing the system autonomous, see for instance [136, sec. VII.1]. Furthermore, we

omit parameter dependencies while emphasizing that the following considera-

tions may be straightforwardly extended to the case of parameter-dependent

FOMs, cf. [64]. According to (2.23), the ROM right-hand side ˜

F:Rr→Rr

without time and parameter dependencies is defined as ˜

F(˜x):= Φ>F(Φ˜x),

where Fis the FOM right-hand side. In particular, as mentioned in the last

subsection, the evaluation of ˜

Finvolves the evaluation of Fand this may ren-

der the online phase expensive. To circumvent this issue, the DEIM is based

on approximating Fby a suitable linear combination of appropriately chosen

ansatz vectors, i.e., we aim to find an approximation of the form

F(Φ˜x(t)) ≈Ψβ(t)(2.26)

with DEIM mode matrix Ψ=[ψ1··· ψs]∈Rn,s, DEIM coefficient vector

β:I→Rs, and sn. Especially, in [64] the authors propose to determine

the DEIM modes ψ1, . . . , ψsin the offline phase by computing a POD of the

nonlinearity snapshots F(x(t1)), . . . , F(x(tq)). Then, in order to determine

the DEIM coefficients in the online phase, they follow the EIM and suggest

to enforce equality in ssuitably selected rows of (2.26). Thus, the coefficients

can be determined by solving the linear equation system

S>Ψβ(t) = S>F(Φ˜x(t)),

2.6. Port-Hamiltonian Systems

where S= [eπ(1), . . . , eπ(s)]∈Rn,s is a truncated permutation matrix. Here, ei

denotes the ith unit vector of the standard basis of Rnfor i= 1, . . . , n and

π:{1, . . . , s}→{1, . . . , n}is an injective mapping which is determined in the

offline phase. Corresponding algorithms for determining πbased on Ψare for

instance proposed in [64, 83]. Especially, as long as Ψhas full column rank,

the mentioned algorithms ensure the invertibility of S>Ψ. Consequently, the

DEIM results in replacing ˜

F(˜x)=Φ>F(Φ˜x)in the ROM by

Φ>Ψ(S>Ψ)−1S>F(Φ˜x).(2.27)

Here, the matrix Φ>Ψ(S>Ψ)−1can be precomputed in the offline phase and,

moreover, the evaluation of (2.27) only involves the evaluation of srows of the

FOM right-hand side F. As long as the Jacobian of Fis sparse, this allows to

implement the ROM evaluation in an efficient way, cf. [64].

2.6. Port-Hamiltonian Systems

In the following we provide a brief summary of port-Hamiltonian systems with

particular emphasis on the topics which are relevant for this thesis. Especially,

we present different linear and nonlinear pH representations and discuss some

related properties such as stability and passivity in section 2.6.1. Moreover,

we summarize a structure-preserving MOR approach for pH systems based on

a Petrov–Galerkin projection in section 2.6.2.

2.6.1. Formulations and Basic Properties

We start by considering linear time-invariant port-Hamiltonian systems of the

form

˙x(t)=(J−R)Qx(t) + Bu(t), x(t0) = x0,(2.28a)

y(t) = B>Qx(t)(2.28b)

for all t∈I, with time interval I= [t0, tend],t0∈R≥0,tend ∈R>t0, state

x:I→Rn,input port u:R≥0→Rm,output port y:I→Rm, initial value

x0∈Rn,structure matrix J∈Rn,n with J=−J>,dissipation matrix R∈Rn,n

with R=R>≥0,energy matrix Q∈Rn,n with Q=Q>≥0, and input matrix

B∈Rn,m. Associated to this system, we introduce a quadratic Hamiltonian

H:Rn→Rvia H(z):=1

2z>Qz. Furthermore, we emphasize that there are

alternative linear time-invariant pH representations including a leading matrix

Ein front of ˙xin (2.28a) or a feedthrough term, cf. Remark 2.6.1, Remark 2.6.4,

and chapter 5.

As a consequence of the structure in (2.28), one can show that the Hamil-

tonian may only increase if the input uand the output ydo not vanish. To

this end, let ube chosen such that (2.28a) admits a solution xin C1(I,Rn).

2. Preliminaries

Then, due to the symmetry and definiteness properties of Jand Ras well as

the symmetry of Q, this solution satisfies the so-called dissipation inequality

dt(H◦x)(t) = x(t)>Q>˙x(t) = x(t)>Q>((J−R)Qx(t) + Bu(t))

=−x(t)>Q>RQx(t) + y(t)>u(t)≤y>(t)u(t)

(2.29)

for all t∈I. The Hamiltonian often represents the stored energy of the sys-

tem and (2.29) corresponds to a power balance, where the term −x>Q>RQx

describes the internal energy dissipation and y>uthe energy exchange with

the environment or with other subsystems, see for instance [278]. Moreover,

(2.29) implies that the pH system (2.28) is passive, i.e., there exists a storage

function V:Rn→R≥0satisfying V(0) = 0 and a dissipation inequality of

the form (2.29) with V=H, see for instance [53] for a formal definition of

passivity.

In the special case where Qis even positive definite and uvanishes, the

Hamiltonian is a globally quadratic Lyapunov function in the sense of Defi-

nition 2.4.9 with H=V,z=∇H,E=In,F(t, x):= (J−R)Qx,c1= 0,

c2=σmin(Q), and c3=σmax(Q). Thus, by Theorem 2.4.10(i) we infer that the

pH structure in (2.28) with the additional assumption Q > 0implies that the

system (2.28a) with u= 0 has a uniformly stable equilibrium point at 0, see

also [193].

Remark 2.6.1 (Port-Hamiltonian systems with feedthrough term).The port-

Hamiltonian structure (2.28) may be extended to systems with feedthrough

terms via

˙x(t)=(J−R)Qx(t)+(G−P)u(t), x(t0) = x0

y(t)=(G+P)>Qx(t)+(S+N)u(t),

for all t∈I, with G, P ∈Rn,m and S, N ∈Rm,m. In this case, we require the

coefficient matrices to satisfy Q=Q>≥0,J=−J>,N=−N>, and

"R P

P>S#="R P

P>S#>≥0,

cf. [276, sec. 6.1]. In a similar way, one may also extend the upcoming nonlinear

pH structure (2.30) to include feedthrough terms, cf. [196]. ¨

Remark 2.6.2 (Hamiltonian and dissipative Hamiltonian systems).In the re-

mainder of this thesis, we refer to the special case of a port-Hamiltonian sys-

tem without external port variables, i.e., with B= 0 in (2.28), as a dissipative

Hamiltonian system. Similarly, we call a dissipative Hamiltonian system with-

out dissipation, i.e, with R= 0 in (2.28), a Hamiltonian system. Moreover,

we use these terms not only in the context of linear time-invariant systems,

but also for nonlinear systems as discussed in the remainder of this section.

We emphasize that especially our use of the term Hamiltonian system is less

2.6. Port-Hamiltonian Systems

restrictive than it is common in the differential equations literature, where the

term is rather used for the case where Jhas a special canonical structure, see

for instance [35, Def. 11.1]. ¨

In [196] the authors introduce a class of nonlinear port-Hamiltonian descrip-

tor systems. The special case without feedthrough term and with pointwise

square Ematrix function reads

E(t, x(t)) ˙x(t) + r(t, x(t)) = (J(t, x(t)) −R(t, x(t)))z(t, x(t)) + B(t, x(t))u(t),

(2.30a)

y(t) = B>(t, x(t))z(t, x(t)) (2.30b)

for all t∈I, with E, J, R ∈C(R≥0×Rn,Rn,n),r, z ∈C(R≥0×Rn,Rn), and

B∈C(R≥0×Rn,Rn,m). Here, the second term on the left-hand side of (2.30a)

reflects an explicit time dependency of the Hamiltonian. More precisely, we

consider an associated Hamiltonian H ∈ C1(R≥0×Rn)and require that E,J,

R,r,zsatisfy pointwise

J=−J>, R =R>≥0, E>z=∇xH, z>r=∂tH.(2.31)

Then, by means of a similar calculation as in (2.29), cf. [196, sec. IIB], one can

show that, for any continuously differentiable solution xof (2.30a), we have a

dissipation inequality of the form

dHs

dt(t)≤y(t)>u(t)for all t∈I,(2.32)

where Hs:R≥0→Ris defined via Hs(t):=H(t, x(t)). This dissipation in-

equality is an important property when investigating stability as well as the

existence and uniqueness of solutions of the state equation (2.30a) with u= 0

and pointwise invertible E.

Theorem 2.6.3 (Stability of (2.30a)).Consider the system (2.30a) with van-

ishing input u= 0,J, R ∈C1(R≥0×Rn,Rn,n),r, z ∈C1(R≥0×Rn,Rn), and

pointwise invertible E∈C1(R≥0×Rn,Rn,n). Furthermore, let (2.31) be sat-

isfied pointwise for some Hamiltonian H ∈ C1(R≥0×Rn), which additionally

fulfills condition (ii) in Definition 2.4.9 with H=V. Then, the following

assertions hold.

(i) For each initial value x0∈Rnand for any time interval I= [t0, tend]

with t0∈R≥0and tend ∈R>t0, the initial value problem associated with

(2.30a),u= 0, and x(t0) = x0has a unique solution on I.

(ii) If r(t, 0) = (J(t, 0) −R(t, 0))z(t, 0) is satisfied for all t∈R≥0, then

(2.30a) with u= 0 has a uniformly stable equilibrium point at the origin.

Proof. (i) Since the assumptions of Theorem 2.4.6 are satisfied, we conclude

that, for a given initial value x0∈Rnand initial time t0∈R≥0, the

2. Preliminaries

corresponding initial value problem associated with (2.30a) and u= 0 is

either uniquely solvable on any time interval I= [t0, tend]with tend ∈R>t0

or there is a maximal existence interval [t0, δmax)with δmax ∈R>t0and

lim

t%δmax kx(t)k=∞.(2.33)

Let us assume that for some (t0, x0)∈R≥0×Rnthe latter statement

is true. Then, since the Hamiltonian satisfies condition (ii) in Defini-

tion 2.4.9, (2.33) implies

lim

t%δmax H(t, x(t)) = ∞.

However, this is a contradiction to the inequality H(t, x(t)) ≤ H(t0, x0),

which holds for any t≥t0and follows from the dissipation inequality

(2.32) in the case u= 0. Thus, we infer that for any initial value x0∈Rn

and initial time t0∈R≥0, the corresponding initial value problem asso-

ciated with (2.30a) and u= 0 is uniquely solvable on any time interval

I= [t0, tend]with tend ∈R>t0.

(ii) First, we note that the equation r(·,0) = (J(·,0) −R(·,0))z(·,0) implies

that 0∈Rnis an equilibrium point of (2.30a) with u= 0. Furthermore,

due to (i), the requirements of Theorem 2.4.10 are satisfied. Besides,

using (2.31) we infer that the Hamiltonian satisfies not only condition

(ii) in Definition 2.4.9, but also condition (i) with c1= 0. Thus, the

Hamiltonian is a globally quadratic Lyapunov function of (2.30a) with

u= 0 and, hence, the claim follows by applying Theorem 2.4.10(i).

The classes of port-Hamiltonian systems considered in (2.28) and (2.30) are

formulated in a continuous-time setting and this is also true for the corre-

sponding dissipation inequalities (2.29) and (2.32). However, the simulation

of a pH system usually involves a discretization in time and there is in general

no guarantee that the resulting time-discrete system inherits a correspond-

ing dissipation inequality. In [169, 196] it is shown that for the special case

where the Hamiltonian is independent of tand quadratic with respect to x,

Gauss–Legendre collocation methods may be used to obtain a dissipation in-

equality also on the discrete-time level. For instance, the Hamiltonians of the

full-order models considered in chapter 6 satisfy this requirement and thus

we obtain a dissipation inequality on the time-discrete level by using the im-

plicit midpoint rule, which is the simplest Gauss–Legendre collocation method,

cf. [80, sec. 6.3.2]. However, the corresponding reduced-order models are port-

Hamiltonian systems with non-quadratic Hamiltonian and, therefore, we partly

use a different time integration scheme based on discrete gradient pairs for the

ROMs, cf. chapter 6 and appendix C.

2.6. Port-Hamiltonian Systems

2.6.2. Structure-Preserving Model Order Reduction

In this subsection we summarize the approach presented in [65] for structure-

preserving model reduction for nonlinear port-Hamiltonian systems of the form

˙x(t) = (J−R)∇H(x(t)) + Bu(t),(2.34a)

y(t) = B>∇H(x(t)) (2.34b)

for all t∈I. In particular, this system class is a special case of (2.30) with

E=In,H ∈ C1(Rn),z=∇H,r= 0, and constant J,R,B.

When applying a classical Galerkin projection to (2.34), cf. section 2.5.3, it is

in general not clear if the resulting reduced-order model is a port-Hamiltonian

system as well. Therefore, in [65] the authors propose a model reduction

scheme based on a suitable Petrov–Galerkin projection. For this purpose, two

subspaces of dimension rnare determined in the offline phase: The trial

space is chosen such that the full-order state xmay be well approximated within

this subspace, whereas the choice of the test space targets a good approximation

of ∇H ◦ x. Especially, both subspaces may for instance be obtained via the

POD method based on snapshots of xand ∇H ◦x, respectively. This yields

matrices Vr∈Rn,r and Wr∈Rn,r whose columns span the trial and the test

space, respectively. Moreover, it is assumed that the two identified subspaces

satisfy a compatibility condition similar to (2.25), which allows to choose Vr

and Wrsuch that W>

rVr=Irholds.

Based on the matrices Vrand Wr, a ROM is constructed as follows. First, a

Petrov–Galerkin projection as outlined at the end of section 2.5.3 is performed,

which results in the reduced system

˜x(t) = W>

r(J−R)∇H(Vr˜x(t)) + W>

rBu(t),

˜y(t) = B>∇H(Vr˜x(t)) (2.35)

for all t∈I, with reduced state ˜x:I→Rrand output ˜y:I→Rm. Especially,

Vr˜xcorresponds to an approximation of the FOM state xand, accordingly, we

define the Hamiltonian of the reduced system ˜

H:Rr→Rvia ˜

H(˜x):=H(Vr˜x).

Furthermore, since Vr˜xapproximates xand, by assumption, ∇H ◦ xmay be

well approximated by elements of im(Wr), it is reasonable to assume that also

∇H(Vr˜x(·)) may be well approximated by elements of im(Wr), cf. [65, sec. 2].

Consequently, we may assume that for each t∈Ithere exists β(t)∈Rrwith

∇H(Vr˜x(t)) ≈Wrβ(t).

Especially, using V>

rWr= (W>

rVr)>=Irwe obtain

∇H(Vr˜x(t)) ≈WrV>

rWrβ(t)≈WrV>

r∇H(Vr˜x(t)) = Wr∇˜

H(˜x(t)).

2. Preliminaries

Then, formally replacing ∇H(Vr˜x)in (2.35) by Wr∇˜

H(˜x)yields the ROM

˜x(t)=(˜

J−˜

R)∇˜

H(˜x(t)) + ˜

Bu(t),

˜y(t) = ˜

B>∇˜

H(˜x(t)) (2.36)

for all t∈I, with ˜

J:=W>

rJWr,˜

R:=W>

rRWr, and ˜

B:=W>

rB. In particular,

Jis skew-symmetric and ˜

Ris symmetric and positive semi-definite and, hence,

the ROM (2.36) has the same pH structure as the FOM (2.34).

Remark 2.6.4 (Structure-preserving MOR for linear time-invariant pH sys-

tems).In the special case of linear time-invariant pH systems of the form

(2.28) with Q=Q>>0and Hamiltonian H(x):=1

2x>Qx, the model reduc-

tion approach explained in this subsection greatly simplifies. In particular, we

may choose Wrvia Wr=QVr(V>

rQVr)−1, see also [130]. Moreover, in this case

we have

∇H(Vr˜x) = QVr˜x=QVr(V>

rQVr)−1V>

rQVr˜x=Wr∇˜

H(˜x)

for all ˜x∈Rrand, thus, the ROM (2.35) obtained via Petrov–Galerkin pro-

jection coincides with the port-Hamiltonian ROM (2.36).

Another important class of linear time-invariant port-Hamiltonian systems

is given by

E˙x(t) = (J−R)x(t) + Bu(t),

y(t) = B>x(t),(2.37)

for all t∈I, with E∈Rn,n satisfying E=E>>0, associated Hamiltonian

H:Rn→Rdefined via H(x):=1

2x>Ex, and J,R,Bas in (2.28). Here, the

structure-preserving model reduction problem becomes even simpler, since a

classical Galerkin projection with Wr=Vrautomatically preserves the port-

Hamiltonian structure, see for instance [230]. Furthermore, applying a state

space transformation of the form ˜x:=Ex to (2.37) leads to a port-Hamiltonian

system of the form (2.28) with positive definite Q=E−1. Consequently, the

stability and passivity properties mentioned at the beginning of section 2.6.1

also apply to pH systems of the form (2.37). ¨

3. Mode Identification

As mentioned at the beginning of chapter 1 and outlined in section 2.5, model

reduction schemes are often separated into an offline and an online phase. The

offline phase includes all steps for constructing a ROM and, for instance when

using a POD-based approach, this typically includes the simulation of the

FOM for different input or parameter configurations as well as a subsequent

determination of suitable ansatz functions based on the snapshot data of the

FOM state. This chapter is devoted to the latter task when using approxima-

tion ansatzes based on transformed modes as in (1.4) or (1.8). To this end,

we present in section 3.1 a framework which is inspired by the POD optimiza-

tion problem, cf. section 2.5.1. In particular, given a trajectory of the FOM

state x, a suitable family of transformations T, and paths p1, . . . , pr, the new

framework aims for determining modes φ1, . . . , φrand amplitudes α1, . . . , αr

such that the error of the approximation

x(·)≈

i=1

αi(·)T(pi(·)) φi,(3.1)

is minimized, cf. (1.4) and (1.7) in section 1.1. Based on the framework from

section 3.1, we demonstrate in section 3.2 how the POD-greedy algorithm

discussed in section 2.5.2 may be extended to approximations based on trans-

formed modes. This yields an adaptive way of gradually sampling the param-

eter domain for cases where the FOM is parameter-dependent.

While we consider a quite general class of transformation families Tin sec-

tions 3.1 and 3.2, in most examples of this thesis we consider the special case

where Tcorresponds to a family of some kind of translation operators on a one-

dimensional spatial domain Ω. The application of such a translation operator

is straightforward in the case Ω = Ror for problems with periodic boundary

conditions. On the contrary, when considering problems on bounded domains

with non-periodic boundary conditions, there are several ways of defining suit-

able translation operators and some of them are presented in section 3.3. Fi-

nally, in section 3.4 we mention some other methods which construct mode

decompositions of the form (3.1) or similar and discuss their advantages and

disadvantages in comparison to the method presented in section 3.1.

3. Mode Identification

3.1. Residual Minimization

In this subsection, we consider a given trajectory x∈L2(I,W)with I= [0, tend]

and tend ∈R>0, where Wis a subspace of a real Hilbert space X. Usually, this

trajectory xis the result of solving an evolution equation of the form (1.1), for

instance via a numerical simulation. The goal of this subsection is to present a

framework for determining an optimal approximation of xusing an ansatz of

the form (3.1). To this end, we assume that we are given a real Banach space V

and a suitable family T:R→ L(V,X)of transformation operators together

with suitable paths p1, . . . , pr∈L2(I). For deriving a corresponding numerical

algorithm, the special case X=W=Rn,V=Rdφwith n, dφ∈Nplays an

important role and may for instance correspond to a numerical approximation

of x. Moreover, in many situations the trajectory xis only known at discrete

time points, which is also reflected in the algorithms presented in this section.

We note that in contrast to the parameter-dependent ansatz (1.7), in this

section we focus on trajectories which are only dependent on time, but not on

additional parameters. However, we emphasize that the presented framework

may be generalized to the parameter-dependent case and we briefly address

this in Remark 3.1.4.

As a starting point, we recall that the POD method is based on the mini-

mization problem (2.15), cf. section 2.5.1. Consequently, the POD method is

optimal in the sense that the POD modes φ1, . . . , φrand the corresponding

coefficients α1, . . . , αrare chosen such that the squared L2(I,X)norm of the

residual is minimized. To obtain a similar optimality for the decomposition

(3.1), we consider the minimization problem

min

φ1,...,φr∈Y, α1,...,αr∈L2(I)

tend

Z0x(t)−

i=1

αi(t)T(pi(t)) φi

dt, (3.2)

where Y⊆Vis defined as in (1.5). The major two differences of the mini-

mization problem (3.2) to the one in (2.15) are the additional transformation

operators and the omission of the orthogonality constraints for the modes φi.

In the POD case, where an optimal time-independent linear subspace is sought,

we can restrict ourselves to searching for an orthonormal basis of the subspace

without affecting the approximation quality of the optimizer. On the other

hand, the approximation ansatz (3.1) is not just given by a linear combination

of modes, but instead also involves transformation operators, which act on the

modes and are parametrized by the time-dependent paths pi. Consequently,

we may not use the same arguments as in the POD case for justifying an or-

thonormality constraint for the modes. In fact, Example 3.1.1 illustrates that

even linearly dependent modes may be optimal when considering an approx-

imation ansatz of the form (3.1). This is in contrast to methods based on

time-independent linear subspaces, where a linearly dependent set of modes

corresponds to a redundancy and allows to remove modes without increasing

3.1. Residual Minimization

the approximation error.

Example 3.1.1 (Linear dependence of optimal modes).We revisit Exam-

ple 1.2.3 for the special case where the initial value of the velocity is zero, i.e.,

v0= 0. Furthermore, we only consider the analytical solution for the density,

which is given by

ρ(t, ξ) = ρref (ϑr(ξ−ct) + ϑl(ξ+ct)) for all (t, ξ)∈I×Ω,

cf. (1.15). Since v0is zero, the Riemann invariants ϑrand ϑlcoincide and are

given by the (b−a)-periodic continuation of 1

2ρref ρ0, cf. (1.16). As pointed out

in Example 1.2.3, the analytical solution may be described by an ansatz of the

form (3.1) with

r= 2, φ1=φ2=ρref ϑr|Ω,T=Tper,

α1(t) = α2(t)=1, p1(t) = −p2(t) = ct for all t∈I,(3.3)

cf. (1.17). Moreover, since the approximation error is zero, the modes and

amplitudes in (3.3) are a solution of the corresponding optimization problem

(3.2) with x=ρand X=Y=L2(Ω). We emphasize that the modes are

optimal and at the same time they coincide, i.e., they are linearly dependent.

However, since they are transformed by different time-dependent transforma-

tion operators Tper(ct)and Tper(−ct), this linear dependence does not imply

that we can remove one of them without introducing an error. All in all, this

example demonstrates that even linearly dependent modes may be optimal

when using an approximation ansatz of the form (3.1). l

Remark 3.1.2 (Determination of the paths).We emphasize that the cost func-

tional in (3.2) is only optimized over the modes and amplitudes, whereas the

paths piare fixed parameters. The simultaneous optimization of the modes,

amplitudes, and paths is not within the scope of this thesis, but is instead ad-

dressed in [40] in a very similar setting. Further data-driven techniques for es-

timating the paths based on snapshots of xare discussed in [199, 241, 242, 259].

Approaches which do not only determine the paths but also the family of trans-

formation operators are for instance presented in [152, 171, 243, 270]. They

are based on describing a transport map via a suitable linear combination of

ansatz functions, which are determined based on snapshot data. ¨

Remark 3.1.3 (Existence and uniqueness of solutions).Throughout this thesis,

we assume that the integral in (3.2) exists and that the minimization problem

(3.2) has a solution, whereas the analysis of the existence of solutions is not

discussed in this thesis. We refer to [37, sec. 4], where these questions have been

discussed in a very similar setting. If a solution of the minimization problem

(3.2) exists, then it is in general not unique. For instance, multiplying an

amplitude with any non-zero constant and dividing the corresponding mode

by the same constant changes the amplitude and the mode, but not the value of

the cost functional. This source of non-uniqueness can be avoided by enforcing

3. Mode Identification

the modes to be normalized, cf. [37]. But even then, the minimizer is in

general not unique, since we can for instance multiply an amplitude and the

corresponding mode by −1without changing the cost functional. As pointed

out in Remark 2.5.3, a similar kind of non-uniqueness also applies to the POD

optimization problem. ¨

Remark 3.1.4 (Parameter-dependent trajectories).If we consider a trajectory

which does not only depend on time but also on a parameter vector µ∈

M, then the cost functional in (3.2) has to be adapted to also account for

the parameter dependency. To this end, we assume that Mis the closure

of an open, non-empty subset of Rnp. Then, we search for solutions of the

minimization problem

min

φ∈Yr, α∈(L2(I×M))r

tend

Z0x(t;µ)−

i=1

αi(t;µ)T(pi(t;µ)) φi

dtdµ.

In a fully discrete setting, the parameter dependency can be treated analo-

gously as the time dependency by using appropriate weights for the different

snapshots, cf. (3.6). A theoretical discussion of the classical POD optimization

problem for parameter-dependent elliptic PDEs is presented in [156]. An alter-

native treatment of the parameter dependency is given by a greedy procedure

as discussed in section 3.2. ¨

As illustrated in Example 3.1.1, it is advisable to omit orthogonality con-

straints for the modes when using an ansatz of the form (3.1), at least in the

case where more than one transformation operator is involved. However, the

fact that the transformed modes are in general not orthonormal prevents a

connection to the singular value decomposition as in the POD case. In the

following, we consider a special case where the minimization problem (3.2)

may be reduced to one of the form (2.16), which may be solved via the POD

method. To this end, we consider the ansatz (1.8) with nt= 1 and assume that

Vis a Hilbert space and that Tis pointwise unitary. Then, since all modes are

affected by the same transformation and since this transformation is unitary,

orthonormality of the modes is preserved by the transformation. Thus, in this

special case it is reasonable to enforce the modes to be orthonormal and the

corresponding minimization problem reads











min

φ1,...,φr∈Y

tend

Z0x(t)−

i=1 hx(t),T(p(t)) φiiXT(p(t)) φi

dt,

s.t. hφi, φjiV=δij for i, j = 1, . . . , r.

(3.4)

Here, we exploited the orthonormality of the transformed modes, which allows

to replace the coefficients αiby the optimal ones hx, T(p)φiiXobtained via

orthogonal projection, cf. section 2.5.1. Furthermore, by using again the fact

that Tis pointwise unitary, the minimization problem (3.4) can be shown to

3.1. Residual Minimization

be equivalent to











min

φ1,...,φr∈Y

tend

Z0T∗(p(t))x(t)−

i=1 hT∗(p(t))x(t), φiiVφi

dt,

s.t. hφi, φjiV=δij for i, j = 1, . . . , r.

(3.5)

If we additionally assume that Vis separable, Y⊆Vis a dense subspace,

and the transformed trajectory T∗(p)xis in L2(I,Y), then Theorem 2.5.2

yields that a solution of this minimization problem is given by the first rPOD

modes of T∗(p)x. This relation has been used for instance in [54] and it has

been formally proven in a slightly different setting in [37, sec. 4]. An example

where the mentioned assumptions on V,Y, and Tare satisfied is given by

V=X=L2(Ω),W=H1

per (Ω) with some spatial domain Ω = (a, b),a∈R,

b∈R>a, and T=Tper, cf. Definition 1.2.2. Due to this special choice of W, the

subspace Yas defined in (1.5) may be shown to coincide with W=H1

per (Ω).

Furthermore, H1

per (Ω) is indeed a dense subspace of V=L2(Ω), which follows

from [46, Cor. 4.23] and the fact that the space of infinitely differentiable

functions with compact support in Ωis a subspace of H1

per (Ω).

Apart from the special case considered in the last paragraph, it is in general

not clear if the minimization problem (3.2) may be reduced to a corresponding

POD optimization problem as in (2.16). Instead, we propose to discretize

(3.2) and solve the resulting finite-dimensional nonlinear optimization problem

numerically. For this purpose, we consider the special case X=W=Rnand

assume that we have access to a finite number of samples of the trajectory

x. Furthermore, we discretize the time integral in (3.2) and consider the fully

discrete problem

min

φ1,...,φr∈Rdφ, a1,1,...,ar,q∈R

i=1

ωi

2[X]i−

j=1 T(gj,i)aj,iφj

,(3.6)

where we also assume a finite-dimensional space V=Y=Rdφfor the modes.

Here, the coefficients ωi∈R>0,i= 1, . . . , q, are weighting factors originating

from a quadrature approximation of the time integral in (3.2), for instance,

using a composite trapezoidal scheme. Furthermore, we use a weighted norm

k·kWwith symmetric and positive definite W∈Rn,n to be able to include for

example a discretization of the L2(Ω) norm with spatial domain Ω, cf. chap-

ter 6. Besides, X∈Rn,q denotes a snapshot matrix which is assumed to be

given and whose columns correspond to samples of the trajectory x, cf. (2.20).

Especially, in order for these samples to be well-defined, we implicitly assume

that xis an element of a subspace of L2(I,Rn)for which point-wise evaluations

are well-defined, as for instance C(I,Rn). A similar assumption applies also to

the amplitudes and the paths, whose discrete analogues αj(ti)and pj(ti)are

denoted by aj,i ∈Rand gj,i ∈R, respectively, for j= 1, . . . , r and i= 1, . . . , q.

3. Mode Identification

Remark 3.1.5 (Alternative for discretizing the cost functional).The discretiza-

tion step leading from the infinite-dimensional problem (3.2) to the finite-

dimensional one (3.6) usually involves replacing the family of transformation

operators Tby a suitable family of matrices, which approximate the action

of Ton the discrete level. This is demonstrated in appendix D.1 for the

translation operators used in chapter 6. In [40, sec. 4], the authors present

an alternative discretization which is based on approximating the modes by

linear combinations of FEM basis functions. This method has the advantage

that it does not require a discretization of the family of transformations T,

but instead it is based on an analytical computation of the occurring inner

products of transformed FEM basis functions, cf. [40, Ex. 3]. However, first

investigations indicate that the resulting discretized cost function cannot be

simply brought into the same form as (3.6) without a significant computa-

tional overhead. Consequently, it is not clear if an efficient reduction of the

cost function as discussed in the upcoming section 3.1.2 is possible when using

the alternative discretization from [40]. This is why we instead focus here on

the discrete cost function as given in (3.6). ¨

Before we discuss strategies for solving the optimization problem (3.6), we

introduce the vectors

φ:=





φ1

φr





∈Rrdφ, a :=









∈Rrq, ai:=





a1,i

ar,i





∈Rr(3.7)

as well as the matrix functions K:Rrdφ→Rqn,rq and Ki:Rrdφ→Rn,r defined

via

K(φ):=





K1(φ)...

Kq(φ)





, Ki(φ):=hT(g1,i)φ1··· T (gr,i)φri,

(3.8)

for i= 1, . . . , q. This allows to reformulate the minimization problem (3.6) as

min

φ∈Rrdφ, a∈Rrq

2kvec(X)−K(φ)ak2

W,(3.9)

where the weighting matrix ˆ

W∈Rnq,nq is defined via

W:=





ω1...

ωq





⊗W. (3.10)

In particular, we observe that the minimization problem (3.9) can be written

3.1. Residual Minimization

as a weighted separable nonlinear least squares problem of the form

min

β∈Rs,γ∈Rp

2kb−A(β)γk2

|{z }

=:Jf(β,γ)

(3.11)

with given matrix function A:Rs→Rk,p, symmetric and positive definite

weighting matrix ˜

W∈Rk,k, and vector b∈Rk. In (3.9), the quantities

φ, a,K,vec(X), and ˆ

Wcorrespond to β,γ,A,b, and ˜

W, respectively, in

the general problem formulation (3.11). The characteristic feature of such

separable nonlinear least squares problems is that the unknowns are separated

into two block components βand γand that A(β)γis linear with respect to the

γcomponent and possibly nonlinear with respect to the βcomponent. These

kinds of problems have been extensively investigated in the past decades. For

instance, in [120] the authors have introduced the variable projection method,

which we follow in section 3.1.2. Constrained separable nonlinear least squares

problems have for example been addressed in [159, 221]. Furthermore, for the

treatment of ill-conditioned problems, some regularizations of Tikhonov type

have been proposed, see e.g. [66, 70, 288]. These and other extensions like the

punishment of discretized derivatives of the modes or the amplitudes can be

added to the minimization problem (3.9) to take additional requirements for

the solution into account. However, these extensions are not within the scope

of this thesis.

In the following, we discuss two different strategies to numerically solve the

minimization problem (3.9). The first approach relies on solving the problem

simultaneously in both variables φand a, whereas the latter one uses the

variable projection method in order to reduce (3.9) to a problem which only

depends on φ.

Remark 3.1.6 (Choice of the mode number).Regardless of whether the full

optimization problem is solved as in section 3.1.1 or the variable projection

method is applied as in section 3.1.2, we usually assume the mode number rto

be given in advance. For simple examples like the linear advection or the linear

wave equation as considered in section 1.2.1, physical insights may be used to

properly choose r. However, when considering more involved problems, it may

be much more difficult or even impossible to determine a suitable value for r

just based on physical considerations. If at least the most relevant transports

have been identified, i.e., nt,T, and p1, . . . , pntin (1.8) may be assumed to be

given, then the mode numbers r1, . . . , rntcould be for instance chosen based

on a greedy algorithm as discussed in section 3.2. A similar procedure for

gradually increasing the mode numbers is proposed in [259, Alg. 1], but in

contrast to the approach in section 3.2 this is not a greedy procedure, but

instead all modes are optimized each time the mode numbers are increased. A

completely different strategy is suggested in [240], where the author proposes

a cost function based on a Schatten 1-norm and argues that the corresponding

heuristic rank minimization property may be used to remove redundant modes

3. Mode Identification

or even redundant transports, i.e., this approach could be used for decreasing

ntand r1, . . . , rntin case that these numbers are chosen larger than necessary

for obtaining a reasonable approximation. ¨

3.1.1. Solving the Full Optimization Problem

We start by considering the full optimization problem (3.9) and aim to solve it

directly without any reduction of the optimization parameters. To this end, we

intend to use gradient-based optimization techniques and, thus, first compute

the partial derivatives of the cost function in (3.9) with respect to φand a, see

Lemma 3.1.7. A corresponding algorithm for evaluating the cost function and

its gradient is provided in Algorithm 3.1. The expression [∇aJfull]ioccurring in

the last line of the for-loop denotes the ith block row of the partial gradient with

respect to a, cf. (3.13) in Lemma 3.1.7. In section 6.1 we use this algorithm

together with a gradient-based optimization solver in the context of a wave

equation test case.

Lemma 3.1.7 (Partial derivatives of cost function in (3.9)).Let q, n, dφ∈N

and r∈N≤nas well as T:R→Rn,dφ,X∈Rn,q, and g1,1, . . . , gr,q ∈Rbe

given. Furthermore, let ˆ

W∈Rnq,nq be as defined in (3.10) with ω1, . . . , ωq∈

R>0and symmetric positive definite matrix W∈Rn,n. Besides, let K:Rrdφ→

Rqn,rq be as defined in (3.8) and let Jfull :Rrdφ×Rrq →Rdenote the cost

function in (3.9), i.e.,

Jfull(φ, a):=1

2kvec(X)−K(φ)ak2

Then, Jfull is continuously differentiable and its partial derivatives are given by

∂φJfull(φ, a) =

i=1

ωi(Ki(φ)ai−[X]i)>Wha1,iT(g1,i)··· ar,iT(gr,i)i,(3.12)

∂aJfull(φ, a) = 





ω1K1(φ)>W(K1(φ)a1−[X]1)

ωqKq(φ)>W(Kq(φ)aq−[X]q)







,(3.13)

where we use the notation from (3.7).

Proof. Since Kdefined in (3.8) is a linear mapping, Jfull is not only a quadratic

function in abut also in φand, hence, continuously differentiable. Before we

compute its partial derivatives, we determine those of K, which are given by

∂K

∂[φk]j

(φ) = 





∂K1

∂[φk]j(φ)

...

∂Kq

∂[φk]j(φ)





=





[T(gk,1)]je>

k...

[T(gk,q)]je>







(3.14)

3.1. Residual Minimization

for k= 1, . . . , r,j= 1, . . . , dφ, where ekdenotes the kth unit vector of the

standard basis of Rr. Consequently, using the product rule we obtain the first

set of partial derivatives of Jfull as

∂Jfull

∂[φk]j

(φ, a) = − ∂K

∂[φk]j

(φ)a!>ˆ

W(vec(X)−K(φ)a)

=





[T(gk,1)]jak,1

[T(gk,q)]jak,q







>





ω1W...

ωqW











K1(φ)a1−[X]1

Kq(φ)aq−[X]q







i=1

ωi(Ki(φ)ai−[X]i)>Wak,i[T(gk,i)]j

for k= 1, . . . , r,j= 1, . . . , dφ. The concatenation of these partial derivatives

into one row vector results in (3.12). Finally, for obtaining (3.13) we compute

∂aJfull(φ, a) = −(vec(X)−K(φ)a)>ˆ

WK(φ)

=





K1(φ)a1−[X]1

Kq(φ)aq−[X]q







>





ω1W...

ωqW











K1(φ)...

Kq(φ)







=





ω1K1(φ)>W(K1(φ)a1−[X]1)

ωqKq(φ)>W(Kq(φ)aq−[X]q)







3.1.2. Using Variable Projection

In the following, we summarize the variable projection method introduced in

[120] for separable nonlinear least squares problems of the form (3.11) and

afterwards discuss its application to the minimization problem (3.9). In [120]

the authors consider a special case of (3.11) where the weighting matrix ˜

equals the identity matrix. However, based on a decomposition of ˜

Wof the

form ˜

W=R>

WR˜

Wwith R˜

W∈Rk,k, the cost function in (3.11) can be written

2kb−A(β)γk2

W=1

2(b−A(β)γ)>R>

WR˜

W(b−A(β)γ) = 1

2˜

b−˜

A(β)γ2

with ˜

b:=R˜

Wband ˜

A:Rs→Rk,p defined via ˜

A(η):=R˜

WA(η). Consequently,

we may assume without loss of generality ˜

W=Ikin (3.11) and apply the

approach from [120].

For a fixed β∈Rs, the minimization problem (3.11) is a linear least squares

problem for γand, hence, the optimal value ˆγneeds to satisfy the necessary

3. Mode Identification

Algorithm 3.1 Evaluation of the cost function and its gradient for (3.9)

Inputs:

•snapshot matrix X∈Rn,q,n, q ∈N

•modes φ1, . . . , φr∈Rdφwith r∈N≤n,dφ∈N

•amplitudes ai,j ∈Rfor i= 1, . . . , r,j= 1, . . . , q

•transformation family T:R→Rn,dφ

•path values gi,j ∈Rfor i= 1, . . . , r,j= 1, . . . , q

•time weights ω1, . . . , ωq∈R>0

•symmetric and positive definite spatial weighting matrix W∈Rn,n

Outputs:

•value of the cost function Jfull as in Lemma 3.1.7 evaluated at (φ, a) =

([φ>

1. . . φ>

r]>,[a1,1··· ar,1a1,2··· ar,2··· a1,q ··· ar,q]>)

•gradient of the cost function Jfull evaluated at (φ, a)

1: Jfull ←0

2: ∇φJfull ←0

3: for i←1to qdo

4: R←Pr

j=1 T(gj,i)φjaj,i −[X]i

5: Jfull ←Jfull +1

2ωiR>WR

6: ∇φJfull ← ∇φJfull +ωiha1,iT(g1,i)··· ar,iT(gr,i)i>WR

7: [∇aJfull]i←ωihT(g1,i)φ1··· T (gr,i)φri>WR

8: end for

9: ∇Jfull ←[∇φJ>

full ∇aJ>

full]>

optimality conditions given by the so-called normal equations

A>(β)A(β)ˆγ=A>(β)b, (3.15)

see for instance [111, sec. 6.2]. We note that the Hessian of the cost function

is A>(β)A(β)and, hence, symmetric and positive semi-definite. Thus, by

Theorem 2.2.6 the cost function is convex and by Theorem 2.2.5 every solution

of (3.15) is a global minimum point of the minimization problem (3.11) with

fixed β. If A(β)has full column rank, then the Gram matrix A>(β)A(β)is

even positive definite and the minimizer ˆγis uniquely determined. In general,

all solutions of the normal equations (3.15) can be characterized by

ˆγ=A(β)+b+v, (3.16)

where v∈ker(A(β)) ⊆Rpcan be chosen arbitrarily and A(β)+denotes the

Moore–Penrose pseudoinverse of A(β), cf. section 2.1.

3.1. Residual Minimization

Substituting (3.16) into the minimization problem (3.11), we obtain

min

β∈Rs

2b−A(β)A(β)+b2

|{z }

=:Jr(β)

.(3.17)

Thus, by using the optimality condition (3.15) for γ, the minimization problem

(3.11) can be reduced to the minimization problem (3.17), where the only

remaining unknown is β. Theorem 3.1.8, which is a modified version of a

part of Theorem 2.1 from [120], establishes a relation between the solutions

of the original problem (3.11) and the reduced one (3.17). Theorem 2.1 from

[120] additionally contains a relation between the critical points of (3.11) and

(3.17), provided that Ahas constant rank and is continuously differentiable

with respect to βin an open set containing the critical points. While we do

not need that part of [120, Thm. 2.1] in this thesis, we exploit the formula for

the partial derivatives of the reduced cost function (3.17) provided in [120],

which is given by

∂Jr

∂βi

(β) = −b>(Ik−A(β)A(β)+)∂A

∂βi

(β)A(β)+bfor i= 1, . . . , s. (3.18)

Theorem 3.1.8 (Relation between (3.11) and (3.17)).Consider the mini-

mization problems (3.11) and (3.17) with b∈Rk,A:Rs→Rk,p,˜

W=Ik, and

k, s, p ∈N. Then, the following assertions hold.

(i) Let (ˆ

β, ˆγ)∈Rs×Rpbe a global minimum point of (3.11). Then, ˆ

βis a

global minimum point of (3.17).

(ii) Let ˆ

βbe a global or local minimum point of (3.17). Then, (ˆ

β, A(ˆ

β)+b)is

a global or local minimum point of (3.11), respectively.

Proof. A very similar version of the statements concerning the global mini-

mum points is included in Theorem 2.1 in [120], which is based on the as-

sumptions that Ais continuously differentiable and has constant rank in an

open set containing the minimum points. However, the part of the proof of

[120, Thm. 2.1] which concerns the global minimum points does not use these

additional assumptions on A. For the sake of self-containedness, we repeat the

main arguments in the following. Both directions are based on the property

b−A(β)A(β)+b2≤ kb−A(β)γk2for all γ∈Rp,(3.19)

which holds for any β∈Rsand follows from the theory of linear least squares

problems, cf. the discussion before (3.17).

(i) Since (ˆ

β, ˆγ)∈Rs×Rpis a global minimum point of (3.11), we have

b−A(ˆ

β)ˆγ2≤ kb−A(β)γk2for all (β, γ)∈Rs×Rp.

3. Mode Identification

Let us assume that ˆ

βis not a global minimum point of (3.17), i.e., there

exists ˜

β∈Rswith

b−A(˜

β)A(˜

β)+b2<b−A(ˆ

β)A(ˆ

β)+b2.

However, due to (3.19) this implies

b−A(˜

β)A(˜

β)+b2<b−A(ˆ

β)ˆγ2,

which is a contradiction to the assumption that (ˆ

β, ˆγ)is a global minimum

point of (3.11). Thus, ˆ

βhas to be a global minimum point of (3.17).

(ii) If ˆ

βis a local minimum point of (3.17), then there exists ∈R>0with

b−A(ˆ

β)A(ˆ

β)+b2≤b−A(β)A(β)+b2for all β∈B(ˆ

β),(3.20)

where B(ˆ

β)denotes the open ball in Rswith radius and center ˆ

β. Let

us assume that (ˆ

β, A(ˆ

β)+b)is not a local minimum point of (3.11), i.e.,

for any ˜∈R>0there exists (˜

β, ˜γ)∈Rs×Rpwith

b−A(˜

β)˜γ2<b−A(ˆ

β)A(ˆ

β)+b2and "˜

β−ˆ

˜γ−A(ˆ

β)+b#<˜.

However, due to (3.19) this implies

b−A(˜

β)A(˜

β)+b2<b−A(ˆ

β)A(ˆ

β)+b2.

Thus, for instance by choosing ˜=, this yields a contradiction to (3.20)

since we also have

˜

β−ˆ

β≤"˜

β−ˆ

˜γ−A(ˆ

β)+b#<˜=.

Consequently, (ˆ

β, A(ˆ

β)+b)needs to be a local minimum point of (3.11).

The proof for global minimum points follows the same lines by omitting

the restrictions to neighborhoods.

Theorem 3.1.8 provides a theoretical justification for replacing the full opti-

mization problem (3.11) by the reduced one (3.17). The assertions (i) and (ii)

establish a one-to-one correspondence between the global minimum points of

the two optimization problems, except for a potential non-uniqueness of the

γcomponent of the original problem (3.11). Furthermore, the local minimum

point property from (ii) implies that the reduced optimization problem (3.17)

has only local minimum points which correspond to local minimum points of

the original problem (3.11). Example 3.1.9 illustrates that the converse in gen-

3.1. Residual Minimization

eral is not true, i.e., there may be local minimum points of the original problem

which do not correspond to local minimum points of the reduced problem. In

summary, this means that when we determine a global or local minimum point

of the reduced problem (3.17), we can be sure that it also corresponds to a

global or local minimum point of the original problem. Furthermore, as il-

lustrated in Example 3.1.9, the reduced optimization problem (3.17) may in

general have less local minimum points than the original problem. This is in

general an advantage when using for instance gradient-based methods for com-

puting a minimum point, since there might be less local minimum points where

the solver may get stuck. The only difficulty which may arise when going from

the original to the reduced problem is that the reduced problem may have in

general more points where the cost function is not differentiable, namely at

points where Achanges rank. In the context of the optimization problem (3.9)

this happens, for instance, at points where the transformed modes become lin-

early dependent, since this leads to a rank drop in K(φ). In fact, in this case

the cost function of the reduced problem may even be discontinuous as illus-

trated in Example 3.1.10, see also Example 3.1.9. This is also reflected in the

fact that the cost function in (3.17) coincides up to the prefactor 1

2with the

squared norm of the orthogonal projection of bonto the kernel of A(β)>. When

the rank of Achanges, then also the dimension of the kernel of A>changes,

which may in turn result in a discontinuity of the cost function. Thus, in order

to numerically solve the reduced nonlinear least squares problem (3.17), it may

be advantageous to use solvers which may handle non-smooth cost functions,

see for instance [75] and the references therein.

Example 3.1.9 (Local minimum points of (3.11) and (3.17)).To illustrate

that the statement (i) in Theorem 3.1.8 may in general not be extended to

local minimum points, we consider the special case given by k=s=p= 1,

b= 1,˜

W= 1, and A:R→Rwith A(β):=β2. The corresponding cost

function of the full optimization problem (3.11) reads

Jf(β, γ) = 1

21−β2γ2.

In particular, it has a local minimum point at (0,−1), since for all (β, γ)∈

B1(0,−1), where B1(0,−1) denotes the open ball in R2with radius 1and

center (0,−1), we have

Jf(β, γ) = 1

21−β2γ2≥1

2=Jf(0,−1).

On the other hand, the corresponding cost function of the reduced problem

(3.17) reads

Jr(β) = 





2,if β= 0,

0,otherwise

and hence has no local minimum point at 0. This demonstrates that Theo-

3. Mode Identification

rem 3.1.8(i) may in general not be extended to local minimum points. l

Example 3.1.10 (Discontinuous cost function).We consider the cost function

J:R4→Rdefined via

J(φ):=1

2I2−K(φ)K(φ)+vec(X)2

as a special case for the reduced cost function corresponding to (3.9), where

X∈R2and K:R4→R2,2are given by

X= vec(X) = "1

0#and K(φ):="φ1φ3

φ2φ4#.

In particular, we investigate the continuity at ˆ

φ= [0 1 0 1]>as an example

for a point where K(ˆ

φ)is rank-deficient and where Xis not contained in the

column span of K(ˆ

φ). The limit considerations

lim

→0Jh1 0 1i>=1

2lim

→0

I2−"0

1 1#"0

1 1#+

"1

0#

= 0,

lim

→0Jh0 1 + 0 1i>=1

2lim

→0

I2−"0 0

1 + 1#" 0 0

1 + 1#+

"1

0#

2lim

→0 I2−"0 0

0 1#!"1

0#

reveal that limφ→ˆ

φJ(φ)does not exist and, thus, Jis not continuous at ˆ

φ.l

In the remainder of this subsection, we present an algorithm for computing

the cost function and the gradient of (3.17), as these two ingredients are usually

required in gradient-based optimization solvers. Since we aim to determine

a solution of the minimization problem (3.9) in the end, we are especially

interested in the special case

b=Rˆ

Wvec(X), A =Rˆ

WK, (3.21)

where Rˆ

W∈Rnq,nq is given by

Rˆ

W=





√ω1...√ωq





⊗RW

and RW∈Rn,n is chosen such that W=R>

WRWis satisfied. In particular,

Rˆ

Wthen satisfies ˆ

W=R>

WRˆ

The cost function in (3.17) can be further simplified as demonstrated in

Lemma 3.1.11. In the special case (3.21), the reduced version of (3.9) is hence

3.1. Residual Minimization

equivalent to minimizing

JvarPro(φ):=−1

2vec(X)>R>

WRˆ

WK(φ) (Rˆ

WK(φ))+Rˆ

Wvec(X)

=−1

2vec(X)>ˆ

WK(φ) (Rˆ

WK(φ))+Rˆ

Wvec(X)

=−1

i=1

ωi[X]>

iWKi(φ)

|{z }

=:[B(φ)]>

(√ωiRWKi(φ))+√ωiRW[X]i

|{z }

=:[C(φ)]i

=−1

2hB(φ), C(φ)iF.

(3.22)

Here, h·,·iFdenotes the Frobenius inner product, cf. section 2.1, and moreover

we used the fact that the Moore–Penrose pseudoinverse of a block diagonal ma-

trix may be obtained by computing the pseudoinverses of the diagonal blocks,

see for instance [57, Thm. 3.4.1]. In addition, we note that the computation

of Cstill involves computing the pseudoinverses of the matrices √ωiRWKi(φ),

which typically involves computing SVDs of √ωiRWKi(φ)for i= 1, . . . , q,

cf. [57, ch. 12]. However, these matrices are usually tall and skinny and, thus,

using Lemma 3.1.12 allows us to significantly reduce the computational effort

by instead calculating eigenvalue decompositions of the r×rmatrices

(√ωiRWKi(φ))>√ωiRWKi(φ) = ωiKi(φ)>WKi(φ)

for i= 1, . . . , q. The calculation of the gradient of JvarPro is carried out based

on (3.18), which together with (3.21) and (3.14) yields

∂JvarPro

∂[φi]j

(φ)

= vec(X)>R>

WRˆ

WK(φ) (Rˆ

WK(φ))+Rˆ

∂K

∂[φi]j

(φ) (Rˆ

WK(φ))+Rˆ

Wvec(X)

−vec(X)>R>

WRˆ

∂K

∂[φi]j

(φ) (Rˆ

WK(φ))+Rˆ

Wvec(X)

= vec(X)>R>

WRˆ

WK(φ) (Rˆ

WK(φ))+>Rˆ

∂K

∂[φi]j

(φ) (Rˆ

WK(φ))+Rˆ

Wvec(X)

−vec(X)>ˆ

W∂K

∂[φi]j

(φ) (Rˆ

WK(φ))+Rˆ

Wvec(X)

=(Rˆ

WK(φ))+Rˆ

Wvec(X)>K(φ)>ˆ

W∂K

∂[φi]j

(φ) (Rˆ

WK(φ))+Rˆ

Wvec(X)

−vec(X)>ˆ

W∂K

∂[φi]j

(φ) (Rˆ

WK(φ))+Rˆ

Wvec(X)

k=1

ωk(Kk(φ)[C(φ)]k−[X]k)>W[T(pi,k)]je>

i[C(φ)]k

3. Mode Identification

k=1

ωk[C(φ)]i,k[T(pi,k)]>

jW(Kk(φ)[C(φ)]k−[X]k).

The details of the computational steps for evaluating JvarPro and its gradient

are summarized in Algorithm 3.2. Especially, we note that the algorithm only

requires the weighting matrix W, whereas the factor RWis not needed.

Lemma 3.1.11 (Alternative formulation of (3.17)).For given b∈Rkand

A:Rs→Rk,p with k, p, s ∈N, the vector ˆ

β∈Rsis a solution of the mini-

mization problem (3.17) if and only if it is a solution of

min

β∈Rs−1

2b>A(β)A(β)+b.(3.23)

Proof. Using the general identities (CC+)>=CC+and CC+C=C, which

hold for any real-valued matrix C, cf. [57, Thm. 1.1.1], we obtain

2b−A(β)A(β)+b2

2kbk2−b>A(β)A(β)+b+1

2b>(A(β)A(β)+)>A(β)A(β)+b

2kbk2−b>A(β)A(β)+b+1

2b>A(β)A(β)+A(β)A(β)+b

2kbk2−1

2b>A(β)A(β)+b.

This calculation shows that the cost functions in (3.17) and (3.23) only differ

by the constant 1

2kbk2, which yields the claim.

Lemma 3.1.12 (Pseudoinverse in terms of an eigenvalue decomposition).

Let A∈Rm,n with m, n ∈Nbe given with r:= rank(A). Furthermore, let

A>A=USU>be an eigenvalue decomposition of A>Awith orthogonal ma-

trix U= [U1U2]∈Rn,n with U1∈Rn,r, U2∈Rn,n−rand diagonal matrix

S= diag(S1,0) ∈Rn,n with S1∈Rr,r. Then, the Moore–Penrose pseudoin-

verse of Ais given by A+=U1S−1

1U>

1A>.

Proof. By [57, Thm. 1.1.1], A+is the Moore–Penrose pseudoinverse of Aif

and only if AA+and A+Aare symmetric and AA+A=Aand A+AA+=A+

are satisfied. The first property holds since AU1S−1

1U>

1A>is symmetric and

the second one follows from

U1S−1

1U>

1A>A=U1S−1

1U>

1U1S1U>

1=U1U>

1=U1U>

1>.

For the other two properties, we exploit the fact that the definition of U1

implies im(U1) = im(A>A) = im(A>)and, thus, obtain

AU1S−1

1U>

1A>A=AU1S−1

1U>

1U1S1U>

1=AU1U>

1=A,

U1S−1

1U>

1A>AU1S−1

1U>

1A>=U1S−1

1U>

1U1S1U>

1U1S−1

1U>

1A>=U1S−1

1U>

1A>,

3.1. Residual Minimization

which concludes the proof. We note that the claim may alternatively be shown

using the construction of the Moore–Penrose pseudoinverse via an SVD, see

for instance [28, Cor. 6.2.1].

Algorithm 3.2 Evaluation of the cost function (3.22) and its gradient

Inputs:

•snapshot matrix X∈Rn,q with n, q ∈N

•modes φ1, . . . , φr∈Rdφwith r∈N≤n,dφ∈N

•transformation family T:R→Rn,dφ

•path values gi,j ∈Rfor i= 1, . . . , r,j= 1, . . . , q

•time weights ω1, . . . , ωq∈R>0

•spatial weighting matrix W∈Rn,n with W=W>>0

Outputs:

•value of the cost function JvarPro evaluated at φ:= [φ>

1. . . φ>

r]>

•gradient of the cost function JvarPro evaluated at φ

1: ∇JvarPro(φ)←0

2: for i←1to qdo

3: ˜

K←hT(g1,i)φ1··· T (gr,i)φri

4: [B]i←ωi˜

K>W[X]i

5: Compute an eigendecomposition ωi˜

K>W˜

K=USU>with orthogonal

U∈Rr,r and diagonal S∈Rr,r with descending diagonal entries

6: Determine the rank of S

7: ˜

U←h[U]1··· [U]rank(S)i

8: ˜

S←diag([S]1,1,[S]2,2,...,[S]rank(S),rank(S))

9: Solve the linear equation system ˜

S˜a=˜

U>[B]ifor ˜a

10: [C]i←˜

U˜a

11: v←ωih[C]1,iT(g1,i)··· [C]r,iT(gr,i)i>W(˜

K[C]i−[X]i)

12: ∇JvarPro(φ)← ∇JvarPro(φ) + v

13: end for

14: JvarPro(φ)← −1

2hB, CiF

Remark 3.1.13 (Alternative algorithms for solving separable least squares prob-

lems).We note that there are also efficient implementations of the variable

projection algorithm available in the literature, see for instance [221]. Never-

theless, the corresponding implementations do usually not exploit the block

structure as it occurs in the specific separable nonlinear least squares problem

(3.9). An adaptation of the algorithms available in the literature to this block-

structured setting would be certainly of interest and may have the potential

to further speed up the computations. Also contributions from the literature

about bilinear least squares problems could perhaps be used for developing

3. Mode Identification

more efficient algorithms, cf. [18, 91]. However, such considerations are not

within the scope of this thesis. ¨

In summary, we conclude that the variable projection approach allows to

replace the optimization problem (3.9) with r(dφ+q)unknowns by a reduced

optimization problem where the number of optimization parameters is rdφ. A

drawback of the variable projection is that it may introduce points where the

cost function is discontinuous, which may result in a need for employing opti-

mization techniques that can also handle non-smooth problems. A numerical

comparison of the two approaches is presented in section 6.1 by means of a

wave equation test case.

3.2. Greedy Algorithm based on Transformed

Modes

While we discussed in the previous section mainly the case where the trajec-

tory xonly depends on time but not additionally on a parameter vector, we

focus in this section on the case where xoriginates from a time- and parameter-

dependent FOM as in (1.6). More precisely, the goal of this section is to extend

the POD-greedy algorithm as presented in section 2.5.2 to the setting of trans-

formed modes. To this end, we use the approximation ansatz (1.8) with x,ˆx,

pi, and αi,j depending on tand on a parameter vector µ∈Mfor j= 1, . . . , ri,

i= 1, . . . , nt. Furthermore, we assume for simplicity that the considered time

interval Idoes not depend on µ. In particular, we propose Algorithm 3.3 and

discuss its properties as well as its peculiarities in comparison to the standard

POD-greedy algorithm, cf. Algorithm 2.1, in the following.

Algorithm 3.3 Greedy algorithm based on transformed modes

Inputs:

•FOM as in (1.6) with X=W=Rn,M⊆Rnp,n, np∈N

•discrete time points t1, . . . , tq∈Iwith 0 = t1< t2< . . . < tq=tend

and q∈N≥2

•parameter training set Mtrain ⊂Mwith |Mtrain|<∞

•transformation family T:R→Rn,dφwith dφ∈N

•offline path estimates ˆp:{t1, . . . , tq}×Mtrain →Rntwith nt∈N

•initial ansatz vectors

Φ0= ({φ1,1, . . . , φ1,r1,0}, . . . , {φnt,1, . . . , φnt,rnt,0})∈(Pf(Rdφ))nt

with r1,0, . . . , rnt,0∈N∩{0}and Pnt

j=1 rj,0≤n

•error estimator ε:Mtrain ×(Pf(Rdφ))nt→R≥0

•error tolerance tol ∈R>0

3.2. Greedy Algorithm based on Transformed Modes

•maximum number of iterations imax ∈N

•spatial weighting matrix W=W>>0∈Rn,n

•time weights ω1, . . . , ωq∈R>0

Output:

•ansatz vectors

Φk= ({φ1,1, . . . , φ1,r1,k }, . . . , {φnt,1, . . . , φnt,rnt,k })∈(Pf(Rdφ))nt

with k∈N∩ {0}and r1,k ∈N≥r1,0,. . .,rnt,k ∈N≥rnt,0, satisfying

max

µ∈Mtrain

ε(µ, Φk)≤tol or k=imax

1: for i←1to imax do

2: Evaluate the error estimator ε(·,Φi−1)on Mtrain

3: if maxµ∈Mtrain ε(µ, Φi−1)≤tol then return Φi−1

4: end if

5: Solve the FOM for a parameter value µmax ∈arg max

µ∈Mtrain

ε(µ, Φi−1)

6: for `←1to qdo

7: Project the FOM snapshot x(t`;µmax)onto the orthogonal comple-

ment of

span nT(ˆp1(t`;µmax)) φ1,1, . . . , T(ˆp1(t`;µmax)) φ1,r1,i−1, . . . ,

T(ˆpnt(t`;µmax)) φnt,1, . . . , T(ˆpnt(t`;µmax)) φnt,rnt,i−1o

and denote the projected snapshot as v`

8: end for

9: for `←1to ntdo

10: Solve the minimization problem ˜

φ`∈arg min

φ∈Rdφ

J`(φ)with

J`(φ):=

j=1

ωj

2In−T (ˆp`(tj;µmax)) φ(T(ˆp`(tj;µmax)) φ)+vj2

11: end for

12: Determine `opt ∈arg min

`∈{1,...,nt}

J`(˜

φ`)

13: Set r`opt,i :=r`opt,i−1+ 1 and r`,i :=r`,i−1for all `∈ {1, . . . , nt}\{`opt}

14: Add φ`opt,r`opt,i :=˜

φ`opt to Φi−1and thereby obtain Φi

15: end for

The major differences between Algorithm 3.3 and the POD-greedy algorithm

presented in section 2.5.2 are as follows. First, we note that the list of input

parameters in Algorithm 3.3 differs from the one in Algorithm 2.1. For ex-

ample, since the approximation ansatz (1.8) does not only involve modes and

3. Mode Identification

corresponding coefficients but also transformation operators parametrized by

paths, Algorithm 3.3 requires a transformation family as well as estimators

for the paths as additional input parameters. Especially, the paths may for

instance be estimated based on snapshot data as mentioned in Remark 3.1.2.

Another difference to the input parameters of Algorithm 2.1 is that the modes

in Algorithm 3.3 are clustered according to the different transformation opera-

tors, which is due to the special structure of the ansatz (1.8) and results in two

indices for the modes. Moreover, Pf(Rdφ)in Algorithm 3.3 denotes the set of

finite subsets of Rdφ, cf. section 2.1. Besides, in contrast to Algorithm 2.1, the

initial set of ansatz vectors in Algorithm 3.3 does not need to be orthonormal,

see also appendix B.

The next considered class of pH systems is a special case of the port-

Hamiltonian DAE systems introduced in [196] and consists of nonlinear time-

invariant systems of the form

E(x(t)) ˙x(t)=(J(x(t)) −R(x(t)))z(x(t)) + B(x(t))u(t),(5.6a)

y(t) = B(x(t))>z(x(t)) (5.6b)

for all t∈I, with associated Hamiltonian H ∈ C1(Rn)and coefficient func-

tions E, J, R ∈C(Rn,Rn,n),z∈C(Rn,Rn), and B∈C(Rn,Rn,m)satisfying

pointwise

J=−J>, R =R>≥0,and E>z=∇H.(5.7)

Finally, we also consider nonlinear time-varying port-Hamiltonian systems

of the form

E(t, x(t)) ˙x(t) + r(t, x(t)) = (J(t, x(t)) −R(t, x(t)))z(t, x(t)) + B(t, x(t))u(t),

(5.8a)

y(t) = B(t, x(t))>z(t, x(t)) (5.8b)

for all t∈I, with E, J, R ∈C(R≥0×Rn,Rn,n),r, z ∈C(R≥0×Rn,Rn), and

B∈C(R≥0×Rn,Rn,m). Here, E,J,R,r, and zare required to satisfy

142

pointwise

J=−J>, R =R>≥0, E>z=∇xH,and z>r=∂tH,(5.9)

where H ∈ C1(R≥0×Rn)is the associated Hamiltonian, cf. [196]. We note

in particular that the pH structure given by (5.8)–(5.9) generalizes the non-

linear time-invariant structure (5.6)–(5.7), but not the linear time-varying pH

structure (5.3)–(5.4), since for instance Jdoes not need to be pointwise skew-

symmetric to fulfill (5.4), see also [197, sec. 4.2]. This incompatibility issue is

addressed in more detail in section 5.2, where we also propose weaker condi-

tions than (5.9), which are especially satisfied if (5.4) holds.

From the point of view of structure-preserving model order reduction, an

important special case of (5.8)–(5.9) is when z(t, x(t)) can be written as

Q(t, x(t))x(t)with Q∈C(R≥0×Rn,Rn,n), i.e.,

E(·, x) ˙x+r(·, x)=(J(·, x)−R(·, x))Q(·, x)x+B(·, x)u, (5.10a)

y=B(·, x)>Q(·, x)x, (5.10b)

where the coefficients satisfy

E(t, x)>Q(t, x)x=∇xH(t, x), x>Q(t, x)>r(t, x) = ∂tH(t, x)

J(t, x) = −J(t, x)>, R(t, x) = R(t, x)>≥0for all (t, x)∈R≥0×Rn.

(5.11)

Remark 5.0.1 (Preservation of algebraic constraints).In this thesis, we only

focus on preserving a port-Hamiltonian structure, whereas the preservation

of algebraic constraints is not considered here. The treatment of algebraic

constraints in the context of model reduction methods for general unstructured

DAEs is for instance discussed in [29], whereas structure-preserving model

reduction for port-Hamiltonian DAEs is addressed in [27, 43, 88, 138, 197,

210, 262], see also section 1.3.2. ¨

In the remainder of this chapter, we discuss structure-preserving model re-

duction approaches for port-Hamiltonian systems based on different linear and

nonlinear model reduction ansatzes. In particular, classical linear model re-

duction is addressed in section 5.1, whereas we consider two nonlinear ap-

proximation ansatzes in sections 5.2 and 5.3. We emphasize that even the

most general ansatz considered in section 5.3 is based on a special nonlin-

earity, whereas more general nonlinear approximation ansatzes are not within

the scope of this thesis. However, in the following remark we briefly address

one possibility for ensuring a dissipation inequality for the ROM Hamiltonian

when using a general nonlinear ansatz of the form x≈f(˜x).

Remark 5.0.2 (Ensuring a dissipation inequality for the ROM Hamiltonian

via constrained residual minimization).In the following we demonstrate how

constrained residual minimization may be used to obtain a ROM based on a

general nonlinear ansatz of the form x≈f(˜x)with f∈C1(Rr,Rn), while

143

5. Structure-Preserving Model Reduction for Port-Hamiltonian Systems

ensuring that the associated ROM Hamiltonian satisfies a dissipation inequal-

ity. To this end, we consider a FOM of the form (5.1)–(5.2). We propose to

construct the ROM by choosing the time derivative ˙

˜x(t)of the reduced state

for all t∈Isuch that it solves the constrained minimization problem

min

η∈Rr

2kR(η, ˜x(t), u(t))k2s.t. f(˜x(t))>Q>R(η, ˜x(t), u(t)) = 0.(5.12)

Here, R:Rr×Rr×Rm→Rnis defined via

R(η1, η2, η3):=Ef0(η2)η1−(J−R)Qf(η2)−Bη3,

where f0denotes the derivative of f, cf. section 2.1. Thus, Ris defined such

that R(˙

˜x(t),˜x(t), u(t)) coincides with the residual at t∈I. We emphasize that

the constraint in (5.12) is chosen such that the ROM Hamiltonian ˜

H ∈ C1(Rr)

defined via

H(z):=H(f(z)) = 1

2f(z)>E>Qf(z)

satisfies the dissipation inequality

dt(˜

H◦ ˜x)(t) = f(˜x(t))>Q>Ef0(˜x(t)) ˙

˜x(t)≤u(t)>˜y(t)for all t∈I

with ˜y:=B>Qf(˜x). Furthermore, applying standard theory for optimiza-

tion problems with equality constraints to (5.12) leads to a DAE system for

the reduced state ˜xand a Lagrange multiplier λ. Moreover, the dissipation

inequality for ˜

His directly encoded in the algebraic constraint of this DAE

system. A detailed investigation of this approach including an explicit port-

Hamiltonian representation of the ROM and an analysis of its solvability will

be subject to future research. ¨

5.1. Linear Approximation Ansatz

In this section we consider structure-preserving model order reduction methods

based on linear approximation ansatzes. In particular, most of this section is

devoted to the special case of a linear time-invariant ansatz of the form x≈Vr˜x

with Vr∈Rn,r, whereas the more general case with time-dependent Vris briefly

addressed at the end of this section. In the following, we discuss corresponding

structure-preserving MOR schemes for different classes of port-Hamiltonian

FOMs. Especially, all ROMs proposed in the following are based on enforcing

the residual to be orthogonal to the span of QVr, where Qmay be time- or state-

dependent according to the corresponding FOM. In the special case Q=In

this corresponds to a classical Galerkin projection, see also Remark 2.6.4. We

begin with considering a linear time-invariant full-order model of the form

(5.1), cf. Theorem 5.1.1. The first assertion of Theorem 5.1.1 is well-known,

see for instance [197, Rem. 8.3], whereas the second assertion regarding the

144

5.1. Linear Approximation Ansatz

residual minimization property is new. Especially, since Eand Qare assumed

to be invertible and E>Qto be symmetric and positive semi-definite, we have

that E−>Q>=E−>(Q>E)E−1is symmetric and positive definite and, thus,

k·kE−>Q>in (5.15) is indeed a norm. We omit the proof of Theorem 5.1.1,

since it is a special case of the upcoming Theorem 5.1.5.

Theorem 5.1.1 (Structure-preserving MOR for (5.1) using a linear time-in-

variant approximation ansatz).Consider the port-Hamiltonian system (5.1)

with E, J, R, Q satisfying (5.2), i.e.,

J=−J>, R =R>≥0, E>Q=Q>E≥0,

and let Eand Qbe invertible. Furthermore, let Vr∈Rn,r with r∈N≤nbe a

matrix with full column rank and let

E˙

˜x(t)=(˜

J−˜

R)˜

Q˜x(t) + ˜

Bu(t),(5.13a)

˜y(t) = ˜

B>˜

Q˜x(t)(5.13b)

for all t∈Ibe a corresponding ROM with coefficient matrices

E=V>

rQ>EVr,˜

J=V>

rQ>JQVr,˜

R=V>

rQ>RQVr,

Q=Ir,˜

B=V>

rQ>B. (5.14)

Besides, we introduce the mapping R:Rr×Rr×Rm→Rnvia

R(η1, η2, η3):=EVrη1−(J−R)QVrη2−Bη3,

i.e., Ris defined such that R(˙

˜x(t),˜x(t), u(t)) coincides with the residual at

t∈I. Then, the following assertions hold.

(i) The ROM coefficient matrices satisfy

J=−˜

J>,˜

R=˜

R>≥0,and ˜

E>˜

Q=˜

Q>˜

E > 0,

i.e., the ROM (5.13) inherits the pH structure from the FOM.

(ii) The ROM (5.13) is optimal in the sense that any solution ˜xof (5.13a)

satisfies

˜x(t)∈arg min

η1∈Rr

2kR(η1,˜x(t), u(t))k2

E−>Q>(5.15)

for all t∈Iand for any input signal u:R≥0→Rmwhich admits a

solution of the ROM state equation (5.13a).

Remark 5.1.2 (Another motivation for using the E−>Q>-norm in (5.15)).In

Theorem 5.1.1 it is stated that the ROM obtained by minimizing the residual in

the E−>Q>-norm inherits the port-Hamiltonian structure from the correspond-

ing FOM. An alternative motivation for the use of the E−>Q>-norm is given by

145

5. Structure-Preserving Model Reduction for Port-Hamiltonian Systems

residual-based error bounds as addressed in Theorem 4.1.9 and Remark 4.1.10.

To this end, we first note that in the case where Eand Qare invertible and

E>Qis symmetric and positive semi-definite, the Hamiltonian coincides up

to the prefactor 1

2with the squared E>Q-norm, i.e., H(x) = 1

2kxk2

E>Qfor

all x∈Rn. If we choose this norm for measuring the error, then by similar

arguments as in the proof of Theorem 4.1.9, see also Remark 4.1.10, we obtain

the error bound

kε(t)kE>Q≤Meωt 

kε(0)kE>Q+

e−ωs R(˙

˜x(s),˜x(s), u(s))E−> Q>ds



for all t∈I, where Mand ωsatisfy eE−1(J−R)QtE>Q≤Meωt for all t∈

I. Hence, this error bound, where the error is measured in the E>Q-norm,

motivates for minimizing the residual in the E−>Q>-norm. ¨

We note that the structure of the ROM (5.13) in Theorem 5.1.1 is the

same as in (2.37) and, thus, the ROM is stable and passive. Furthermore, we

emphasize that it is essential in Theorem 5.1.1 that a weighted norm of the

residual is minimized with weighting matrix E−>Q>. The following example

illustrates that the dynamics of the resulting ROM can be completely different

when we instead use another norm for the residual minimization.

Example 5.1.3 (Standard vs. structure-preserving residual minimization).

We demonstrate the importance of the special choice of the weighted norm

in Theorem 5.1.1 by means of a simple mass-spring system presented in [278,

Ex. 2.1]. The state is given by x= [q p]>∈R2, where qdenotes the elongation

of the spring and pthe momentum of the mass. The dynamics is described by

˙x=JQx with J="0 1

−1 0#, Q ="k0

2m#,(5.16)

where k∈R>0denotes the spring constant and m∈R>0the mass. Since

Jis skew-symmetric and Qsymmetric and positive definite, this system has

a Hamiltonian structure with Hamiltonian H:R2→Rdefined via H(x):=

2x>Qx and, thus, the system is stable, cf. section 2.6.1.

With the objective of comparing different residual-minimization-based MOR

approaches, we use the special linear approximation ansatz

x≈1

√2"1

1#˜x. (5.17)

In the following, we consider two different variants of constructing a ROM

based on this ansatz: The first one is based on a Galerkin projection, which

corresponds to minimizing the residual in the Euclidean norm, cf. section 2.5.3.

The second approach is based on a Petrov–Galerkin projection as in Theo-

rem 5.1.1, which corresponds to minimizing the residual in the norm k·kQ. In

146

5.1. Linear Approximation Ansatz

the Galerkin projection case, we obtain the ROM

˜xG=1

21

2m−k˜xG,(5.18)

whereas the Petrov–Galerkin projection yields the ROM

˜xPG = 0.(5.19)

We observe that the Hamiltonian structure is preserved by the Petrov–Galerkin

projection, whereas this is in general not true for the Galerkin projection. In

fact, the Galerkin ROM is Hamiltonian if and only if k=1

2m, cf. (5.18). In that

case, Qis a multiple of the identity matrix, see (5.16), and, hence, in that case

the ROMs obtained by the Galerkin and by the Petrov–Galerkin projection

coincide. For all other combinations of the spring constant kand the mass

m, the ROMs (5.18) and (5.19) have in general different solutions. Moreover,

if 2km < 1holds, then the Galerkin ROM (5.18) is unstable and, thus, ˜xG

grows exponentially, provided that the initial value is non-zero. Consequently,

also the residual and the error associated with ˜xGgrow exponentially. On

the other hand, the solution ˜xPG of the Petrov–Galerkin ROM (5.19) remains

constant independently of kand mand, hence, the corresponding residual

and error remain bounded. Thus, this example illustrates that the choice of

the norm for the residual minimization may play an important role not only

for structure preservation, but also for the error. Furthermore, this example

shows that a ROM derived via minimizing the residual locally in time does

not necessarily yield a trajectory which leads to a small residual over the

complete time interval, see also the related discussion in Remark 4.1.4 about

the difference between instantaneous and cumulative residual minimization.

While Theorem 5.1.1 is formulated based on constant FOM coefficient ma-

trices, we emphasize that its assertions are still valid even if J,R, or Bde-

pend on time or on the state. An example for such a system is provided

in Example 5.1.4, where we consider a dissipative Hamiltonian system with

state-dependent Jand R, which arises from the modeling of a wildland fire.

However, even though the structure preservation may be achieved by a Petrov–

Galerkin projection as in Theorem 5.1.1, the state dependency of the coefficient

matrices may require to also apply a hyperreduction method to the resulting

ROM to achieve an efficient offline/online decomposition. Structure-preserving

hyperreduction methods are not within the scope of this thesis, but instead

we refer to [65] for a modification of the DEIM which allows to preserve a pH

structure, see also section 1.3.2.

Example 5.1.4 (Wildfire model).As example for a dissipative Hamiltonian

system with state-dependent Jand R, we consider a semi-discretized wild-

land fire model as used for the numerical experiments in [39]. The governing

147

5. Structure-Preserving Model Reduction for Port-Hamiltonian Systems

equations on the infinite-dimensional level and on a one-dimensional spatial

domain are given by

∂tT=k∂ξξT−w∂ξT+α(Sv(T, β)−γT),

∂tS=−ζSv(T, β)(5.20)

with unknowns T, S :I×Ω→R, spatial domain Ω=(a, b)with a∈R,

b∈R>a, time interval I= [0, tend]with tend ∈R>0,v:R×R→Rdefined via

v(T, β):=





exp −β

T,if T > 0,

0,otherwise,

and given constants k, α, β, γ, ζ ∈R>0,w∈R. Furthermore, the system (5.20)

is closed via appropriate initial conditions and periodic boundary conditions,

cf. [39]. For the physical meaning of the coefficients, unknowns, and model

equations we refer to [39] as well as to [188] where the model has been originally

proposed.

After a semi-discretization of (5.20) in space by a central finite difference

scheme using an equidistant grid with grid size h=b−a

N+1 with N∈N, we

obtain a finite-dimensional system of the form

"˙x1(t)

˙x2(t)#="kD2−wD1−αγIN+1 αV (x1(t), β)

0−ζV (x1(t), β)#"x1(t)

x2(t)#,(5.21)

where x1, x2:I→RN+1 correspond to approximations of Tand Sat the

spatial grid points ih for i= 1, . . . , N +1, while D1=−D>

1and D2=D>

2≤0

are finite difference approximations of the first and second spatial derivative,

respectively. Furthermore, the function V:RN+1 ×R→RN+1,N+1 is given by

V(x1, β):= diag (v([x1]1, β), . . . , v([x1]N+1, β)) .

By introducing η:=α

4γζ , we obtain that (5.21) is equivalent to

"IN+1 0

0ηIN+1#

|{z }

=:E

"˙x1(t)

˙x2(t)#

=

"−wD10

0 0#

|{z }

=:J1

+α

2"0V(x1(t), β)

−V(x1(t), β) 0 #

|{z }

=:J2(x1(t))



"x1(t)

x2(t)#

−

k"−D20

0 0#

|{z }

=:R1

+"αγIN+1 −α

2V(x1(t), β)

−α

2V(x1(t), β)ηζV (x1(t), β)#

|{z }

=:R2(x1(t))



"x1(t)

x2(t)#.

(5.22)

148

5.1. Linear Approximation Ansatz

The facts that Eis symmetric and positive definite, J1is skew-symmetric, J2is

pointwise skew-symmetric, and R1is symmetric and positive semi-definite fol-

low from the symmetry and definiteness properties of D1,D2, and V(x1(t), β)

as well as from the positive signs of ηand k. To investigate the definiteness

properties of R2, we consider for arbitrary vectors z= [p>q>]>∈R2(N+1)

with p, q ∈RN+1 and u∈RN+1 the product

z>R2(u)z=αγp>p+

N+1

i=1 ηζv(ui, β)q2

i−αv(ui, β)piqi

=α

N+1

i=1 γp2

i−v(ui, β)piqi+1

4γv(ui, β)q2

|{z }

=:si

.

We continue by investigating the sign of the summands si. If uiis smaller

than or equal to 0for some i∈ {1, . . . , N + 1}, then v(ui, β)is zero and siis

non-negative. If on the other hand uiis positive, we have

si=γp2

i−exp −β

ui!piqi+1

4γexp −β

ui!q2

≥γp2

i−exp −β

ui!piqi+1

4γexp −2β

ui!q2

=γp2

i−exp −β

ui!piqi+1

4γ exp −β

ui!!2

= √γpi−1

2√γexp −β

ui!qi!2

≥0.

Thus, in total we infer that R2is pointwise symmetric and positive semi-

definite and, thus, (5.22) is a dissipative Hamiltonian system with J:=J1+J2

and R:=R1+R2depending on the state. l

Next, we consider the case, where the FOM has a linear time-varying port-

Hamiltonian structure as in (5.3)–(5.4). Also in this case, we may obtain a

ROM which is pH and optimal in the sense of weighted residual minimization.

Theorem 5.1.5 (Structure-preserving MOR for (5.3) using a linear time-in-

variant approximation ansatz).Consider the port-Hamiltonian system (5.3)

with E, K, J, R, Q satisfying pointwise (5.4), i.e.,

E>Q=Q>E≥0, Q>RQ =Q>R>Q≥0,

dt(Q>E) = Q>(K−JQ)+(K−JQ)>Q

and let Eand Qbe pointwise invertible. Furthermore, let Vr∈Rn,r with

149

5. Structure-Preserving Model Reduction for Port-Hamiltonian Systems

r∈N≤nbe a matrix with full column rank and let

E(t)˙

˜x(t) + ˜

K(t)˜x(t)=(˜

J(t)−˜

R(t)) ˜

Q˜x(t) + ˜

B(t)u(t),(5.23a)

˜y(t) = ˜

B(t)>˜

Q˜x(t)(5.23b)

for all t∈Ibe a corresponding ROM with coefficients ˜

E, ˜

K, ˜

J, ˜

R:R≥0→Rr,r,

Q∈Rr,r, and ˜

B:R≥0→Rr,m defined as

E(t):=V>

rQ(t)>E(t)Vr,˜

K(t):=V>

rQ(t)>K(t)Vr,

J(t):=V>

rQ(t)>J(t)Q(t)Vr,˜

R(t):=V>

rQ(t)>R(t)Q(t)Vr,

Q:=Ir,˜

B(t):=V>

rQ(t)>B(t).

(5.24)

Besides, we introduce the residual mapping R:R≥0×Rr×Rr×Rm→Rnvia

R(t, η1, η2, η3):=E(t)Vrη1+ (K(t)−(J(t)−R(t))Q(t))Vrη2−B(t)η3.

Then, the following assertions hold.

(i) The ROM matrices satisfy pointwise

E>˜

Q=˜

Q>˜

E > 0,d

dt(˜

Q>˜

E) = ˜

Q>(˜

K−˜

J˜

Q)+(˜

K−˜

J˜

Q)>˜

and ˜

Q>˜

R˜

Q=˜

Q>˜

R>˜

Q≥0,

i.e., the ROM (5.23) inherits the pH structure from the FOM (5.3).

(ii) The ROM state equation (5.23a) is optimal in the sense that any solution

˜xsatisfies

˜x(t)∈arg min

η1∈Rr

2kR(t, η1,˜x(t), u(t))k2

E(t)−>Q(t)>(5.25)

for all t∈Iand for any input signal u:R≥0→Rmwhich admits a

solution of the ROM state equation.

Proof. (i) The fact that ˜

Q>˜

E=˜

Edefined in (5.24) is pointwise symmetric

and positive definite follows from the assumptions that E>Qis pointwise

symmetric and positive semi-definite, that Eand Qare pointwise in-

vertible, and that Vrhas full column rank. Furthermore, using the third

equation in (5.4), we obtain

dt(˜

Q>˜

E) = V>

dt(Q>E)Vr=V>

rQ>(K−JQ)+(K−JQ)>QVr

=˜

Q>(˜

K−˜

J˜

Q)+(˜

K−˜

J˜

Q)>˜

Finally, the pointwise symmetry and positive semi-definiteness of ˜

Q>˜

R˜

follow from the corresponding properties of Q>RQ.

150

5.1. Linear Approximation Ansatz

(ii) For fixed t∈I, the first-order necessary optimality condition for (5.25)

reads

rQ(t)>E(t)Vrη1+V>

rQ(t)>K(t)Vr˜x(t)

=V>

rQ(t)>(J(t)−R(t))Q(t)Vr˜x(t) + V>

rQ(t)>B(t)u(t)

and this condition is even sufficient since the corresponding Hessian

rQ(t)>E(t)Vr=˜

E(t)does not depend on η1and is symmetric and

positive definite, cf. section 2.2. Finally, comparing the first-order opti-

mality condition with (5.23a) yields the claim.

Remark 5.1.6 (Structure-preserving MOR for the linear time-varying pH struc-

ture from [25]).The structure preservation and the residual minimization

stated in Theorem 5.1.5 may also be shown when considering a linear time-

varying port-Hamiltonian system of the form (5.3)–(5.4) where Kis replaced

by EK, as considered in [25]. In this case, one obtains ˜

Eas specified in (5.24)

and ˜

Kas ˜

E−1V>

rQ>EKVr. On the other hand, an advantage of the port-

Hamiltonian formulation (5.3)–(5.4) as considered in Theorem 5.1.5 is that

the invertibility of ˜

Eis formally not required for the structure preservation,

i.e., Theorem 5.1.5(i) with ˜

Q>˜

E≥0instead of ˜

Q>˜

E > 0holds even if Eor Q

are not invertible or if Vrhas not full column rank. ¨

Stability of linear time-varying port-Hamiltonian systems of the form (5.3) is

discussed in appendix B, in particular in Theorem B.3. The following corollary

states that if the FOM satisfies the assumptions in Theorem B.3, then the

ROM state equation (5.23a) with u= 0 has a uniformly stable equilibrium

point in the origin. Hence, in this case the model reduction scheme considered

in Theorem 5.1.5 is not only structure-preserving, but also stability-preserving.

We omit the proof of Corollary 5.1.7, since it is a special case of the upcoming

Corollary 5.3.5 by choosing Vrto be constant.

Corollary 5.1.7 (Stability of (5.23a)).Let the assumptions of Theorem 5.1.5

be satisfied and let additionally E,K,J,R, and Qbe continuously differen-

tiable. Furthermore, let there exist constants ˜c1,˜c2∈R>0with

σmax(E(t)>Q(t)) ≤˜c1and σmin(E(t)>Q(t)) ≥˜c2for all t∈R≥0.(5.26)

Then, (5.23a) with u= 0 has a uniformly stable equilibrium point at 0∈Rr.

We proceed by considering structure-preserving MOR for nonlinear time-

varying pH systems of the form (5.10), cf. Theorem 5.1.8. The residual min-

imization properties of the structure-preserving ROMs presented in Theo-

rems 5.1.1 and 5.1.5 rely on the assumption that E>Qis (pointwise) positive

definite. Since E>Qcoincides with the Hessian of the Hamiltonian with respect

to the state, the (pointwise) positive definiteness of E>Qcorresponds to the

assumption that the Hamiltonian is equivalent to a squared norm of the state,

cf. Remark 5.1.2. Also when considering a nonlinear port-Hamiltonian FOM

151

5. Structure-Preserving Model Reduction for Port-Hamiltonian Systems

of the form (5.10), one could derive a residual minimization property, provided

that E>Qis pointwise symmetric and positive definite. However, in contrast to

the linear case, the matrix function E>Qdoes in general not coincide with the

Hessian of the Hamiltonian associated with the nonlinear pH system (5.10).

Consequently, assuming E>Qto be pointwise positive definite in the context

of the nonlinear pH structure (5.10)–(5.11) appears to be less natural than in

the context of linear pH systems of the forms (5.1) or (5.3). For this reason, we

do not make this assumption in Theorem 5.1.8 and only focus on the structure

preservation without addressing the question whether the ROM is optimal in

some sense. Furthermore, we omit the proof of Theorem 5.1.8, since it is a

special case of the upcoming Theorem 5.3.6 by choosing Vrto be constant.

Accordingly, we may use Corollary 5.3.7 for obtaining sufficient conditions for

the stability of the ROM state equation (5.27a) with u= 0.

Theorem 5.1.8 (Structure-preserving MOR for (5.10) using a linear time-in-

variant approximation ansatz).Consider the port-Hamiltonian system (5.10)

with E, r, J, R, Q and the associated Hamiltonian Hsatisfying (5.11), i.e.,

E(t, x)>Q(t, x)x=∇xH(t, x), x>Q(t, x)>r(t, x) = ∂tH(t, x)

J(t, x) = −J(t, x)>, R(t, x) = R(t, x)>≥0for all (t, x)∈R≥0×Rn.

Furthermore, let Vr∈Rn,r be a given matrix with r∈N≤nand let

E(t, ˜x(t)) ˙

˜x(t) + ˜

r(t, ˜x(t)) = ( ˜

J(t, ˜x(t)) −˜

R(t, ˜x(t))) ˜

Q˜x(t) + ˜

B(t, ˜x(t))u(t),

(5.27a)

˜y(t) = ˜

B(t, ˜x(t))>˜

Q˜x(t),(5.27b)

for all t∈Ibe a corresponding ROM with coefficients ˜

E, ˜

J, ˜

R:R≥0×Rr→Rr,r,

r:R≥0×Rr→Rr,˜

Q∈Rr,r, and ˜

B:R≥0×Rr→Rr,m defined as

E(t, ˜x):=V>

rQ(t, Vr˜x)>E(t, Vr˜x)Vr,

J(t, ˜x):=V>

rQ(t, Vr˜x)>J(t, Vr˜x)Q(t, Vr˜x)Vr,

R(t, ˜x):=V>

rQ(t, Vr˜x)>R(t, Vr˜x)Q(t, Vr˜x)Vr,˜

Q:=Ir

r(t, ˜x):=V>

rQ(t, Vr˜x)>r(t, Vr˜x),˜

B(t, ˜x):=V>

rQ(t, Vr˜x)>B(t, Vr˜x).

(5.28)

Moreover, we define the associated ROM Hamiltonian ˜

H:R≥0×Rr→Rvia

H(t, ˜x):=H(t, Vr˜x). Then, ˜

His continuously differentiable and the ROM

coefficients satisfy

J(t, ˜x) = −˜

J(t, ˜x)>,˜

R(t, ˜x) = ˜

R(t, ˜x)>≥0,

E(t, ˜x)>˜

Q˜x=∇˜x˜

H(t, ˜x),˜x>˜

Q>˜

r(t, ˜x) = ∂t˜

H(t, ˜x)

for all (t, ˜x)∈R≥0×Rr, i.e., the ROM (5.27) inherits the port-Hamiltonian

structure from the FOM (5.10).

152

5.1. Linear Approximation Ansatz

Remark 5.1.9 (Structure-preserving MOR for pH systems of the form (5.6)).

The structure preservation in Theorem 5.1.8 relies on the special structure of

the FOM (5.10)–(5.11), which is a special case of (5.8)–(5.9) where zmay be

factorized as z(t, x) = Q(t, x)x. Similarly, one may exploit such a factorization

in the time-invariant case (5.6). However, if such a factorization of zis not

available, we cannot readily apply the Petrov–Galerkin projection techniques

considered in this section. To still obtain structure-preserving schemes in this

case, one could for instance make use of the ideas of [65], see also section 2.6.2.

The model reduction method presented in [65] considers the special case E=In

and an extension to the case of a more general state-dependent Ewould be

desirable to treat general nonlinear pH systems of the form (5.6). However,

such an extension appears not to be straightforward, especially in the case

of a singular Ematrix. Moreover, since zis a linear function of the state in

all examples considered in this thesis, we refrain from discussing the general

nonlinear case in detail here. ¨

While classical model reduction schemes are based on ansatz functions which

do not depend on time, the most general linear approximation ansatz allows

also for a time dependency in the modes. Accordingly, we close this section by

considering an approximation ansatz of the form

x(t)≈Vr(t)˜x(t),(5.29)

where the mapping Vr:R≥0→Rn,r is assumed to be given. This ansatz

is particularly important for transport-dominated systems as considered in

the previous chapters. Especially, in the case where the paths piare known

functions of time, the approximation ansatz (1.4) corresponds in the finite-

dimensional setting to a special case of (5.29).

In the following, we restrict ourselves to structure-preserving MOR for linear

time-varying port-Hamiltonian FOMs of the form (5.3), cf. Theorem 5.1.10.

The case of linear time-invariant FOMs is obtained as a special case of this

theorem and the case of nonlinear FOMs may be treated analogously as out-

lined in section 5.3. Moreover, we may use the upcoming Corollary 5.3.5 to

obtain sufficient conditions for the stability of the ROM state equation (5.30a).

Especially, the assumption on the invertibility of ˜

Ein Corollary 5.3.5 is auto-

matically satisfied if Vrdoes not depend on the ROM state.

Theorem 5.1.10 (Structure-preserving MOR for (5.3) using a linear time–

varying approximation ansatz).Consider the port-Hamiltonian system (5.3)

with E, K, J, R, Q satisfying pointwise (5.4), i.e.,

E>Q=Q>E≥0, Q>RQ =Q>R>Q≥0,

dt(Q>E) = Q>(K−JQ)+(K−JQ)>Q

and let Eand Qbe pointwise invertible. Furthermore, let Vr∈C1(R≥0,Rn,r)

153

5. Structure-Preserving Model Reduction for Port-Hamiltonian Systems

with r∈N≤nhave pointwise full column rank and let

E(t)˙

˜x(t) + ˜

K(t)˜x(t)=(˜

J(t)−˜

R(t)) ˜

Q˜x(t) + ˜

B(t)u(t),(5.30a)

˜y(t) = ˜

B(t)>˜

Q˜x(t),(5.30b)

for all t∈Ibe a corresponding ROM with coefficients ˜

E, ˜

K, ˜

J, ˜

R:R≥0→Rr,r,

Q∈Rr,r, and ˜

B:R≥0→Rr,m defined as

E:=V>

rQ>EVr,˜

K:=V>

rQ>KVr+E˙

Vr,˜

J:=V>

rQ>JQVr,

R:=V>

rQ>RQVr,˜

Q:=Ir,˜

B:=V>

rQ>B.

(5.31)

Besides, we introduce the residual mapping R:R≥0×Rr×Rr×Rm→Rnvia

R(t, η1, η2, η3)

:=E(t)Vr(t)η1+E(t)˙

Vr(t)+(K(t)−(J(t)−R(t))Q(t))Vr(t)η2−B(t)η3.

Then, the following assertions hold.

(i) The ROM matrices satisfy pointwise

E>˜

Q=˜

Q>˜

E > 0,d

dt(˜

Q>˜

E) = ˜

Q>(˜

K−˜

J˜

Q)+(˜

K−˜

J˜

Q)>˜

and ˜

Q>˜

R˜

Q=˜

Q>˜

R>˜

Q≥0,

i.e., the ROM (5.30) inherits the port-Hamiltonian structure from the

FOM (5.3).

(ii) The ROM state equation (5.30a) is optimal in the sense that any solution

˜xsatisfies

˜x(t)∈arg min

η1∈Rr

2kR(t, η1,˜x(t), u(t))k2

E(t)−>Q(t)>(5.32)

for all t∈Iand for any input signal u:R≥0→Rmwhich admits a

solution of the ROM state equation.

Proof. (i) The pointwise symmetry and definiteness properties of ˜

Q>˜

Eand

Q>˜

R˜

Qfollow by similar arguments as in the proof of Theorem 5.1.5.

Furthermore, exploiting (5.4) and the definitions of the ROM coefficient

matrices provided in (5.31), we obtain

dt(˜

Q>˜

E) = ˙

Vr>Q>EVr+V>

dt(Q>E)Vr+V>

rQ>E˙

=˙

Vr>Q>EVr+V>

rQ>(K−JQ)+(K−JQ)>QVr+V>

rQ>E˙

=˜

Q>(˜

K−˜

J˜

Q)+(˜

K−˜

J˜

Q)>˜

154

5.2. Nonlinear Separable Approximation Ansatz

(ii) The proof follows along the lines of the proof of Theorem 5.1.5, where

the major difference is that the first-order necessary optimality condition

of the minimization problem (5.32) for fixed t∈Iis given by

Vr(t)>Q(t)>E(t)Vr(t)η1+Vr(t)>Q(t)>E(t)˙

Vr(t) + K(t)Vr(t)˜x(t)

=Vr(t)>Q(t)>(J(t)−R(t))Q(t)Vr(t)˜x(t) + Vr(t)>Q(t)>B(t)u(t).

5.2. Nonlinear Separable Approximation Ansatz

In this section we consider a nonlinear approximation ansatz of the form

x(t)≈Vr(p(t))α(t),(5.33)

where the mapping Vr:Rrp→Rn,rαis assumed to be given and where the

reduced state

˜x="α

p#(5.34)

consists of the variables p:I→Rrpand α:I→Rrαwith r:=rα+rp. Since

the ansatz (5.33) is linear in αand possibly nonlinear in p, we call this a sepa-

rable approximation ansatz, since it is the same kind of nonlinearity as in the

separable nonlinear least-squares problem (3.11). In fact, this class of nonlin-

ear ansatzes is of particular importance for this thesis since the ansatz (1.4) is

a special case of (5.33) in the finite-dimensional setting, cf. Remark 4.1.3.

We start by considering structure-preserving MOR for the case of a linear

time-invariant FOM of the form (5.1)–(5.2) with invertible Eand Q. By using

the approximation ansatz (5.33) and minimizing the residual in the weighted

E−>Q>-norm as in the previous section, we obtain the ROM

E(˜x(t)) ˙

˜x(t) = ( ˜

J(˜x(t)) −˜

R(˜x(t))) ˜

Q˜x(t) + ˜

B(˜x(t))u(t),(5.35a)

˜y(t) = ˜

B(˜x(t))>˜

Q˜x(t),(5.35b)

for all t∈I, where ˜

J, ˜

R, ˜

E:Rr→Rr,r,˜

Q∈Rr,r, and ˜

B:Rr→Rr,m are

defined as1

J(˜x):="˜

J11(p)−˜

J21(α, p)>

J21(α, p) 0 #,

J11(p):=Vr(p)>Q>JQVr(p)∈Rrα,rα,

J21(α, p):=c

Vr(p)α>Q>JQVr(p)∈Rrp,rα,

(5.36a)

1I owe special thanks to Riccardo Morandin who made me aware that the ROM may be

written in the form (5.35) by using a singular ˜

Qmatrix and by introducing a suitable

second block column in the definitions of ˜

Jand ˜

R. In an earlier version, I used a more

general definition of pH systems to show that the ROM is pH, but this is not necessary.

155

5. Structure-Preserving Model Reduction for Port-Hamiltonian Systems

R(˜x):="˜

R11(p)˜

R21(α, p)>

R21(α, p)˜

R22(α, p)#,

R11(p):=Vr(p)>Q>RQVr(p)∈Rrα,rα,

R21(α, p):=c

Vr(p)α>Q>RQVr(p)∈Rrp,rα,

R22(α, p):=c

Vr(p)α>Q>RQc

Vr(p)α∈Rrp,rp

(5.36b)

E(˜x):="˜

E11(p)˜

E12(α, p)

E12(α, p)>˜

E22(α, p)#,

E11(p):=Vr(p)>Q>EVr(p)∈Rrα,rα,

E12(α, p):=Vr(p)>Q>Ec

Vr(p)α∈Rrα,rp,

E22(α, p):=c

Vr(p)α>Q>Ec

Vr(p)α∈Rrp,rp,

(5.36c)

Q:="Irα0

0 0#,(5.36d)

B(˜x):="˜

B1(p)

B2(α, p)#,

B1(p):=Vr(p)>Q>B∈Rrα,m,

B2(α, p):=c

Vr(p)α>Q>B∈Rrp,m.

(5.36e)

Here, we use the notation for the block components of ˜xas in (5.34) and,

besides, c

Vr:Rrp→ L(Rrα,Rn,rp)is defined via

Vr(η1)(η2)η3:=V0

r(η1)(η3)η2for all (η1, η2, η3)∈Rrp×Rrα×Rrp,(5.37)

where V0

rdenotes the derivative of Vr, cf. section 2.1. Moreover, we note that

(5.35)–(5.36) is obtained by enforcing the residual to be orthogonal to the

column span of Q[Vr(p)c

Vr(p)α]. As a consequence, the ROM (5.35) is port-

Hamiltonian and optimal in the sense of residual minimization as stated in the

following.

Theorem 5.2.1 (Structure-preserving MOR for (5.1) using a separable ap-

proximation ansatz).Consider the pH system (5.1) with E, J, R, Q satisfying

(5.2), i.e.,

J=−J>, R =R>≥0, E>Q=Q>E≥0,

and let Eand Qbe invertible. Furthermore, let Vr∈C1(Rrp,Rn,rα)with

rα, rp∈Nand r:=rα+rp≤nbe given. We consider the corresponding

reduced-order model (5.35) with coefficients ˜

Q∈Rr,r,˜

E, ˜

J, ˜

R:Rr→Rr,r,

and ˜

B:Rr→Rr,m as defined in (5.36). Besides, we define the Hamiltonian

156

5.2. Nonlinear Separable Approximation Ansatz

H:Rr→Rassociated with (5.35) via

H(˜x):=1

2α>Vr(p)>E>QVr(p)α, (5.38)

where we use the notation from (5.34) for the block components of ˜x. Then,

the following assertions hold.

(i) The ROM Hamiltonian ˜

His continuously differentiable and the ROM

coefficients satisfy

J(˜x) = −˜

J(˜x)>,˜

R(˜x) = ˜

R(˜x)>≥0,˜

E(˜x)>˜

Q˜x=∇˜

H(˜x)

for all ˜x∈Rr, i.e., the ROM (5.35) has a nonlinear time-invariant port-

Hamiltonian structure as in (5.6)–(5.7).

(ii) The ROM state equation (5.35a) is optimal in the sense that any solution

˜x= [α>p>]>satisfies

˜x(t) = "˙α(t)

˙p(t)#∈arg min

(η1,η2)∈Rrα×Rrp

2kR(η1, η2, α(t), p(t), u(t))k2

E−>Q>

(5.39)

for all t∈Iand for any input signal u:R≥0→Rmwhich admits a

solution of the ROM state equation (5.35a). Here, the residual mapping

R:Rrα×Rrp×Rrα×Rrp×Rm→Rnis defined via

R(η1, η2, η3, η4, η5)

:=EVr(η4)η1+EV 0

r(η4)(η2)η3−(J−R)QVr(η4)η3−Bη5.

Proof. (i) The pointwise skew-symmetry of ˜

Jfollows from J=−J>and

(5.36a). Regarding ˜

R, we observe that it may be factorized as

R(˜x) = hVr(p)c

Vr(p)αi>Q>RQ hVr(p)c

Vr(p)αi

for all ˜x∈Rrand, thus, its pointwise symmetry and positive semi-

definiteness follow from the corresponding properties of R. For the Hamil-

tonian ˜

H, we note that it is continuously differentiable due to the con-

tinuous differentiability of Vrand we compute its partial derivatives as

∂α˜

H(˜x) = α>Vr(p)>E>QVr(p),

∂p˜

H(˜x)ζ=α>Vr(p)>E>QV 0

r(p)(ζ)α=α>Vr(p)>E>Qc

Vr(p)(α)ζ

for all (˜x, ζ)∈Rr×Rrp. Consequently, we obtain the gradient of ˜

Has

∇˜

H(˜x) = 

Vr(p)>Q>EVr(p)α

c

Vr(p)α>Q>EVr(p)α

=˜

E(˜x)>˜

Q˜x.

157

5. Structure-Preserving Model Reduction for Port-Hamiltonian Systems

(ii) For fixed t∈Iand by computing the partial derivatives of the cost func-

tion in (5.39), we obtain the first-order necessary optimality conditions

Vr(p(t))>Q>EVr(p(t))η1+Ec

Vr(p(t))(α(t))η2

=Vr(p(t))>Q>((J−R)QVr(p(t))α(t) + Bu(t)) ,

c

Vr(p(t))α(t)>Q>EVr(p(t))η1+Ec

Vr(p(t))(α(t))η2

=c

Vr(p(t))α(t)>Q>((J−R)QVr(p(t))α(t) + Bu(t)) ,

(5.40)

where we used the definition of c

Vrin (5.37). Moreover, the corresponding

Hessian is given by



Vr(p(t))>Q>EVr(p(t)) Vr(p(t))>Q>Ec

Vr(p(t))α(t)

c

Vr(p(t))α(t)>Q>EVr(p(t)) c

Vr(p(t))α(t)>Q>Ec

Vr(p(t))α(t)



|{z }

=˜

E(˜x(t))

which is independent of (η1, η2)and symmetric positive semi-definite due

to the assumptions on E>Q. Thus, the necessary conditions (5.40) are

also sufficient, cf. section 2.2. The claim then follows from observing that

(5.40) coincides with the ROM state equation (5.35a) with η1= ˙α(t)and

η2= ˙p(t).

In contrast to the setting considered in Theorem 5.1.1 from section 5.1, the

ROM Hamiltonian in Theorem 5.2.1 in general is not a quadratic function of

the ROM state, cf. (5.38). This prevents us from using similar arguments as in

section 5.1 to show that the ROM state equation (5.35a) is stable. Moreover,

whenever the first block component αof the ROM state ˜xattains the value 0,

the second block row of (5.35a) reduces to the trivial equation 0 = 0. Conse-

quently, an initial value problem associated with (5.35a), u= 0, and an initial

value (α(t0), p(t0)) with α(t0)=0is not uniquely solvable. Thus, since the

stability notions introduced in Definition 2.4.8 require unique solvability of the

initial value problem for any initial condition, the ROM state equation (5.35a)

cannot possess any stable equilibrium point in the sense of Definition 2.4.8. In

order to still obtain at least a statement about the boundedness of solutions, we

observe that the ROM Hamiltonian ˜

Hdefined in (5.38) is a quadratic function

with respect to α. Hence, under additional assumptions on Vr, we may at least

show that αremains bounded by exploiting the dissipation inequality for ˜

This is detailed in Corollary 5.2.2. Similarly as in the proof of Corollary 5.2.2,

one can show that even if the condition (5.42) on the singular values of Vris

not satisfied, we have at least the bound

kVr(p(t))α(t)k ≤ qκ(E>Q)kVr(p(t0))α(t0)kfor all t∈I(5.41)

for the approximation Vr(p)αof the FOM state, where κ(E>Q)denotes the

158

5.2. Nonlinear Separable Approximation Ansatz

condition number of E>Q, cf. section 2.1.

Corollary 5.2.2 (Boundedness of part of the state vector in (5.35a)).Let

the assumptions of Theorem 5.2.1 be satisfied and let there additionally exist

constants ˆc1,ˆc2∈R>0with

σmax(Vr(η)) ≤ˆc1and σmin(Vr(η)) ≥ˆc2for all η∈Rrp.(5.42)

Furthermore, let ˜x= [α>p>]>∈C1(I,Rrα+rp)satisfy pointwise the ROM state

equation (5.35a) with u= 0 on the time interval I= [t0, tend]with t0∈R≥0

and tend ∈R>t0. Then, there exists a constant c∈R>0which is independent

of t0and tend and satisfies

kα(t)k ≤ ckα(t0)kfor all t∈I.

Proof. By Theorem 5.2.1(i), the ROM Hamiltonian ˜

Hdefined in (5.38) is

continuously differentiable and, since also ˜xis continuously differentiable, we

infer that ˜

H◦ ˜x∈C1(I,R)holds. Furthermore, the fact that the ROM (5.35)

has a port-Hamiltonian structure as stated in Theorem 5.2.1(i), yields the

dissipation inequality

dt(˜

H◦ ˜x)(t)≤0for all t∈I,

where we have used u= 0. Consequently, we obtain

(˜

H◦ ˜x)(t)≤(˜

H◦ ˜x)(t0)for all t∈I.

Using this inequality and the bounds (5.42) as well as [34, Lem. 8.4.3] and [34,

Fact 9.13.1], we arrive at

kα(t)k2≤1

σmin(Vr(p(t)))2kVr(p(t))α(t)k2

≤1

σmin(E>Q)ˆc2

α(t)>Vr(p(t))>E>QVr(p(t))α(t)

σmin(E>Q)ˆc2

H(˜x(t)) ≤2

σmin(E>Q)ˆc2

H(˜x(t0))

≤κ(E>Q)

ˆc2

2kVr(p(t0))α(t0)k2≤κ(E>Q)σmax(Vr(p(t0)))2

ˆc2

2kα(t0)k2

≤κ(E>Q)ˆc2

ˆc2

2kα(t0)k2

for all t∈I, which yields the assertion.

Next, we focus on deriving a structure-preserving ROM using a separable

nonlinear approximation ansatz of the form (5.33) for the case where the FOM

has a linear time-varying port-Hamiltonian structure as in (5.3). Again, based

159

5. Structure-Preserving Model Reduction for Port-Hamiltonian Systems

on a suitably weighted residual minimization approach, we propose the ROM

E(t, ˜x(t)) ˙

˜x(t) + ˜

r(t, ˜x(t)) = ( ˜

J(t, ˜x(t)) −˜

R(t, ˜x(t))) ˜

Q˜x(t) + ˜

B(t, ˜x(t))u(t),

(5.43a)

˜y(t) = ˜

B(t, ˜x(t))>˜

Q˜x(t),(5.43b)

for all t∈I, where ˜

E, ˜

J, ˜

R:R≥0×Rr→Rr,r,˜

r:R≥0×Rr→Rr,˜

Q∈Rr,r,

and ˜

B:R≥0×Rr→Rr,m are defined as

E(t, ˜x):="˜

E11(t, p)˜

E12(t, α, p)

E12(t, α, p)>˜

E22(t, α, p)#,

E11(t, p):=Vr(p)>Q(t)>E(t)Vr(p)∈Rrα,rα,

E12(t, α, p):=Vr(p)>Q(t)>E(t)c

Vr(p)α∈Rrα,rp,

E22(t, α, p):=c

Vr(p)α>Q(t)>E(t)c

Vr(p)α∈Rrp,rp,

(5.44a)

J(t, ˜x):="˜

J11(t, p) 0

J21(t, α, p) 0#,

J11(t, p):=Vr(p)>Q(t)>J(t)Q(t)Vr(p)∈Rrα,rα,

J21(t, α, p):=c

Vr(p)α>Q(t)>J(t)Q(t)Vr(p)∈Rrp,rα,

(5.44b)

R(t, ˜x):="˜

R11(t, p) 0

R21(t, α, p) 0#,

R11(t, p):=Vr(p)>Q(t)>R(t)Q(t)Vr(p)∈Rrα,rα,

R21(t, α, p):=c

Vr(p)α>Q(t)>R(t)Q(t)Vr(p)∈Rrp,rα,

(5.44c)

r(t, ˜x):="˜

r1(t, ˜x)

r2(t, ˜x)#,

r1(t, ˜x):=Vr(p)>Q(t)>K(t)Vr(p)α∈Rrα,

r2(t, ˜x):=c

Vr(p)α>Q(t)>K(t)Vr(p)α∈Rrp,

(5.44d)

Q:="Irα0

0 0#,(5.44e)

B(t, ˜x):="˜

B1(t, p)

B2(t, α, p)#,

B1(t, p):=Vr(p)>Q(t)>B(t)∈Rrα,m,

B2(t, α, p):=c

Vr(p)α>Q(t)>B(t)∈Rrp,m.

(5.44f)

Here, we use the notation from (5.34) for the block components of ˜x, whereas

160

5.2. Nonlinear Separable Approximation Ansatz

Vris as defined in (5.37).

The residual minimization property which leads to the ROM state equation

(5.43a) is stated in Theorem 5.2.3(ii), whereas some structural properties of

the ROM coefficients are addressed in (i). Especially, we emphasize that the

nonlinear time-varying ROM (5.43) does in general not fit into the class of

port-Hamiltonian systems of the form (5.8)–(5.9), for instance, since ˜

Jis in

general not pointwise skew-symmetric. It appears that the reason for this

incompatibility of the pH structure (5.8)–(5.9) and the ROM (5.43) originates

from the fact that the nonlinear time-varying pH structure (5.8)–(5.9) is not a

consistent generalization of the linear time-varying pH structure (5.3)–(5.4) of

the corresponding FOM. The resulting structure of the ROM (5.43) motivates

to generalize the port-Hamiltonian structure (5.8)–(5.9) via replacing (5.9) by

the less restrictive conditions

z(t, x)>R(t, x)z(t, x)≥0,∇xH(t, x) = E(t, x)>z(t, x),

∂tH(t, x) = z(t, x)>(r(t, x)−J(t, x)z(t, x)) (5.45)

for all (t, x)∈R≥0×Rn. First, we note that these conditions are indeed less

restrictive than (5.9), since the first property in (5.45) holds especially if Ris

pointwise positive semi-definite and since the last equation in (5.45) is satisfied

if Jis pointwise skew-symmetric and ∂tH=z>rholds. However, we emphasize

that (5.8) with (5.45) is not a generalization of the port-Hamiltonian structure

introduced in [196]. Instead, one would also have to include a feedthrough

term and allow for over- and underdetermined systems to obtain a proper

generalization of the structure in [196], but this is not within the scope of this

thesis.

Nevertheless, the nonlinear time-varying pH structure (5.8) with (5.45) is a

consistent generalization of the linear time-varying pH structure (5.3)–(5.4).

In fact, the latter one is a special case of (5.8), (5.45) with z(t, x) = Q(t)x,

r(t, x) = K(t)x, and the Hamiltonian as in (5.5), which follows from the cal-

culation

z(t, x)>R(t)z(t, x) = x>Q(t)>R(t)Q(t)x≥0,

∇xH(t, x) = E(t)>Q(t)x=E(t)>z(t, x)

as well as

∂tH(t, x) = 1

2x>d

dt(E>Q)(t)x=1

2x>d

dt(Q>E)(t)x

2x>Q(t)>(K(t)−J(t)Q(t)) + (K(t)−J(t)Q(t))>Q(t)x

=x>Q(t)>(K(t)−J(t)Q(t))x=z(t, x)>(r(t, x)−J(t)z(t, x))

for all (t, x)∈R≥0×Rn. Moreover, we note that the nonlinear time-varying

pH structure (5.8) with (5.45) leads to a dissipation inequality for the corre-

161

5. Structure-Preserving Model Reduction for Port-Hamiltonian Systems

sponding Hamiltonian. More precisely, for a given solution x∈C1(I,Rn)of

(5.8a), we obtain that the function Hs:I→Rdefined via Hs(t):=H(t, x(t))

satisfies

dHs

dt(t) = ∂tH(t, x(t)) + ∇xH(t, x(t))>˙x(t)

=z(t, x(t))>(r(t, x(t)) −J(t, x(t))z(t, x(t))) + z(t, x(t))>E(t, x(t)) ˙x(t)

=−z(t, x(t))>R(t, x(t))z(t, x(t)) + z(t, x(t))>B(t, x(t))u(t)

=−z(t, x(t))>R(t, x(t))z(t, x(t)) + y(t)>u(t)≤y(t)>u(t)

for all t∈I. Finally, we note that Theorem 5.2.3(i) implies that the ROM

(5.43) has a port-Hamiltonian structure of the form (5.8) with (5.45) and,

hence, its structure implies a dissipation inequality for the associated Hamil-

tonian ˜

Hdefined in (5.46). Similarly as in Corollary 5.2.2, this allows to derive

a bound for the αcomponent of the ROM state, cf. Corollary 5.2.4. Further-

more, similarly as in (5.41) we also obtain a bound for the approximation of

the FOM state, even if the condition (5.42) on the singular values of Vris not

satisfied. Especially, we obtain the bound

kVr(p(t))α(t)k ≤ s˜c1

˜c2kVr(p(t0))α(t0)kfor all t∈I

with ˜c1and ˜c2as in Corollary 5.2.4.

Theorem 5.2.3 (Structure-preserving MOR for (5.3) using a separable ap-

proximation ansatz).Consider the pH system (5.3) with E, K, J, R, Q satisfy-

ing pointwise (5.4), i.e.,

E>Q=Q>E≥0, Q>RQ =Q>R>Q≥0,

dt(Q>E) = Q>(K−JQ)+(K−JQ)>Q

and let Eand Qbe pointwise invertible. Furthermore, let Vr∈C1(Rrp,Rn,rα)

with rα, rp∈Nand r:=rα+rp≤nbe given. Moreover, we consider

the reduced-order model (5.43) with coefficients ˜

E,˜

r,˜

J, ˜

R, ˜

Q, ˜

Bas defined in

(5.44). Besides, we define the Hamiltonian ˜

H:R≥0×Rr→Rassociated with

(5.43) via

H(t, ˜x):=1

2α>Vr(p)>E(t)>Q(t)Vr(p)α, (5.46)

where we use the notation from (5.34) for the block components of ˜x. Then,

the following assertions hold.

(i) The ROM Hamiltonian ˜

His continuously differentiable and the ROM

coefficients satisfy

Q>˜

R(t, ˜x)˜

Q=˜

Q>˜

R(t, ˜x)>˜

Q≥0,˜

E(t, ˜x)>˜

Q˜x=∇˜x˜

H(t, ˜x),

162

5.2. Nonlinear Separable Approximation Ansatz

∂t˜

H(t, ˜x) = ˜x>˜

Q>˜

r(t, ˜x)−˜

J(t, ˜x)˜

Q˜x

for all (t, ˜x)∈R≥0×Rr.

(ii) The ROM state equation (5.43a) is optimal in the sense that any solution

˜x= [α>p>]>satisfies

˜x(t) = "˙α(t)

˙p(t)#∈arg min

(η1,η2)∈Rrα×Rrp

2kR(t, η1, η2, α(t), p(t), u(t))k2

E(t)−>Q(t)>

(5.47)

for all t∈Iand for any input signal u:R≥0→Rmwhich admits a

solution of the ROM state equation (5.43a). Here, the residual mapping

R:R≥0×Rrα×Rrp×Rrα×Rrp×Rm→Rnis defined via

R(t, η1, η2, η3, η4, η5):=E(t) (Vr(η4)η1+V0

r(η4)(η2)η3)

+ (K(t)−(J(t)−R(t))Q(t))Vr(η4)η3−B(t)η5.

Proof. (i) The pointwise symmetry and positive semi-definiteness of ˜

Q>˜

R˜

with ˜

Rand ˜

Qas defined in (5.44) follow from the assumption that Q>RQ

is pointwise symmetric and positive semi-definite. For the ROM Hamil-

tonian ˜

H, we first observe that it is continuously differentiable due to the

continuous differentiability of Vrand E>Qand by following the lines of

the proof of Theorem 5.2.1 we obtain the equation

∇˜x˜

H(t, ˜x) = 

Vr(p)>

c

Vr(p)α>

Q(t)>E(t)Vr(p)α=˜

E(t, ˜x)>˜

Q˜x

for all (t, ˜x)∈R≥0×Rr. For the partial derivative of the ROM Hamilto-

nian with respect to t, we calculate

∂t˜

H(t, ˜x) = 1

2α>Vr(p)>d

dt(E>Q)(t)Vr(p)α

=α>Vr(p)>Q(t)>(K(t)−J(t)Q(t)) Vr(p)α

= ˜x>˜

Q>˜

r(t, ˜x)−˜

J(t, ˜x)˜

Q˜x

for all (t, ˜x)∈R≥0×Rr.

(ii) The proof follows along the lines of the proof of Theorem 5.2.1, where

the major difference is that the first-order necessary optimality condition

of the minimization problem (5.47) for fixed t∈Iis given by

Vr(p(t))>Q(t)>E(t)Vr(p(t))η1+c

Vr(p(t))(α(t))η2

=Vr(p(t))>Q(t)>(((J(t)−R(t))Q(t)−K(t)) Vr(p(t))α(t)−B(t)u(t)) ,

c

Vr(p(t))α(t)>Q(t)>E(t)Vr(p(t))η1+c

Vr(p(t))(α(t))η2

163

5. Structure-Preserving Model Reduction for Port-Hamiltonian Systems

=c

Vr(p(t))α(t)>Q(t)>((J(t)−R(t))Q(t)−K(t)) Vr(p(t))α(t)

−c

Vr(p(t))α(t)>Q(t)>B(t)u(t).

Corollary 5.2.4 (Boundedness of part of the state in (5.43a) with (5.44)).

Let the assumptions of Theorem 5.2.3 be satisfied and let there additionally

exist constants ˜c1,˜c2,ˆc1,ˆc2∈R>0such that the singular value bounds (5.26)

and (5.42) are satisfied. Furthermore, let ˜x= [α>p>]>∈C1(I,Rrα+rp)be a

solution of the ROM state equation (5.43a) with u= 0 and coefficients as in

(5.44) on the time interval I= [t0, tend]with t0∈R≥0and tend ∈R>t0. Then,

there exists a constant c∈R>0which is independent of t0and tend and satisfies

kα(t)k ≤ ckα(t0)kfor all t∈I.

Proof. From the continuous differentiability of ˜xand ˜

H, cf. Theorem 5.2.3(i),

we infer that the function ˜

Hs:I→Rdefined via ˜

Hs(t):=˜

H(t, ˜x(t)) is contin-

uously differentiable as well. Furthermore, as mentioned before Theorem 5.2.3,

Hssatisfies the dissipation inequality

Hs(t)≤0for all t∈I,

where we have used u= 0. Consequently, we obtain

H(t, ˜x(t)) = ˜

Hs(t)≤˜

Hs(t0) = ˜

H(t0,˜x(t0)) for all t∈I.

Using this inequality, the bounds (5.26) and (5.42), and similar arguments as

in the proof of Corollary 5.2.2, we arrive at

kα(t)k ≤ ˆc1

ˆc2v

tσmax(E(t0)>Q(t0))

σmin(E(t)>Q(t)) kα(t0)k ≤ ˆc1

ˆc2s˜c1

˜c2kα(t0)k

for all t∈I.

At the end of this section, we discuss the case of a nonlinear time-varying

FOM of the form (5.10) with coefficients satisfying (5.11). Similarly as in the

previous section, we only consider the task of structure preservation, whereas

it is in general not clear how to achieve residual minimization at the same time,

unless E>Qhappens to be pointwise symmetric and positive definite, cf. the

discussion before Theorem 5.1.8. Especially, based on enforcing the residual

to be orthogonal to the column span of

Q(t, Vr(p(t))α(t))[Vr(p(t)) c

Vr(p(t))α(t)]

for all t∈I, we propose a ROM of the form (5.43) where the coefficients are

164

5.2. Nonlinear Separable Approximation Ansatz

not specified as in (5.44), but instead given by

E(t, ˜x):="˜

E11(t, ˜x)˜

E12(t, ˜x)

E21(t, ˜x)˜

E22(t, ˜x)#,

E11(t, ˜x):=Vr(p)>Q(t, Vr(p)α)>E(t, Vr(p)α)Vr(p)∈Rrα,rα,

E12(t, ˜x):=Vr(p)>Q(t, Vr(p)α)>E(t, Vr(p)α)c

Vr(p)α∈Rrα,rp,

E21(t, ˜x):=c

Vr(p)α>Q(t, Vr(p)α)>E(t, Vr(p)α)c

Vr(p)α∈Rrα,rp,

E22(t, ˜x):=c

Vr(p)α>Q(t, Vr(p)α)>E(t, Vr(p)α)c

Vr(p)α∈Rrp,rp,

(5.48a)

J(t, ˜x):="˜

J11(t, ˜x)−˜

J21(t, ˜x)>

J21(t, ˜x) 0 #,

J11(t, ˜x):=Vr(p)>Q(t, Vr(p)α)>J(t, Vr(p)α)Q(t, Vr(p)α)Vr(p)∈Rrα,rα,

J21(t, ˜x):=c

Vr(p)α>Q(t, Vr(p)α)>J(t, Vr(p)α)Q(t, Vr(p)α)Vr(p)∈Rrp,rα,

(5.48b)

R(t, ˜x):="˜

R11(t, ˜x)˜

R21(t, ˜x)>

R21(t, ˜x)˜

R22(t, ˜x)#,

R11(t, ˜x):=Vr(p)>Q(t, Vr(p)α)>R(t, Vr(p)α)Q(t, Vr(p)α)Vr(p)∈Rrα,rα,

R21(t, ˜x):=c

Vr(p)α>Q(t, Vr(p)α)>R(t, Vr(p)α)Q(t, Vr(p)α)Vr(p)∈Rrp,rα,

R22(t, ˜x):=c

Vr(p)α>Q(t, Vr(p)α)>R(t, Vr(p)α)Q(t, Vr(p)α)c

Vr(p)α∈Rrp,rp,

(5.48c)

r(t, ˜x):="˜

r1(t, ˜x)

r2(t, ˜x)#,

r1(t, ˜x):=Vr(p)>Q(t, Vr(p)α)>r(t, Vr(p)α)∈Rrα,

r2(t, ˜x):=c

Vr(p)α>Q(t, Vr(p)α)>r(t, Vr(p)α)∈Rrp,

(5.48d)

Q:="Irα0

0 0#,(5.48e)

B(t, ˜x):="˜

B1(t, ˜x)

B2(t, ˜x)#,

B1(t, ˜x):=Vr(p)>Q(t, Vr(p)α)>B(t, Vr(p)α)∈Rrα,m,

B2(t, ˜x):=c

Vr(p)α>Q(t, Vr(p)α)>B(t, Vr(p)α)∈Rrp,m.

(5.48f)

Here, we use the notation from (5.34) for the block components of ˜x, whereas c

is as defined in (5.37). We note in particular that in contrast to the definitions

165

5. Structure-Preserving Model Reduction for Port-Hamiltonian Systems

in (5.36) and (5.44), the matrix function ˜

Edefined in (5.48a) is not necessarily

pointwise symmetric.

In Theorem 5.2.5 it is stated that the ROM (5.43) with coefficients as de-

fined in (5.48) has a nonlinear time-varying pH structure as the corresponding

FOM. Under additional assumptions on the FOM Hamiltonian and Vr, one can

show that the αcomponent of the ROM state is bounded, cf. Corollary 5.2.6.

Furthermore, following the same arguments as in the proof of Corollary 5.2.6,

we may infer the bound

kVr(p(t))α(t)k ≤ sc3

c2kVr(p(t0))α(t0)kfor all t∈I

for the approximation Vr(p)αof the FOM state, even if the singular values of

Vrare not uniformly bounded as in (5.42). Here, the constants c2and c3are

as specified in Corollary 5.2.6 and Definition 2.4.9.

Theorem 5.2.5 (Structure-preserving MOR for (5.10) using a separable ap-

proximation ansatz).Consider the pH system (5.10) with E, r, J, R, Q and the

associated Hamiltonian Hsatisfying (5.11), i.e.,

E(t, x)>Q(t, x)x=∇xH(t, x), x>Q(t, x)>r(t, x) = ∂tH(t, x)

J(t, x) = −J(t, x)>, R(t, x) = R(t, x)>≥0for all (t, x)∈R≥0×Rn.

Furthermore, let Vr∈C1(Rrp,Rn,rα)with rα, rp∈Nand r:=rα+rp≤nbe

given. Moreover, we consider the reduced-order model (5.43) with coefficients

E,˜

r,˜

J, ˜

R, ˜

Q, ˜

Bas defined in (5.48) and associated Hamiltonian ˜

H:R≥0×Rr→

Rdefined via ˜

H(t, ˜x):=H(t, Vr(p)α). Then, ˜

His continuously differentiable

and the ROM coefficients satisfy

J(t, ˜x) = −˜

J(t, ˜x)>,˜

R(t, ˜x) = ˜

R(t, ˜x)>≥0,˜

E(t, ˜x)>˜

Q˜x=∇˜x˜

H(t, ˜x),

∂t˜

H(t, ˜x) = ˜x>˜

Q>˜

r(t, ˜x)for all (t, ˜x)∈R≥0×Rr,

i.e., the ROM inherits the port-Hamiltonian structure from the FOM.

Proof. The properties of ˜

Jand ˜

Rfollow by similar arguments as in the proof

of Theorem 5.2.1. To check the properties associated with ˜

H, we first note

that ˜

His continuously differentiable due to the continuous differentiability of

Hand Vr. Furthermore, its partial derivatives with respect to the ROM state

variables are given by

∂α˜

H(t, ˜x) = ∂xH(t, Vr(p)α)Vr(p),

∂p˜

H(t, ˜x)ζ=∂xH(t, Vr(p)α)V0

r(p)(ζ)α=∂xH(t, Vr(p)α)c

Vr(p)(α)ζ

for all (t, ˜x, ζ)∈R≥0×Rr×Rrpand, hence, we obtain

∇˜x˜

H(t, ˜x) = h∂α˜

H(t, ˜x)∂p˜

H(t, ˜x)i>=hVr(p)c

Vr(p)αi>∇xH(t, Vr(p)α)

166

5.3. Nonlinear Factorizable Approximation Ansatz

=hVr(p)c

Vr(p)αi>E(t, Vr(p)α)>Q(t, Vr(p)α)Vr(p)α=˜

E(t, ˜x)>˜

Q˜x

for all (t, ˜x)∈R≥0×Rr. Finally, we compute the partial derivative of ˜

Hwith

respect to tand arrive at

∂t˜

H(t, ˜x) = ∂tH(t, Vr(p)α) = α>Vr(p)>Q(t, Vr(p)α)>r(t, Vr(p)α) = ˜x>˜

Q>˜

r(t, ˜x)

for all (t, ˜x)∈R≥0×Rr.

Corollary 5.2.6 (Boundedness of part of the state in (5.43a) with (5.48)).Let

the assumptions of Theorem 5.2.5 be satisfied and let there additionally exist

constants ˆc1,ˆc2∈R>0such that the singular value bounds in (5.42) hold. Fur-

thermore, let the FOM Hamiltonian Hsatisfy condition (ii) in Definition 2.4.9

with V=H. In addition, let ˜x= [α>p>]>∈C1(I,Rrα+rp)be a solution of

the ROM state equation (5.43a) with u= 0 and coefficients as in (5.48) on the

time interval I= [t0, tend]with t0∈R≥0and tend ∈R>t0. Then, there exists a

constant c∈R>0which is independent of t0and tend and satisfies

kα(t)k ≤ ckα(t0)kfor all t∈I.

Proof. By using similar arguments as in the proof of Corollary 5.2.4, we infer

that the function ˜

Hs:I→Rdefined via ˜

Hs(t):=˜

H(t, ˜x(t)) is continuously

differentiable and satisfies

H(t, ˜x(t)) = ˜

Hs(t)≤˜

Hs(t0) = ˜

H(t0,˜x(t0)) for all t∈I.

Using this inequality, the singular value bounds (5.42), the assumption that

Hsatisfies condition (ii) in Definition 2.4.9 with constants c2, c3∈R>0, and

similar arguments as in the proof of Corollary 5.2.2, we arrive at

kα(t)k2≤1

σmin(Vr(p(t)))2kVr(p(t))α(t)k2≤1

c2ˆc2

2H(t, Vr(p(t))α(t))

c2ˆc2

H(t, ˜x(t)) ≤1

c2ˆc2

H(t0,˜x(t0)) = 1

c2ˆc2

2H(t0, Vr(p(t0))α(t0))

≤c3

c2ˆc2

2kVr(p(t0))α(t0)k2≤c3σmax(Vr(p(t0)))2

c2ˆc2

2kα(t0)k2

≤c3ˆc2

c2ˆc2

2kα(t0)k2,

for all t∈I, which yields the assertion.

5.3. Nonlinear Factorizable Approximation Ansatz

In section 5.2, we have considered a special nonlinear approximation ansatz

which is given by a linear combination of state-dependent ansatz vectors corre-

167

5. Structure-Preserving Model Reduction for Port-Hamiltonian Systems

sponding to the columns of Vr. In particular, this ansatz includes a splitting of

the ROM state into two block components αand p, where the αcomponent rep-

resents the coefficients of the linear combination, whereas the pcomponent is

used for parametrizing the ansatz vectors. As demonstrated in Theorems 5.2.1

and 5.2.3, this special structure still allows to obtain structure-preserving and

residual-minimizing ROMs. In this section, we consider a more general non-

linear approximation ansatz, which is based on a linear combination of ansatz

vectors which depend on time and on the ROM state, but in contrast to the

previous section we do not assume a separation of the ROM state variables.

Thus, we consider a so-called factorizable approximation ansatz of the form

x(t)≈Vr(t, ˜x(t))˜x(t)(5.49)

with Vr:R≥0×Rr→Rn,r. Unfortunately, an extension of the residual mini-

mization property as in Theorems 5.2.1 and 5.2.3 to the more general ansatz

(5.49) appears to be challenging and is left for future research. Instead, we

only focus on the structure preservation in the following. In particular, this is

similarly as in section 5.1 achieved by enforcing the residual to be orthogonal

to the span of QVr, where Qmay be time- or state-dependent according to

the corresponding FOM. For the special case of a linear time-invariant port-

Hamiltonian FOM, a corresponding statement is provided in Theorem 5.3.1.

Its proof is omitted since it is a special case of the upcoming Theorem 5.3.6,

as the linear time-invariant pH structure (5.1)–(5.2) is a special case of the

nonlinear time-varying pH structure (5.10)–(5.11).

Theorem 5.3.1 (Structure-preserving MOR for (5.1) using a factorizable ap-

proximation ansatz).Consider the pH system (5.1) with E, J, R, Q satisfying

(5.2), i.e.,

J=−J>, R =R>≥0, E>Q=Q>E≥0.

Furthermore, for given Vr∈C1(R≥0×Rr,Rn,r)we consider the ROM

E(t, ˜x(t)) ˙

˜x(t) + ˜

K(t, ˜x)˜x(t)=(˜

J(t, ˜x(t)) −˜

R(t, ˜x(t))) ˜

Q˜x(t) + ˜

B(t, ˜x(t))u(t),

(5.50a)

˜y(t) = ˜

B(t, ˜x(t))>˜

Q˜x(t),(5.50b)

for all t∈I, with coefficients ˜

E, ˜

K, ˜

J, ˜

R:R≥0×Rr→Rr,r,˜

Q∈Rr,r, and

B:R≥0×Rr→Rr,m defined via

E(t, ˜x):=Vr(t, ˜x)>Q>EVr(t, ˜x) + Vr(t, ˜x)>Q>Ec

Vr(t, ˜x)˜x, (5.51a)

K(t, ˜x):=Vr(t, ˜x)>Q>E∂tVr(t, ˜x),˜

J(t, ˜x):=Vr(t, ˜x)>Q>JQVr(t, ˜x),

(5.51b)

R(t, ˜x):=Vr(t, ˜x)>Q>RQVr(t, ˜x),˜

Q=Ir,˜

B(t, ˜x):=Vr(t, ˜x)>Q>B.

(5.51c)

168

5.3. Nonlinear Factorizable Approximation Ansatz

Here, c

Vr:R≥0×Rr→ L(Rr,Rn,r)is defined via

Vr(t, η1)(η2)η3:=∂˜xVr(t, η1)(η3)η2(5.52)

for all (t, η1, η2, η3)∈R≥0×Rr×Rr×Rr. Moreover, we define the associated

Hamiltonian ˜

H ∈ C1(R≥0×Rr)via ˜

H(t, ˜x):=1

2˜x>Vr(t, ˜x)>E>QVr(t, ˜x)˜x.

Then, ˜

His continuously differentiable and the ROM coefficients satisfy

∂t˜

H(t, ˜x) = ˜x>˜

Q>˜

K(t, ˜x)˜x, ∇˜x˜

H(t, ˜x) = ˜

E(t, ˜x)>˜

Q˜x,

J(t, ˜x) = −˜

J(t, ˜x)>,˜

R(t, ˜x) = ˜

R(t, ˜x)>≥0

for all (t, ˜x)∈R≥0×Rr, i.e., the ROM (5.50)–(5.51) has a nonlinear port-

Hamiltonian structure as in (5.10)–(5.11).

In section 5.1, we have argued that the state equation (5.13a) of the port-

Hamiltonian ROM considered in Theorem 5.1.1 is stable by exploiting that

Vrhas full column rank and that Eand Qare invertible and satisfy E>Q=

Q>E≥0. Especially, these assumptions allow to infer that ˜

Edefined in

(5.14) is symmetric and positive definite. However, we may not use the same

arguments for the ROM (5.50) presented in Theorem 5.3.1. This is mainly

due to the second term in the definition of ˜

Ein (5.51), which results in ˜

not necessarily being pointwise positive definite even in the case where E>Q

is positive definite and Vrhas pointwise full column rank. Nevertheless, if

Eis at least invertible, we may obtain a stability result for the ROM (5.50)

as detailed in Corollary 5.3.2. Its proof is omitted, since it is a special case

of Corollary 5.3.7, where especially the assumption that the origin 0∈Rn

is an equilibrium point of the FOM with u= 0 is automatically satisfied.

Moreover, the invertibility of Eand Qimplies together with (5.2) that E>Q

is positive definite and, hence, the FOM Hamiltonian satisfies condition (ii) in

Definition 2.4.9.

Corollary 5.3.2 (Stability of (5.50a) with coefficients as in (5.51)).Let the

assumptions of Theorem 5.3.1 be satisfied and let additionally Eand Qbe

invertible. Furthermore, let Vrbe twice continuously differentiable and let Vr,

E, and Qbe such that ˜

Eas defined in (5.51) is pointwise invertible, cf. Re-

mark 5.3.3. Besides, let there exist constants ˆc1,ˆc2∈R>0with

σmax(Vr(t, ˜x)) ≤ˆc1and σmin(Vr(t, ˜x)) ≥ˆc2for all (t, ˜x)∈R≥0×Rr.

(5.53)

Then, the ROM state equation (5.50a) with u= 0 and coefficients as in (5.51)

has a uniformly stable equilibrium point at 0∈Rr.

Remark 5.3.3 (Invertibility of ˜

E).In the special case where Vris constant

with respect to its second argument, we have that ˜

Eas defined in (5.51)

is pointwise symmetric and positive definite, provided that E>Qis positive

definite and Vrhas pointwise full column rank, cf. section 5.1. However, as

169

5. Structure-Preserving Model Reduction for Port-Hamiltonian Systems

mentioned before Corollary 5.3.2, this is not necessarily true if ∂˜xVrdoes not

vanish. Nevertheless, if E>Qis positive definite and Vrhas pointwise full

column rank, then we have at least that ˜

E(t, 0) is invertible for all t∈R≥0,

since the second term on the right-hand side of (5.51a) vanishes for ˜x= 0. To

derive general conditions for the pointwise invertibility of ˜

E, we observe that

it may be factorized as

E(t, ˜x):=Vr(t, ˜x)>Q>EVr(t, ˜x) + c

Vr(t, ˜x)˜xfor all (t, ˜x)∈R≥0×Rr.

Thus, using [34, Fact. 2.10.14], we infer

rank ˜

E(t, ˜x)

= rank(Vr(t, ˜x)) −dim ker (Vr(t, ˜x)) ∩im Vr(t, ˜x) + c

Vr(t, ˜x)˜x>E>Q,

= rank Vr(t, ˜x) + c

Vr(t, ˜x)˜x

−dim ker Vr(t, ˜x)>Q>E∩im Vr(t, ˜x) + c

Vr(t, ˜x)˜x

for all (t, ˜x)∈R≥0×Rr. Consequently, ˜

Eis pointwise invertible if and only if

for all (t, ˜x)∈R≥0×Rrwe have

(i) Vr(t, ˜x)has full column rank and

(ii) ker (Vr(t, ˜x)) ∩im Vr(t, ˜x) + c

Vr(t, ˜x)˜x>E>Q={0}

or, equivalently,

(a) Vr(t, ˜x) + c

Vr(t, ˜x)˜xhas full column rank and

(b) ker Vr(t, ˜x)>Q>E∩im Vr(t, ˜x) + c

Vr(t, ˜x)˜x={0}.

Especially, we note that these conditions are rather general and do not require

Eor Qto be invertible or E>Qto be symmetric or positive semi-definite. In

the special case where E>Qis symmetric and positive definite, we have for

instance that (b) is equivalent to the condition that im(Vr(t, ˜x) + c

Vr(t, ˜x)˜x)

has a trivial intersection with the orthogonal complement of im(Vr(t, ˜x)) with

respect to the weighted E>Qinner product. In accordance with the beginning

of this remark, this is for instance satisfied if c

Vror ∂˜xVrvanishes. Finally, we

note that similar arguments allow to obtain analogous conditions for the case,

where Eand Qdepend on tand ˜x.¨

We continue by considering the case of a linear time-varying pH full-order

model of the form (5.3) and present a corresponding port-Hamiltonian ROM

in Theorem 5.3.4. In particular, we emphasize that the ROM coefficients

defined in (5.54) do in general not satisfy the conditions (5.11), but instead

less restrictive conditions of the form (5.45), cf. the discussion after (5.44).

Under additional assumptions including the pointwise invertibility of ˜

E, one

170

5.3. Nonlinear Factorizable Approximation Ansatz

can show that the ROM state equation with u= 0 has a uniformly stable

equilibrium point in the origin, cf. Corollary 5.3.5.

Theorem 5.3.4 (Structure-preserving MOR for (5.3) using a factorizable ap-

proximation ansatz).Consider the pH system (5.3) with E, K, J, R, Q satisfy-

ing pointwise (5.4), i.e.,

E>Q=Q>E≥0, Q>RQ =Q>R>Q≥0,

dt(Q>E) = Q>(K−JQ)+(K−JQ)>Q.

Furthermore, for given Vr∈C1(R≥0×Rr,Rn,r)we consider a ROM of the form

(5.50) with coefficient matrix functions ˜

E, ˜

K, ˜

J, ˜

R:R≥0×Rr→Rr,r,˜

Q∈Rr,r,

and ˜

B:R≥0×Rr→Rr,m defined via

E(t, ˜x):=Vr(t, ˜x)>Q(t)>E(t)Vr(t, ˜x) + c

Vr(t, ˜x)˜x,

K(t, ˜x):=Vr(t, ˜x)>Q(t)>(E(t)∂tVr(t, ˜x) + K(t)Vr(t, ˜x)) ,

J(t, ˜x):=Vr(t, ˜x)>Q(t)>J(t)Q(t)Vr(t, ˜x),

Q:=Ir,

R(t, ˜x):=Vr(t, ˜x)>Q(t)>R(t)Q(t)Vr(t, ˜x),

B(t, ˜x):=Vr(t, ˜x)>Q(t)>B(t)

(5.54)

with c

Vras defined in (5.52). Moreover, we introduce the ROM Hamiltonian

H:R≥0×Rr→Rvia

H(t, ˜x):=H(t, Vr(t, ˜x)˜x) = 1

2˜x>Vr(t, ˜x)>E(t)>Q(t)Vr(t, ˜x)˜x.

Then, ˜

His continuously differentiable and the ROM coefficients satisfy

∂t˜

H(t, ˜x) = ˜x>˜

Q>˜

K(t, ˜x)−˜

J(t, ˜x)˜

Q˜x,

∇˜x˜

H(t, ˜x) = ˜

E(t, ˜x)>˜

Q˜x,

Q>˜

R(t, ˜x)˜

Q=˜

Q>˜

R(t, ˜x)>˜

Q≥0

(5.55)

for all (t, ˜x)∈R≥0×Rr, i.e., the ROM (5.50) with coefficients as in (5.54)

has a nonlinear time-varying port-Hamiltonian structure as in (5.8),(5.45).

Proof. The fact that ˜

Q>˜

R˜

Q=˜

Rdefined in (5.54) is pointwise symmetric

and positive semi-definite follows from the corresponding properties of Q>RQ.

Furthermore, ˜

His continuously differentiable due to the continuous differen-

tiability of E>Qand Vr. Moreover, the partial derivative of ˜

Hwith respect to

˜xis given by

∂˜x˜

H(t, ˜x)ζ=∂xH(t, Vr(t, ˜x)˜x) (Vr(t, ˜x)ζ+∂˜xVr(t, ˜x)(ζ)˜x)

171

5. Structure-Preserving Model Reduction for Port-Hamiltonian Systems

= ˜x>Vr(t, ˜x)>Q(t)>E(t)Vr(t, ˜x) + c

Vr(t, ˜x)˜xζ

for all (t, ˜x, ζ)∈R≥0×Rr×Rrand, hence, we obtain

∇˜x˜

H(t, ˜x) = Vr(t, ˜x) + c

Vr(t, ˜x)˜x>E(t)>Q(t)Vr(t, ˜x)˜x=˜

E(t, ˜x)>˜

Q˜x

for all (t, ˜x)∈R≥0×Rr. Finally, for the partial derivative of ˜

Hwith respect

to t, we compute

∂t˜

H(t, ˜x) = ∂tH(t, Vr(t, ˜x)˜x) + ∂xH(t, Vr(t, ˜x)˜x)∂tVr(t, ˜x)˜x

2˜x>Vr(t, ˜x)>d

dt(E>Q)(t)Vr(t, ˜x)˜x+ ˜x>Vr(t, ˜x)>Q(t)>E(t)∂tVr(t, ˜x)˜x

= ˜x>Vr(t, ˜x)>Q(t)>((K(t)−J(t)Q(t)) Vr(t, ˜x) + E(t)∂tVr(t, ˜x)) ˜x

= ˜x>˜

Q>˜

K(t, ˜x)−˜

J(t, ˜x)˜

Q˜x

for all (t, ˜x)∈R≥0×Rr.

Corollary 5.3.5 (Stability of (5.50a) with coefficients as in (5.54)).Let the

assumptions of Theorem 5.3.4 be satisfied and let there additionally exist con-

stants ˜c1,˜c2,ˆc1,ˆc2∈R>0such that the singular value bounds (5.26) and (5.53)

hold. Furthermore, let E, Q, J, R, K be continuously differentiable, Vrbe twice

continuously differentiable, and Vr,E, and Qbe such that ˜

Eas defined in

(5.54) is pointwise invertible, cf. Remark 5.3.3. Then, the ROM state equa-

tion (5.50a) with u= 0 and coefficients as in (5.54) has a uniformly stable

equilibrium point at 0∈Rr.

Proof. By Theorem 5.3.4 and due to the differentiability assumptions on Vrand

on the FOM coefficients, we obtain that ˜

J,˜

R,˜

K,˜

H, and ˜

Eare continuously

differentiable. Furthermore, due to (5.26) and (5.53) and by using similar

arguments as in the proof of Corollary 5.2.2, we obtain the bounds

H(t, ˜x) = 1

2˜x>Vr(t, ˜x)>E(t)>Q(t)Vr(t, ˜x)˜x≤1

2σmax(E(t)>Q(t)) kVr(t, ˜x)˜xk2

≤1

2σmax(E(t)>Q(t))σmax(Vr(t, ˜x))2k˜xk2≤ˆc2

1˜c1

2k˜xk2=:c3k˜xk2

as well as

H(t, ˜x)≥1

2σmin(E(t)>Q(t)) kVr(t, ˜x)˜xk2

≥1

2σmin(E(t)>Q(t))σmin(Vr(t, ˜x))2k˜xk2≥ˆc2

2˜c2

2k˜xk2=:c2k˜xk2

for all (t, ˜x)∈R≥0×Rr. Thus, ˜

Hsatisfies condition (ii) from Definition 2.4.9.

Moreover, 0∈Rris an equilibrium point of (5.50a) with u= 0 and the claim

follows then by using the same arguments as in the proof of Theorem 2.6.3.

172

5.3. Nonlinear Factorizable Approximation Ansatz

In fact, we note that the only reason which prevents us from directly applying

Theorem 2.6.3 is that ˜

Jis in general not pointwise skew-symmetric, but the

proof of Theorem 2.6.3 works analogously by exploiting (5.55) instead of (2.31).

We close this chapter by considering structure-preserving MOR for nonlin-

ear pH systems of the form (5.10) using a factorizable approximation ansatz

of the form (5.49). A corresponding port-Hamiltonian ROM is presented in

the following theorem and under some additional assumptions including the

pointwise invertibility of ˜

E, we may infer that the ROM state equation (5.56a)

with vanishing input has a uniformly stable equilibrium point at 0, see Corol-

lary 5.3.7.

Theorem 5.3.6 (Structure-preserving MOR for (5.10) using a factorizable

approximation ansatz).Consider the pH system (5.10) with E, r, J, R, Q and

the associated Hamiltonian Hsatisfying (5.11), i.e.,

E(t, x)>Q(t, x)x=∇xH(t, x), x>Q(t, x)>r(t, x) = ∂tH(t, x)

J(t, x) = −J(t, x)>, R(t, x) = R(t, x)>≥0for all (t, x)∈R≥0×Rn.

Furthermore, let Vr:R≥0×Rr→Rn,r with r∈N≤nbe continuously differen-

tiable and let

E(t, ˜x(t)) ˙

˜x(t) + ˜

r(t, ˜x(t)) = ( ˜

J(t, ˜x(t)) −˜

R(t, ˜x(t))) ˜

Q˜x(t) + ˜

B(t, ˜x(t))u(t),

(5.56a)

˜y(t) = ˜

B(t, ˜x(t))>˜

Q˜x(t),(5.56b)

for all t∈Ibe a corresponding ROM with coefficient functions ˜

E, ˜

J, ˜

R:R≥0×

Rr→Rr,r,˜

r:R≥0×Rr→Rr,˜

Q∈Rr,r, and ˜

B:R≥0×Rr→Rr,m defined as

E(t, ˜x):=Vr(t, ˜x)>Q(t, Vr(t, ˜x)˜x)>E(t, Vr(t, ˜x)˜x)Vr(t, ˜x) + c

Vr(t, ˜x)˜x,

J(t, ˜x):=Vr(t, ˜x)>Q(t, Vr(t, ˜x)˜x)>J(t, Vr(t, ˜x)˜x)Q(t, Vr(t, ˜x)˜x)Vr(t, ˜x),

R(t, ˜x):=Vr(t, ˜x)>Q(t, Vr(t, ˜x)˜x)>R(t, Vr(t, ˜x)˜x)Q(t, Vr(t, ˜x)˜x)Vr(t, ˜x),

r(t, ˜x):=Vr(t, ˜x)>Q(t, Vr(t, ˜x)˜x)>(r(t, Vr(t, ˜x)˜x) + E(t, Vr(t, ˜x)˜x)∂tVr(t, ˜x)˜x),

Q:=Ir,

B(t, ˜x):=Vr(t, ˜x)>Q(t, Vr(t, ˜x)˜x)>B(t, Vr(t, ˜x)˜x).

(5.57)

Here, c

Vris defined as in (5.52) and, moreover, we introduce the ROM Hamilto-

nian ˜

H:R≥0×Rr→Rvia ˜

H(t, ˜x):=H(t, Vr(t, ˜x)˜x). Then, ˜

His continuously

differentiable and the ROM coefficients satisfy

J(t, ˜x) = −˜

J(t, ˜x)>,˜

R(t, ˜x) = ˜

R(t, ˜x)>≥0,

E(t, ˜x)>˜

Q˜x=∇˜x˜

H(t, ˜x),˜x>˜

Q>˜

r(t, ˜x) = ∂t˜

H(t, ˜x),

173

5. Structure-Preserving Model Reduction for Port-Hamiltonian Systems

for all (t, ˜x)∈R≥0×Rr, i.e., the ROM (5.56) inherits the port-Hamiltonian

structure from the FOM (5.10).

Proof. The properties of ˜

Jand ˜

Rfollow from the corresponding properties of

Jand R, respectively. Furthermore, ˜

His continuously differentiable due to the

continuous differentiability of Hand Vr. Moreover, the relations concerning

the partial derivatives of ˜

Hfollow from

∂t˜

H(t, ˜x) = ∂tH(t, Vr(t, ˜x)˜x) + ∂xH(t, Vr(t, ˜x)˜x)∂tVr(t, ˜x)˜x

= ˜x>Vr(t, ˜x)>Q(t, Vr(t, ˜x)˜x)>(r(t, Vr(t, ˜x)˜x) + E(t, Vr(t, ˜x)˜x)∂tVr(t, ˜x)˜x)

= ˜x>˜

Q>˜

r(t, ˜x)

and

∂˜x˜

H(t, ˜x)ζ=∂xH(t, Vr(t, ˜x)˜x) (Vr(t, ˜x)ζ+∂˜xVr(t, ˜x)(ζ)˜x)

= ˜x>Vr(t, ˜x)>Q(t, Vr(t, ˜x)˜x)>E(t, Vr(t, ˜x)˜x)Vr(t, ˜x) + c

Vr(t, ˜x)˜xζ

=˜

E(t, ˜x)>˜

Q˜x>ζ

for all (t, ˜x, ζ)∈R≥0×Rr×Rr.

Corollary 5.3.7 (Stability of the ROM state equation (5.56a)).Let the as-

sumptions of Theorem 5.3.6 be satisfied and let additionally E,J,R,Q, and r

be continuously differentiable and Vrbe twice continuously differentiable. Fur-

thermore, let the FOM Hamiltonian Hsatisfy condition (ii) in Definition 2.4.9

with V=Hand let there exist constants ˆc1,ˆc2∈R>0such that the singular

values of Vrsatisfy (5.53). Besides, let E,Q, and Vrbe such that ˜

Eas defined

in (5.57) is pointwise invertible, cf. Remark 5.3.3, and let 0∈Rnbe an equi-

librium point of the FOM state equation (5.10a) with u= 0. Then, the ROM

state equation (5.56a) with u= 0 has a uniformly stable equilibrium point at

0∈Rr.

Proof. First, we note that the differentiability assumptions on the FOM co-

efficient functions and on Vrimply that ˜

E,˜

J,˜

R, and ˜

ras defined in (5.57)

are continuously differentiable. Furthermore, since Hsatisfies condition (ii) in

Definition 2.4.9 with constants c2, c3∈R>0and since the singular values of Vr

are bounded as in (5.53), we infer that also the ROM Hamiltonian ˜

Hsatisfies

condition (ii) in Definition 2.4.9, which follows from the calculation

H(t, ˜x) = H(t, Vr(t, ˜x)˜x)≤c3kVr(t, ˜x)˜xk2≤c3σmax(Vr(t, ˜x))2k˜xk2≤c3ˆc2

1k˜xk2

and ˜

H(t, ˜x)≥c2kVr(t, ˜x)˜xk2≥c2σmin(Vr(t, ˜x))2k˜xk2≥c2ˆc2

2k˜xk2

for all (t, ˜x)∈R≥0×Rr. In addition, the fact that 0∈Rnis an equilibrium

point of (5.10a) with u= 0 implies r(t, 0) = 0 for all t∈R≥0. Thus, we also

have ˜

r(t, 0) = 0 for all t∈R≥0and the claim follows from Theorem 2.6.3.

174

6. Numerical Examples

In this chapter, we illustrate some of the methods discussed in the previous

chapters by means of numerical test cases with a one-dimensional spatial do-

main Ω. In particular, we start in section 6.1 by considering the linear wave

equation with periodic boundary conditions as in Example 1.2.3. This test

case allows for an explicit analytical solution on the PDE level, cf. (1.15), and

this solution may be described by two shifted modes. Moreover, the modes

are shifted in opposite directions and thus we use an approximation ansatz of

the form (1.4) with two distinct path variables.

In section 6.2 we consider a linear advection–diffusion problem with an inho-

mogeneous Robin condition at the left boundary and a homogeneous Neumann

condition at the right boundary. Especially, the left boundary condition re-

sults in a wave entering the computational domain after a certain time period.

In order to reflect this entering wave in the approximation ansatz, we use a

discretized version of the extended domain shift operator introduced in sec-

tion 3.3.1, cf. Example 3.3.5. Furthermore, we demonstrate the performance

of the greedy algorithm presented in section 3.2.

Finally, we consider a nonlinear test case in section 6.3 given by a nonlinear

reaction–diffusion equation with mixed Dirichlet/Neumann boundary condi-

tions. In particular, the nonlinearity originates from the reaction term and

an efficient ROM evaluation is achieved by using the hyperreduction method

introduced in section 4.3.2.

For all three test cases, the space discretization is carried out by a standard

Galerkin finite element scheme based on an equidistant mesh and piecewise

linear basis functions. Moreover, the initial value of the semi-discrete system

is obtained by evaluating the corresponding PDE initial value at the FEM

grid points. For the time discretization, we use the implicit midpoint rule

and an equidistant time grid. Since each of the considered FOMs allows for

a port-Hamiltonian representation with quadratic Hamiltonian, the implicit

midpoint rule guarantees that a dissipation inequality is satisfied after time

discretization, cf. section 2.6.1. Besides, whenever a nonlinear equation system

needs to be solved during the simulation of the FOM or the ROM, we use

MATLAB’s fsolve function with default settings, except for the numerical

experiments in section 6.3 where the parameter OptimalityTolerance is set

to 10−8.

In the offline phase, the determination of the modes follows the approach

presented in section 3.1. More precisely, we solve minimization problems of the

form (3.9)–(3.10), where the time weights are chosen based on the composite

175

6. Numerical Examples

trapezoidal rule as ω1=ωq=1

2,ω2=. . . =ωq−1= 1 and the spatial weighting

matrix Wis chosen as the corresponding FEM mass matrix. Unless stated oth-

erwise, we employ the variable projection approach outlined in section 3.1.2 for

reducing the optimization problems and we use the non-commercial software

GRANSO (version 1.6.4) as optimization solver, cf. [75].

The ROMs are constructed based on residual minimization as outlined in

section 4.1, where we use the finite-dimensional systems obtained after space

discretization as full-order models. However, in contrast to section 4.1, we

use a weighted norm for the residual as in Theorem 5.2.1(ii) to preserve the

port-Hamiltonian structure of the FOM. The time integration of the ROMs is

carried out using either the implicit midpoint rule or a time integration scheme

based on discrete gradient pairs. The latter is detailed in appendix C.2 and

guarantees that a dissipation inequality is satisfied after time discretization,

even if the Hamiltonian is not quadratic.

Unless stated otherwise, the error values specified in the following sections

correspond to the relative error in a discretized L2(I×Ω) norm. To this end,

we use the same discretization as for the mode determination, i.e., the relative

error between the FOM state x:I→Rnand a corresponding approximation

ˆx:I→Rnis computed as

tPq

i=1 ωikx(ti)−ˆx(ti)k2

i=1 ωikx(ti)k2

where t1, . . . , tq∈Iare the grid points of the time integration scheme and the

weights ω1, . . . , ωqand Ware chosen as for the mode determination.

All numerical experiments have been conducted on a laptop with 2.7 GHz

Dual-Core Intel Core i5 processor and 8 GB RAM. Furthermore, we have used

MATLAB R2020b.

6.1. Linear Wave Equation

We revisit Example 1.2.3 and consider the linear acoustic wave equation











∂tρ(t, ξ) = −ρref ∂ξv(t, ξ)for all (t, ξ)∈I×Ω,

∂tv(t, ξ) = −c2

ρref

∂ξρ(t, ξ)for all (t, ξ)∈I×Ω,

ρ(0, ξ) = ρ0(ξ)for all ξ∈Ω,

v(0, ξ) = v0(ξ)for all ξ∈Ω,

ρ(t, 0) = ρ(t, 1) for all t∈I,

v(t, 0) = v(t, 1) for all t∈I,

(6.1)

with spatial domain Ω = (0,1), time interval I= [0, tend], constants c, ρref ∈

R>0, initial values ρ0, v0∈C1

per(Ω), and unknowns ρ, v:I×Ω→R. Fur-

176

6.1. Linear Wave Equation

thermore, we introduce the associated Hamiltonian H:L2(Ω) ×L2(Ω) →R

via

H(ˆρ, ˆv):=1

Ω c2

ρref

ˆρ(ξ)2+ρref ˆv(ξ)2!dξ, (6.2)

which is defined such that Hs:I→Rwith Hs(t):=H(ρ(t, ·), v(t, ·)) corre-

sponds to the sound energy, cf. [177, § 65]. It can be shown that Hsis constant,

i.e., the sound energy is a conserved quantity of (6.1), cf. the upcoming calcu-

lation in (6.4).

In the following, we perform a semi-discretization of (6.1) in space using

a finite element approach and, for this purpose, we start by deriving a corre-

sponding weak formulation. We are interested in obtaining a finite-dimensional

system which is port-Hamiltonian and ideally allows for a pH representation

with an Ematrix as in (2.37), since this is particularly convenient for model

reduction as mentioned in Remark 2.6.4. For this reason and taking into ac-

count the coefficients of the Hamiltonian in (6.2), it turns out to be beneficial

to multiply the first and second equation in (6.1) by c2

ρref and ρref, respectively.

After that, by applying integration by parts similarly as in [211], we obtain a

corresponding weak formulation as follows: Find (ρ, v): I×Ω→R2such that

(i) for all t∈I,ρ(t, ·)and v(t, ·)are in H1

per (Ω) and satisfy

ρref hψρ, ∂tρ(t, ·)iL2(Ω) +ρref hψv, ∂tv(t, ·)iL2(Ω)

=−c2

2hψρ, ∂ξv(t, ·)iL2(Ω) −Dv(t, ·), ψ0

ρEL2(Ω)

+hψv, ∂ξρ(t, ·)iL2(Ω) −hρ(t, ·), ψ0

viL2(Ω) 

(6.3)

for all ψρ, ψv∈H1

per (Ω), where ψ0

ρand ψ0

vdenote the weak derivatives of

ψρand ψv, respectively, and ∂ξρand ∂ξvthe weak partial derivatives of

ρand vwith respect to ξ, respectively,

(ii) for all ξ∈Ω, we have ρ(0, ξ) = ρ0(ξ)and v(0, ξ) = v0(ξ).

Before we proceed with the FEM discretization, we use (6.3) to demonstrate

that the sound energy is indeed a conserved quantity. For this purpose, let

(ρ, v): I×Ω→R2be a solution of the weak formulation of (6.1) and let it be

continuously differentiable with respect to time in the sense that the mapping

t7→ (ρ(t, ·), v(t, ·)) is in C1(I, L2(Ω) ×L2(Ω)). In addition, we note that Hs

as defined after (6.2) represents the Hamiltonian along the solution trajectory.

Then, using the chain rule, [11, Prop. VII.4.6] for the derivative of H, and

177

6. Numerical Examples

(6.3), we obtain

dHs

dt(t) = H0(ρ(t, ·), v(t, ·))(∂tρ(t, ·), ∂tv(t, ·))

=c2

ρref hρ(t, ·), ∂tρ(t, ·)iL2(Ω) +ρref hv(t, ·), ∂tv(t, ·)iL2(Ω)

=−c2

2hρ(t, ·), ∂ξv(t, ·)iL2(Ω) −hv(t, ·), ∂ξρ(t, ·)iL2(Ω)

+hv(t, ·), ∂ξρ(t, ·)iL2(Ω) −hρ(t, ·), ∂ξv(t, ·)iL2(Ω) 

= 0

(6.4)

for all t∈I, i.e., the Hamiltonian is preserved along the solution trajectories.

Here, we exploited that the right-hand side of (6.3) may be described via a

skew-adjoint operator and that (6.3) may be formulated as a Hamiltonian sys-

tem. However, we omit further details here since infinite-dimensional Hamil-

tonian or port-Hamiltonian systems are not within the scope of this thesis.

Instead, we demonstrate the Hamiltonian structure on the finite-dimensional

level after semi-discretization in space.

Applying a standard Galerkin finite element scheme in space based on an

equidistant mesh with mesh size h=1

N+1 ,N∈N, and globally continuous and

piecewise linear ansatz and test functions yields the semi-discretized system

Eh˙xh(t) = Jhxh(t)for all t∈I,(6.5)

where xh:I→R2(N+1) contains the coefficients corresponding to the FEM

ansatz functions and Eh, Jh∈R2(N+1),2(N+1) are given by

Eh:=h

6diag c2

ρref

, ρref!⊗M, M :=







4 1 0 ··· 0 1

1 4 1 ....

0 1 4 ...0.

..........1 0

0··· 0 1 4 1

1 0 ··· 0 1 4







∈RN+1,N+1,

Jh:=−c2

2"0 1

1 0#⊗P, P :=







0 1 0 ··· 0−1

−1 0 1 ....

0−1 0 ...0.

..........1 0

0··· 0−1 0 1

1 0 ··· 0−1 0







∈RN+1,N+1.

(6.6)

We emphasize that Ehis symmetric and positive definite and Jhis skew-

symmetric. Consequently, (6.5) is a Hamiltonian system with Hamiltonian

178

6.1. Linear Wave Equation

0 1

0.2

0.4

ρ(t, ξ)

0 1

−0.2

0.2

v(t, ξ)

Figure 6.1.: Linear wave equation: pseudocolor plots of the FOM solution for the density

(left) and the velocity (right).

Hh:R2(N+1) →Rdefined via Hh(xh):=1

2x>

hEhxhand, in particular, stable,

cf. section 2.6.

For the following numerical experiments, we choose the PDE parameters as

c= 1 and ρref = 1, the final time as tend = 1, and the initial values as v0= 0

and

ρ0(ξ) = 









2exp 1−1

1−(20(ξ−1

2))2,if ξ∈(0.45,0.55),

0,otherwise (6.7)

for all ξ∈Ω, cf. Figure 6.2, left. Furthermore, we divide the spatial domain

into N+ 1 = 2000 equidistant intervals, which corresponds to a mesh size

of h= 5 ·10−4, and for the time discretization we use a step size of 10−4.

This results in a relative error of about 7·10−4when comparing the numerical

solution of the fully discretized system with the analytical solution provided

in Example 1.2.3. Figures 6.1 and 6.2 depict the numerical solution in the

form of pseudocolor plots and by means of plotting some selected snapshots,

respectively. As is typical for the linear wave equation, we observe traveling

waves both in the density and in the velocity snapshots. We note that the

velocity snapshots at t= 0 and t= 0.5in Figure 6.2, right, are zero everywhere

and thus covered by the other graphs.

Before we proceed with constructing a ROM based on transformed modes,

we first consider the singular value decay of the snapshot matrix in Figure 6.3.

As outlined in section 2.5.1, the singular value decay reflects the approxima-

tion quality which may be expected from a classical POD-based approach. We

observe a rather slow singular value decay in Figure 6.3, which is not unusual

considering the transport-dominated nature of the problem and especially the

sharp front profiles of the two traveling waves, cf. section 1.2.1. For instance,

for obtaining a relative offline error of 10−3with respect to the spectral norm

of the snapshot matrix, the POD requires more than 150 modes. While this

still corresponds to a dimension reduction of more than one order of magni-

tude, a description based on more than 150 degrees of freedom appears to be

unnecessarily complex in light of the simple structure of the analytical solution

179

6. Numerical Examples

0 0.5 1

0.2

0.4

ρ(t, ξ)

t= 0 t= 0.1t= 0.3t= 0.5t= 0.8

0 0.5 1

−0.2

0.2

v(t, ξ)

Figure 6.2.: Linear wave equation: some selected snapshots of the FOM solution for the

density (left) and the velocity (right).

100 200 300 400 500 600 700 800 900 1,000

10−7

10−5

10−3

10−1

σi

σ1

Figure 6.3.: Linear wave equation: singular value decay of the snapshot matrix.

provided in (1.15).

In the following, we pursue a nonlinear model reduction approach based

on the approximation ansatz (1.4). To this end, we first determine suitable

modes by following the residual minimization approach discussed in section 3.1,

where we use the spatial weighting matrix W=h

6I2⊗Mwith Mas in (6.6).

Furthermore, for reducing the computational effort, we consider only every

fifth snapshot for the mode determination, which corresponds to an effective

time step size of ∆t= 5 ·10−4. Moreover, we set the number of modes to

r= 2 and the mode dimension to dφ= 2(N+ 1). For the transformations we

employ a discretized version of the family of periodic shift operators defined in

Definition 1.2.2 based on cubic spline interpolation. More precisely, for given

shift value p∈R, we use I2⊗ Tper,h(p)as shift operator, where Tper,h(p)∈

180

6.1. Linear Wave Equation

L(RN+1)is defined via

Tper,h(p)φ:=T(q(p))a(φ)−ζ(p)b(φ) + ζ(p)2c(φ)−ζ(p)3d(φ)

with q(p):=$pmod 1

h%, ζ(p):=pmod h, T(q):="0Iq

IN+1−q0#.

Here, a(φ), b(φ), c(φ), d(φ)∈RN+1 are vectors containing the coefficients of the

spline interpolant of φ, see appendix D.1.1 for their definition and a derivation

of Tper,h from Tper. For the choice of the path values in the offline phase, we

exploit the fact that the wave speeds of (6.1) are known and set p1(t) = −ct

and p2(t) = ct for all t∈I. Alternatively, they could be determined based

on the snapshot data as illustrated in [241, sec. 2.3] or by using one of the

other methods mentioned in Remark 3.1.2. For determining the modes, we

use the variable projection approach discussed in section 3.1.2 and solve the

corresponding optimization problem by means of the GRANSO software with

default settings. As starting values we use the respective first left singular

vector of the shifted snapshot matrices

h(I2⊗Tper,h(−p1(t1))) xh(t1)··· (I2⊗Tper,h(−p1(tq))) xh(tq)i

and h(I2⊗Tper,h(−p2(t1))) xh(t1)··· (I2⊗Tper,h(−p2(tq))) xh(tq)i.

Here, the application of the negative shift corresponds to shifting the snapshots

into the respective co-moving reference frame.

The GRANSO solver terminates after nine iterations, since the line search

algorithm bracketed a minimizer, but fails to satisfy the Wolfe conditions,

see [75] for more details about the employed line search algorithm. It is also

stated that this often indicates that a stationary point has been reached. The

resulting relative offline error is about 3.4·10−4and the determined modes are

depicted in Figure 6.4. On the other hand, we omit a plot of the corresponding

amplitudes, since they are nearly constant with respect to time, which is in

accordance with the corresponding discussion in Example 1.2.3 based on the

analytical solution. Considering the modes depicted in Figure 6.4, we observe

that they agree well with the profiles of the traveling waves except for some

constant offsets in the velocity component. These offsets are actually already

present in the corresponding starting values and the fact that they are not

removed by the optimization procedure may be attributed to a non-uniqueness

of the optimal solution. In fact, since constant functions are invariant under

shifting operations, we may add constant offsets to each of the modes without

changing the value of the cost function, as long as the offsets sum up to zero.

Indeed, by inspecting Figure 6.4, right, we note that the magnitude of the

offsets in the first and the second mode is the same, but they have opposite

signs. A similar observation is made in [241, sec. 3], where such offsets are

encountered in the context of a two-dimensional vortex pair test case.

181

6. Numerical Examples

0 0.5 1

0.04

0.08

φ(ξ)

ρcomponent of φ1

ρcomponent of φ2

0 0.5 1

−0.1

0.1

φ(ξ)

vcomponent of φ1

vcomponent of φ2

Figure 6.4.: Linear wave equation: density (left) and velocity (right) components of the

determined modes.

The results stated in the last paragraph are based on solving a reduced op-

timization problem, which is obtained via variable projection as outlined in

section 3.1.2. In order to assess the benefit of solving the reduced problem

instead of the full one discussed in section 3.1.1, we also solve the full opti-

mization problem via GRANSO, where we use the same settings as before.

Especially, for the starting values we take the same modes as for the reduced

problem, whereas the starting values of the amplitudes are determined based

on solving the corresponding linear least squares problem with fixed modes,

cf. the discussion before (3.17). The total number of unknowns of the full

problem is r(dφ+ 1 + 1

∆t) = 12,002, whereas the reduced problem involves

rdφ= 8000 optimization parameters. For the full problem, the GRANSO

solver terminates after 105 iterations, since the norm of the gradient becomes

smaller than the default tolerance of 10−8. The resulting offline error is the

same as the one obtained using the reduced optimization problem, while the

number of iterations is more than one order of magnitude higher for the full

optimization problem. However, this comparison of the iteration numbers is of

little significance, since for the full problem the solver has reached the tolerance

for the norm of the gradient, whereas for the reduced problem the solver has

terminated prematurely due to a failure of the line search algorithm. In order

to obtain a more meaningful comparison, we specify a target value of 10−3for

the relative error of the resulting snapshot matrix approximation. Solving the

reduced problem, the GRANSO procedure attains an error of approximately

3.6·10−4after three iterations, which involve nine evaluations of the cost func-

tion and its gradient, and requires a computation time of about 16 seconds.

On the other hand, when solving the full problem, GRANSO needs four itera-

tions involving ten function evaluations, takes roughly 23 seconds, and attains

182

6.1. Linear Wave Equation

an error of about 7.1·10−4. The advantage of using the reduced problem

may also be observed when decreasing the target value for the relative error

to 3.5·10−4: Solving the reduced problem, GRANSO needs four iterations,

ten function evaluations, and approximately 19 seconds, whereas it requires

twelve iterations, 31 function evaluations, and about 67 seconds when solving

the full problem. All in all, the numerical experiments indicate an advantage

of using the reduced optimization problem discussed in section 3.1.2.

We proceed to construct a ROM similarly as in (4.1)–(4.2) based on the

determined modes and using the derivative of Tper,h provided in (D.4) in ap-

pendix D.1.1. In fact, we use a slightly modified version of the framework

presented in section 4.1 by minimizing a weighted norm of the residual with

weighting matrix E−1

h. Especially, since (1.4) is a special case of a separable

approximation ansatz, it follows from the considerations in section 5.2 that

the resulting ROM is port-Hamiltonian, cf. Theorem 5.2.1. Actually, since the

FOM is even a Hamiltonian system, this is also true for the ROM, i.e., the

ROM Hamiltonian is a conserved quantity.

Since the FOM is linear, the resulting ROM has the form

"Mα(p)N(p)D(α)

D(α)>N(p)>D(α)>Mp(p)D(α)#"˙α

˙p#="Aα(p)

D(α)>Ap(p)#α, (6.8)

where Dis constructed as in (4.2) and Aαand Apas in (4.37). Also Mα,

N, and Mpare obtained as in (4.2), but the occurring inner products need

to be replaced by weighted inner products with weighting matrix Eh. As

discussed in section 4.3, the evaluation of the ROM still scales with the FOM

dimension, since each entry of Mα,N,Mp,Aα, and Apinvolves products

of high-dimensional matrices and vectors. However, in [38] it is shown for the

linear wave equation that, even though the ROM coefficients depend in general

on the paths, the solution of the ROM remains the same when ignoring the

path dependency of the coefficient matrices. Admittedly, the result in [38] is

only stated for the case where the ROM is directly obtained by projecting the

infinite-dimensional problem (6.1), whereas the ROM (6.8) has been derived

based on the finite-dimensional FOM (6.5). Nevertheless, we have observed

in our numerical experiments that the solution of the ROM (6.8) does not

significantly change when ignoring the path dependencies of Mα,N,Mp,Aα,

and Ap. Hence, we may precompute the evaluations of these matrix functions

at p(0) = 0 and use the resulting constant matrices in the online phase. In

particular, this hyperreduction approach, which consists in approximating the

path-dependent coefficient matrices by constant ones, ensures that the online

phase does not scale with the FOM dimension.

For a comparison of the ROMs with and without hyperreduction, we solve

them numerically using the implicit midpoint rule and the same parameter

values which have been used for generating the FOM snapshot data in the

offline phase. The relative speed-up of the hyperreduced ROM in comparison

to the ROM without hyperreduction is roughly 6, while the corresponding

183

6. Numerical Examples

errors differ only slightly: the ROM without hyperreduction yields a relative

online error of about 3.4·10−4and the error of the hyperreduced ROM is

approximately 3.7·10−4. Consequently, in both cases the online error is in the

same order of magnitude as the offline error and as the error of the FOM with

respect to the analytical solution. Furthermore, both ROMs are faster than

the FOM, which takes about 80 seconds and hence roughly 20 seconds more

than the ROM without hyperreduction. One reason for the ROM without

hyperreduction to be faster than the FOM is that the ROM evaluation only

involves matrix vector products which scale with the FOM dimension, whereas

evaluating the FOM also requires solving a high-dimensional linear equation

system in each time step, since we use an implicit time integration scheme.

Next, we compare the ROM (6.8) and its hyperreduced analogue with a

ROM obtained by using a classical POD approach and a subsequent Galerkin

projection, cf. sections 2.5.1 and 2.5.3. To this end, we construct a ROM based

on 280 POD modes, which results in an online error of about 3.6·10−4, i.e., the

accuracy is comparable to that of the two ROMs addressed in the preceding

paragraph. Even though the number of POD modes is fairly large, the evalua-

tion of the resulting ROM requires only ten seconds, which is about the same

computation time as for the hyperreduced ROM based on two transformed

modes. One reason for this appears to be that the POD-based ROM is linear

and therefore its evaluation only requires to solve a linear equation system of

moderate size in each time step. On the other hand, the ROM (6.8) and its

hyperreduced counterpart are nonlinear and, hence, each time step involves

solving a nonlinear equation system via an iterative procedure. However, we

emphasize that all computation times mentioned so far are based on a time

step size of 10−4, i.e., we have used the same time step size for the ROMs as

for the FOM. When increasing the time step size, the accuracy of the POD-

based ROM decreases significantly. For instance, an increment of the time

step size from 10−4to 2.5·10−4leads to an online error of about 2.7·10−3. In

contrast to this, the accuracy of the hyperreduced ROM based on the nonlin-

ear ansatz (1.4) appears to be largely independent of the time step size. For

example, increasing the time step size to 0.02 still yields an online error of

around 3.7·10−4, while the computation time reduces to less than a second.

This flexibility in choosing the time step size is in fact not surprising when

considering the time dependency of the amplitudes and the paths. Based on

the analytical solution and in accordance with the discussion in Example 1.2.3,

we expect the amplitudes to be nearly constant and the paths to be almost

linear functions of time. This expectation is also confirmed by the numerical

experiments, cf. Figure 6.5, which explains the flexibility in choosing the time

step size, since the implicit midpoint rule is exact for constant and linear func-

tions, see for instance [80, sec. 6.3]. We note that in [37, sec. 7.3] a similar

freedom of choosing the time step size has been observed, while using a time

integration scheme with adaptive time stepping.

The main reason for us choosing the implicit midpoint rule as time inte-

184

6.2. Linear Advection–Diffusion Equation

0 0.5 1

3.3

3.35

3.4

α(t)

α1

α2

0 0.5 1

−1

p(t)

Figure 6.5.: Linear wave equation: online values of the amplitudes (left) and the paths

(right).

gration scheme is that the Hamiltonian of the FOM is a quadratic function

and, thus, the implicit midpoint rule ensures that the Hamiltonian is also a

conserved quantity of the time-discrete system, see for instance [80, sec. 6.3.4].

However, due to the nonlinearity of the approximation ansatz (1.4), the Hamil-

tonian of the ROM (6.8) is in general not a quadratic function, cf. section 5.2.

Consequently, there is no guarantee that the ROM Hamiltonian is a conserved

quantity of the time-discrete ROM when using the implicit midpoint rule. In

fact, the ROM Hamiltonian is slightly increasing as illustrated in Figure 6.6.

On the contrary, when using a time discretization scheme based on the mid-

point discrete gradient pair, cf. appendix C.2, the ROM Hamiltonian is guaran-

teed to be a conserved quantity of the time-discrete ROM. This is also reflected

in the numerical experiments, where the largest occurring deviation from the

initial value of the ROM Hamiltonian is about 2·10−16, see also Figure 6.6.

These results indicate that discrete gradient pair methods as discussed in ap-

pendix C.2 may be a useful tool in applications with state-dependent mass

matrix and non-quadratic Hamiltonian, especially when an exact conservation

of the Hamiltonian is required.

6.2. Linear Advection–Diffusion Equation

In contrast to the preceding section, we consider in the following a test case

with non-periodic boundary conditions. More precisely, we focus on a linear

advection–diffusion equation on the spatial domain Ω = (0,1) with an inhomo-

geneous Robin condition on the left boundary and a homogeneous Neumann

condition on the right boundary. The corresponding governing equations for

185

6. Numerical Examples

0 0.2 0.4 0.6 0.8 1

0.5

1.5·10−11

H(˜x(t)) −˜

H(˜x(0))

implicit midpoint rule

discrete gradient pair method

Figure 6.6.: Linear wave equation: Comparison of the error in conservation of the ROM

Hamiltonian using the implicit midpoint rule and the midpoint discrete gra-

dient pair method outlined in appendix C.2.

the unknown x:I×Ω→Rare given by











∂tx(t, ξ) = −c∂ξx(t, ξ) + d∂ξξx(t, ξ)for all (t, ξ)∈I×Ω,

cx(t, 0) −d∂ξx(t, 0) = cg(t)for all t∈I,

∂ξx(t, 1) = 0 for all t∈I,

x(0, ξ) = x0(ξ)for all ξ∈Ω

(6.9)

with time interval I= [0, tend],tend ∈R>0, advection speed c∈R>0, diffusion

coefficient d∈R>0, Robin boundary value g:R≥0→R, and initial value

x0:Ω→R. The combination of Robin and Neumann boundary conditions

as used in (6.9) is sometimes referred to as Danckwerts boundary conditions,

cf. [5, 77].

Similarly as in section 6.1, we use integration by parts and obtain the fol-

lowing weak formulation: Find x:I×Ω→Rsuch that

(i) for all t∈I,x(t, ·)is in H1(Ω) and satisfies

hψ, ∂tx(t, ·)iL2(Ω) =cψ(0)g(t)−c

2hψ, ∂ξx(t, ·)iL2(Ω) −hψ0, x(t, ·)iL2(Ω)

−dhψ0, ∂ξx(t, ·)iL2(Ω) −c

2(ψ(1)x(t, 1) + ψ(0)x(t, 0))

for all ψ∈H1(Ω),

(ii) for all ξ∈Ω, we have x(0, ξ) = x0(ξ).

Furthermore, we introduce the associated Hamiltonian H:L2(Ω) →Rvia

H(x):=1

2kxk2

L2(Ω) and derive in the following a corresponding dissipation

inequality. To this end, let x:I×Ω→Rbe a solution of the weak formulation

186

6.2. Linear Advection–Diffusion Equation

of (6.9) and let it be continuously differentiable with respect to time in the

sense that the mapping t7→ x(t, ·)is in C1(I, L2(Ω)). In addition, we consider

the function Hs:I→Rwhich is defined via Hs(t):=H(x(t, ·)) and represents

the Hamiltonian along the solution trajectory. Then, by similar arguments as

in (6.4), we obtain the dissipation inequality

dHs

dt(t) = H0(x(t, ·))(∂tx(t, ·)) = hx(t, ·), ∂tx(t, ·)iL2(Ω)

=−dk∂ξx(t, ·)k2

L2(Ω) −c

2x(t, 1)2+x(t, 0)2+cx(t, 0)g(t)

≤cx(t, 0)g(t)

for all t∈I. Thus, in the special case g= 0 the Hamiltonian does not

increase along solution trajectories. This property is also reflected in a port-

Hamiltonian structure on the semi-discretized level as shown in the following.

For the semi-discretization in space, we use the Galerkin FEM with mesh size

h=1

N+1 ,N∈N, analogously as in section 6.1. The resulting semi-discretized

system takes the form

Eh˙xh(t)=(Jh−Rh)xh(t) + Bhu(t)for all t∈I,(6.10)

where xh:I→RN+2 contains the coefficients corresponding to the FEM ansatz

functions, the input u:R≥0→Ris given by u=g, and Eh, Jh, Rh∈RN+2,N+2,

Bh∈RN+2 are defined as

Eh:=h







2 1 0 ··· 0 0

1 4 1 ....

0 1 4 ...0 0

..........1 0

0··· 0 1 4 1

0··· 0 0 1 2







, Bh:=c













,(6.11a)

Jh:=−c

2tridiagN+2(−1,0,1),(6.11b)

Rh:=d







1−1 0 ··· 0 0

−1 2 −1....

0−1 2 ...0 0

..........−1 0

0··· 0−1 2 −1

0··· 0 0 −1 1







2diag(1,0,...,0,1).(6.11c)

Here, Ehis symmetric and positive definite, Jhis skew-symmetric, and Rhis

symmetric and positive semi-definite. Hence, the semi-discrete system (6.10)

corresponds to the state equation of a port-Hamiltonian system of the form

187

6. Numerical Examples

0 0.5 1

0.5

1.5

0.2

0.4

x(t, ξ)

0 0.5 1

0.2

0.4

t= 0 t= 0.15 t= 0.3t= 0.6t= 0.9

Figure 6.7.: Linear advection–diffusion equation: pseudocolor plot of the FOM solution

(left) and some selected snapshots (right).

(2.37) with Hamiltonian Hh(xh) = 1

2x>

hEhxh. In particular, we infer that

(6.10) is stable for u= 0, cf. section 2.6. Moreover, since the Hamiltonian is

quadratic, a time discretization based on the implicit midpoint rule ensures a

dissipation inequality also on the time-discrete level, see section 2.6.1.

For the following numerical experiments, we choose the PDE parameters as

c= 1 and d= 10−3, the final time as tend = 1.5, the boundary value as

g(t) = u(t) = 





2exp 1−1

1−(20(t−0.15·tend))2,if t∈(0.175,0.275),

0,otherwise,(6.12)

for all t∈R≥0, and the initial value as x0=ρ0with ρ0as in (6.7). Besides,

we divide the spatial domain into N+ 1 = 1000 equidistant intervals, which

corresponds to a mesh size of h= 10−3. For the time discretization, we use a

step size of 10−3as well. Figure 6.7 depicts the numerical solution by means of a

pseudocolor plot and some selected snapshots. We observe that the initial wave

profile is transported to the right, while its shape and amplitude changes due

to the diffusion. After a certain time, a second wave enters the computational

domain via the left boundary and is also transported to the right. Eventually,

both waves leave the computational domain via the right boundary.

We proceed by determining suitable modes based on the snapshot data of the

FOM and, for this purpose, we use again the residual minimization approach

from section 3.1. The spatial weighting matrix is chosen as W=Eh, where

Ehdenotes the FEM mass matrix as defined in (6.11a). Furthermore, for

reducing the computational effort, we only use every second snapshot for the

mode determination, which corresponds to an effective time step size of ∆t=

2·10−3. Moreover, we set the number of modes to r= 3 and we use a family

of transformation operators which is inspired by the extended domain shift

operator introduced in Definition 3.3.3. More precisely, we first introduce an

188

6.2. Linear Advection–Diffusion Equation

extended domain similar to (3.32), where we use one common shift pfor all

three modes. In particular, for the offline phase we exploit the fact that the

advection speed of (6.9) is known and set p(t) = ct for all t∈I. The resulting

extended domain is given by b

Ωe= (−ctend,1), i.e., defining the modes φion

Ωeensures the shifted modes φi(ξ−ct)to be well-defined for all (t, ξ)∈I×Ω

and i∈ {1,2,3}. For the discretized modes, this leads to a dimension of

dφ=1 + ctend

h+ 1 = 2501,

where each entry corresponds to a point of the grid which is obtained by di-

viding b

Ωeinto equidistant intervals of length h. However, we emphasize that,

even though the extended domain is sufficiently large for the offline phase

where the shift is prescribed, there is in general no guarantee that the shift

computed in the online phase remains in the interval [0, ctend]. This is due to

the fact that the shift pconstitutes a part of the ROM state, which is a priori

unknown. In order to avoid defining the modes on an even larger domain, we

use a family of transformation operators which combines the extended domain

shift operator introduced in Definition 3.3.3 with the constant extrapolation

shift operator defined in Definition 3.3.8. The resulting family of transfor-

mation operators essentially uses an extended domain shift operator, as long

as the shift values are within the range [0, ctend], and constant extrapolation

otherwise, see appendix D.1.2 for more details. A discretized version of this

transformation family is obtained via cubic spline interpolation and given by

Tce,h :R→ L(Rdφ,RN+2)with

Tce,h(η)φ:=T1(q(η))φ+T2(q(η))a(φ)−ζ(η)b(φ) + ζ(η)2c(φ)−ζ(η)3d(φ),

(6.13a)

q(η):=η

h, ζ(η):=ηmod h, (6.13b)

T1(q):=













1N+2+min(q+1−dφ,0) 0

0 0

∈RN+2,dφ,if q≥dφ−(N+ 2),



0 0

01min(N+2,max(0,−q))

∈RN+2,dφ,otherwise,

(6.13c)

189

6. Numerical Examples

T2(q):=













0 0

Imax(dφ−1−q,0) 0

∈RN+2,dφ−1,if q≥dφ−(N+ 2),



0z1(q)×z2(q)Iz1(q)0

0 0 0

∈RN+2,dφ−1,otherwise,

z1(q):= max(N+ 2 + min(0, q),0),

z2(q):=dφ−1−max(0, N +2+q).

(6.13d)

The unspecified block sizes in the definitions of T1and T2result from the sizes

of the other blocks with the understanding that a block row or column does not

occur if the corresponding number of rows or columns is zero. Furthermore,

the vectors a(φ), b(φ), c(φ), d(φ)∈Rdφ−1contain the spline coefficients of φ,

see appendix D.1.2 for their definition and a derivation of Tce,h.

For the actual computation of the modes, we use the variable projection

approach discussed in section 3.1.2 and solve the corresponding optimization

problem by means of the GRANSO software with default settings. Regarding

the choice of the starting values, we cannot use the same approach as for the

wave equation in section 6.1, since we use a different family of transformation

operators. In fact, our choice for the starting values in section 6.1 is motivated

by the pointwise invertibility of the family of periodic shift operators Tper,

cf. appendix A. On the other hand, the family of extended domain shift opera-

tors Teis not pointwise invertible and neither is Tce,h, which is pointwise a non-

square matrix. However, one can show that Tesatisfies Te(η)Te(η)∗= IdL2(Ω)

for all η∈Rand we observed that also Tce,h(η)Tce,h(η)>≈IN+2 holds, at least

within the considered range of shift values. Accordingly, we propose to use the

first three left singular vectors of the transformed snapshot matrix

hTce,h(p(t1))>xh(t1)··· Tce,h(p(tq))>xh(tq)i

as starting values for the modes. Based on these starting values, the GRANSO

solver terminates after almost 5000 iterations and the resulting relative offline

error is about 7.6·10−3.

Next, we construct a ROM based on the determined modes and using the

derivative of Tce,h provided in (D.10) in appendix D.1.2. Analogously as in

section 6.1, we slightly differ from the setting in section 4.1 and use a weighted

norm for the residual minimization, which ensures that the ROM is port-

Hamiltonian. The resulting ROM has the form

"Mα(p)N(p)α

α>N(p)>α>Mp(p)α#"˙α

˙p#="Aα(p)

α>Ap(p)#α+"Bα(p)

α>Bp(p)#u, (6.14)

where Aα, Ap:R→Rr,r are constructed as in (4.37) with p1=. . . =pr=p

190

6.2. Linear Advection–Diffusion Equation

and F:=Jh−Rh, while Bα, Bp:R→Rrare defined via

Bα(p):=hφ>

iTce,h(p)>Bii, Bp(p):=hφ>

iT0

ce,h(p)>Bii.

Moreover, Mα,N, and Mpare constructed as in (4.15c)–(4.15e), but the oc-

curring inner products need to be replaced by weighted inner products with

weighting matrix Eh.

As discussed in section 4.3, the evaluation of the ROM still scales with the

FOM dimension, since each entry of Mα,N,Mp,Aα,Ap,Bα, and Bpinvolves

products of high-dimensional matrices and vectors. In order to achieve an

efficient offline/online decomposition, we follow the approach outlined in sec-

tion 4.3.1 by sampling these path-dependent matrices and constructing asso-

ciated interpolants. More precisely, we construct piecewise linear interpolants

for each of these matrix functions. For this purpose, we compute correspond-

ing samples within the range [0, ctend]and based on an equidistant grid whose

grid size coincides with the one used for the FEM discretization. Due to the

resulting large number of interpolation intervals, the ROM evaluation formally

still scales with the FOM dimension, since we need to determine the correct

interpolation interval whenever the shift pis updated. However, such a binary

search only scales logarithmically with the full dimension, see for instance [165,

sec. 6.2.1].

For a comparison of the ROMs with and without hyperreduction, we solve

them numerically using the implicit midpoint rule and the same parameter val-

ues which have been used for generating the FOM snapshot data in the offline

phase. The evaluation of the non-hyperreduced ROM takes approximately

eight seconds, whereas the hyperreduced ROM requires only a computation

time of about 4.2seconds. This amounts to a relative speed-up of roughly 1.9,

while both ROMs achieve a relative online error of 9.9·10−3, which is thus

slightly higher than the offline error of 7.6·10−3. However, even if the hyper-

reduction leads to a lower computation time of the ROM, this speed-up is still

not sufficient, since the evaluation of the FOM is already very fast and takes

only about 0.35 seconds. Furthermore, the same computation time is required

by a POD-based ROM with 32 modes, which leads to an online error of about

8.6·10−3. Thus, for the problem at hand, the nonlinear ROMs cannot compete

with the FOM or with the POD-based ROM in terms of the achieved speed-up

and accuracy. One reason for this is that the nonlinear ROMs require non-

linear system solves in each time step, see also the corresponding discussion

for the wave equation test case in section 6.1. Furthermore, the simulation

of the hyperreduced ROM involves the evaluation of multiple interpolants for

assembling the coefficient matrices and this constitutes a computational over-

head, which is neither present in the FOM evaluation nor in the evaluation of

the POD-based ROM. Nevertheless, the disadvantage of the nonlinear ROMs

in terms of computation time may be reduced by exploiting the fact that the

ROMs may be simulated with a larger time step size than the FOM without

191

6. Numerical Examples

a significant loss of accuracy. For instance, when increasing the time step size

tenfold, the computation time of the hyperreduced ROM is about 0.8seconds

and thus at least in the same order of magnitude as the FOM simulation time.

By contrast, the resulting increment of the online error is negligibly small.

On the other hand, this flexibility in increasing the time step size without a

significant loss of accuracy does not equally apply to the POD-based ROM,

for which the tenfold time step size results in a tenfold error.

As mentioned after (6.11), an advantage of using the implicit midpoint rule

is that it ensures a dissipation inequality for the time-discrete FOM, since the

corresponding Hamiltonian is a quadratic function of the state. However, this

is not true for the ROM Hamiltonian and consequently there is no guarantee

that the time-discrete ROM satisfies a dissipation inequality when using the

implicit midpoint rule. Nevertheless, as long as the time step size is suffi-

ciently small, we have observed in our numerical experiments that the implicit

midpoint rule still yields an energy-consistent solution in the sense that the

Hamiltonian only increases within the time interval (0.175,0.275), where the

input is non-zero, cf. (6.12). On the other hand, when using an increased

time step size of 0.08, this is no longer true and we observe a violation of

the power balance, cf. Figure 6.8. In particular, at t= 0.28 the input and

hence the supplied power are zero, whereas the discrete time derivative of the

ROM Hamiltonian obtained via the implicit midpoint rule is positive, i.e., the

Hamiltonian increases despite the vanishing input value. Moreover, we observe

that the discrete time derivative of the ROM Hamiltonian does not match the

corresponding dissipation at t= 0.28 and partly also afterwards. When we use

instead a time discretization scheme based on the midpoint discrete gradient

pair, cf. appendix C.2, Theorem C.2.3 yields that also on the time-discrete level

a dissipation inequality is satisfied. Accordingly, we observe in Figure 6.8 that

the inconsistency of the power balance does not apply to the solution obtained

by the discrete gradient pair approach. Instead, the discrete time derivative of

the Hamiltonian and the corresponding dissipation are in excellent agreement

in that part of the time interval where the supplied power is zero.

While the fact that discrete gradient pair methods lead to an exact power

balance on the time-discrete level is proven in appendix C.2 and illustrated in

Figures 6.6 and 6.8, we have not yet addressed its convergence behavior. To

study the order of convergence numerically, we consider a reference solution

obtained by solving the ROM (6.14) via the RADAU IIA method of order

five, cf. [133, p. 72ff.], with time step size 2·10−6. Furthermore, to diminish

the influence of the accuracy of the nonlinear equation system solver fsolve,

we set the tolerances OptimalityTolerance and FunctionTolerance to 10−13

and 10−8, respectively. Moreover, we consider the reduced time interval [0,1.2]

instead of [0,1.5], since we have observed in our numerical experiments that

the obtained solutions of the nonlinear equation systems become quite sensitive

with respect to the starting point and the fsolve tolerances towards the end

of the time interval [0,1.5], especially when using a coarse time discretization.

192

6.2. Linear Advection–Diffusion Equation

0 0.2 0.4 0.6 0.8 1 1.2 1.4

−0.05

0.05

0.1

d˜

dt(implicit midpoint rule)

d˜

dt(discrete gradient pair method)

dissipation (implicit midpoint rule)

dissipation (discrete gradient pair method)

supplied power (implicit midpoint rule)

supplied power (discrete gradient pair method)

0.26 0.28 0.3

−0.01

0.01

Figure 6.8.: Linear advection–diffusion equation: Comparison of the discrete time deriva-

tive of the ROM Hamiltonian and the corresponding dissipation and supplied

power with time step size ∆t= 0.08 when using the implicit midpoint rule

and the midpoint discrete gradient pair approach from appendix C.2. The

inset highlights an energy inconsistency of the implicit midpoint rule, where

the discrete time derivative of the Hamiltonian is positive despite a vanishing

power supply. Here, t= 0.28 corresponds to the first midpoint where the

input becomes permanently zero.

193

6. Numerical Examples

10−610−510−410−310−210−1100

10−11

10−9

10−7

10−5

10−3

10−1

∆t

relative error

implicit midpoint rule

discrete gradient pair method

(∆t)2

Figure 6.9.: Linear advection–diffusion equation: Convergence of the implicit midpoint

rule and the midpoint discrete gradient pair method from appendix C.2.

Presumably, this is due to the fact that the values of the FOM state become

very small, once the second wave leaves the computational domain, and hence

there may be multiple choices for the path and the amplitudes leading to a

small residual.

Based on the reference solution, we determine the relative errors of the ROM

solutions obtained via the implicit midpoint rule and the discrete gradient

pair approach for various time step sizes ranging from 2·10−6to 217 ·10−6≈

0.13. The specified error values correspond to the relative error with respect

to the Frobenius norm of the ROM state snapshot matrices. The resulting

error decays for both time integration methods are depicted in Figure 6.9

together with a reference line, which corresponds to a convergence order of

two. In particular, we observe that the convergence behavior of both methods

is very similar and the discrete gradient pair approach is almost as accurate

as the implicit midpoint rule. In addition, the numerical results indicate a

convergence order of two as it is to be expected for the implicit midpoint rule,

see for instance [80, sec. 6.3.2].

So far, we have only considered the case where the ROM is evaluated for the

same parameter setting as has been used in the offline phase for generating the

FOM snapshot data. At the end of this section, we also investigate the ability

of the ROM to handle parameter variations. For this purpose, we consider

variations of the diffusion coefficient dwithin the range M= [10−3.5,10−2.5],

see Figure 6.10 for corresponding pseudocolor plots of the FOM solutions for

the extreme values 10−3.5and 10−2.5. We note that the only FOM coefficient

matrix depending on dis Rh, cf. (6.11). Especially, Rhis given by the sum of

two matrices: The first one is linear with respect to d, while the second one

is independent of d. Consequently, the d-dependency of Jh−Rhallows for a

194

6.2. Linear Advection–Diffusion Equation

0 0.5 1

0.5

1.5

0 0.5 1

0.5

1.5

0.2

0.4

x(t, ξ)

Figure 6.10.: Linear advection–diffusion equation: pseudocolor plots of the FOM solution

for d= 10−3.5(left) and d= 10−2.5(right).

10−3.410−3.210−310−2.810−2.6

0.05

0.1

diffusion coefficient d

relative error

Figure 6.11.: Linear advection–diffusion equation: ROM accuracy for different values of

the diffusion coefficient dwhen using a ROM with r= 3 transformed modes

based on FOM snapshots with d= 10−3.

separation similar to (2.24) with K= 2 and thus we may construct parameter-

dependent ROMs while achieving an efficient offline/online decomposition as

mentioned in Remark 4.3.10.

For investigating the parameter-dependent case, we start by considering the

hyperreduced ROM which has been the subject of the discussion after (6.14),

i.e., it is based on r= 3 modes, which have been determined based on snapshots

of the FOM with d= 10−3. As already mentioned, the corresponding relative

online error is 9.9·10−3when using the same parameter value as in the offline

phase. In Figure 6.11 it is depicted how the error changes as the diffusion

coefficient is varied within the range M= [10−3.5,10−2.5]. As expected, the

error is relatively small in the vicinity of d= 10−3but increases towards the

boundaries of M. The error attains its maximum value of about 0.14 at the

right boundary, which corresponds to the highest amount of diffusion and thus

also to the strongest deformation of the wave profiles, cf. Figure 6.10, right.

In the following, we construct an alternative parameter-dependent ROM by

195

6. Numerical Examples

employing the greedy algorithm introduced in section 3.2. To this end, an

error indicator is required to decide in each iteration which parameter value

is used for determining a new mode, cf. Algorithm 3.3. From the perspective

of computational efficiency, it would be in general desirable to use an error

indicator whose evaluation does not scale with the dimension of the FOM.

This is for instance the case when using a residual-based error bound as in

Theorem 4.1.9, provided that the residual norm is approximated by combining

the approaches outlined in Remarks 4.3.10 and 4.3.11. Accordingly, we have

implemented and tested error bounds of the form (4.10) using different methods

for bounding the matrix exponential, cf. Remark 4.1.10. However, the resulting

error bounds appear to be not useful for the problem at hand, since the error

bound is mostly orders of magnitudes larger than the actual error. Moreover,

the parameter dependency of the true error is also not well captured by the

corresponding error bound. Consequently, in the following we consider the true

relative online error as error indicator for the greedy algorithm, while exploiting

that the evaluation of the FOM is already quite fast for the problem at hand,

cf. the discussion after (6.14). Furthermore, we choose the logarithmically

equidistant grid

Mtrain ={10−3.5,10−3.4,...,10−2.5}

as parameter training set and an error tolerance tol = 0.05. Besides, as initial

ansatz vector we choose one mode that is determined based on snapshots of

the FOM with d= 10−3, which corresponds to the logarithmic middle of the

considered parameter domain.

During the greedy procedure, one mode is added after another until the

tolerance is met. For the corresponding ROMs constructed during this iterative

procedure, a plot of the relative online error over the diffusion coefficient is

provided in Figure 6.12. In particular, the black circles indicate the respective

worst-case parameter values, cf. µmax in Algorithm 3.3. The greedy algorithm

terminates after 3iterations, since the resulting ROM with r= 4 achieves a

relative online error of less than tol for all parameter values in the training set

Mtrain. Moreover, we observe that the ROM with r= 3 has a maximum error

of approximately 0.08 and thus outperforms the ROM considered in Figure 6.11

in terms of the maximum error. While both ROMs have the same dimension,

the latter has been constructed using only FOM snapshots with d= 10−3,

whereas the former is based on FOM snapshots with d= 10−3.5,d= 10−3, and

d= 10−2.5.

6.3. Nonlinear Reaction–Diffusion Equation

To illustrate the application of the hyperreduction framework for nonlinear

systems presented in section 4.3.2, we consider a nonlinear reaction–diffusion

problem on the one-dimensional spatial domain Ω = (0,1) and on a time

interval I= [0, tend]with tend ∈R>0. The corresponding governing equations

196

6.3. Nonlinear Reaction–Diffusion Equation

10−3.410−3.210−310−2.810−2.6

tol

0.1

0.15

0.2

0.25

0.3

0.35

diffusion coefficient d

relative error

r= 1

r= 2

r= 3

r= 4

Figure 6.12.: Linear advection–diffusion equation: relative online error for different mode

numbers and values of dwhen using the greedy algorithm from section 3.2 for

determining the modes. The black circles highlight the respective worst-case

parameter values.

read











∂tu(t, ξ) = d∂ξξu(t, ξ) + ζˆv(u(t, ξ), β)for all (t, ξ)∈I×Ω,

u(t, 0) = 1 for all t∈I,

∂ξu(t, 1) = 0 for all t∈I,

u(0, ξ) = u0(ξ)for all ξ∈Ω

with unknown normalized temperature u:I×Ω→Rand given diffusion

parameter d∈R>0, pre-exponential factor ζ∈R>0, Arrhenius coefficient

β∈R>0, initial value u0:Ω→[0,1], and reaction rate function ˆv:R×R→R

defined via

ˆv(u, β):=





0,if u≤0,

(1 −u) exp(−β

u),otherwise. (6.15)

This test case is based on the reaction–diffusion example presented in [118,

eq. (10.67)], but in contrast to [118] we do not explicitly constrain the state

uto only attain values between 0and 1. Nevertheless, we emphasize that

values outside of this range are not meaningful from a physical point of view.

Furthermore, in contrast to [118], we consider a diffusion coefficient dwhich

is not necessarily equal to one in order to obtain a transport-dominated test

case with a sharp reaction front. Finally, inspired by the wildland fire model

197

6. Numerical Examples

considered in Example 5.1.4, we set the threshold value in (6.15) to zero in

order to ensure that ˆvis smooth with respect to ufor any β∈R>0.

In order to obtain a system with equilibrium point at 0and homogeneous

boundary conditions, we introduce the new state variable x:=u−1and obtain

the corresponding governing equations











∂tx(t, ξ) = d∂ξξx(t, ξ)−ζv(x(t, ξ), β)for all (t, ξ)∈I×Ω,

x(t, 0) = 0 for all t∈I,

∂ξx(t, 1) = 0 for all t∈I,

x(0, ξ) = x0(ξ):=u0(ξ)−1for all ξ∈Ω

(6.16)

with v:R×R→Rdefined via

v(x, β):=−ˆv(x+ 1, β) = 





0,if x≤ −1,

xexp(−β

x+1 ),otherwise. (6.17)

Then, using integration by parts, we obtain a corresponding weak formulation

as follows: Find x:I×Ω→Rsuch that

(i) for all t∈I,x(t, ·)is in V:={z∈H1(Ω) |z(0) = 0}and satisfies

hψ, ∂tx(t, ·)iL2(Ω) =−dhψ0, ∂ξx(t, ·)iL2(Ω) −ζhψ, v(x(t, ·), β)iL2(Ω)

for all ψ∈ V,

(ii) for all ξ∈Ω, we have x(0, ξ) = x0(ξ).

Furthermore, similarly as in section 6.2, we introduce the associated Hamil-

tonian H:L2(Ω) →Rvia H(x):=1

2kxk2

L2(Ω) and derive in the following a

corresponding dissipation inequality. To this end, let x:I×Ω→Rbe a solu-

tion of the weak formulation of (6.16) such that the mapping t7→ x(t, ·)is in

C1(I, L2(Ω)). Moreover, analogously as in section 6.2, we consider the func-

tion Hs:I→Rdefined via Hs(t):=H(x(t, ·)) and obtain the corresponding

dissipation inequality

dHs

dt(t) = H0(x(t, ·))(∂tx(t, ·)) = hx(t, ·), ∂tx(t, ·)iL2(Ω)

=−dk∂ξx(t, ·)k2

L2(Ω) −ζhx(t, ·), v(x(t, ·), β)iL2(Ω) ≤0

for all t∈I, where the last inequality follows from the definition of vin (6.17).

Thus, we infer that the Hamiltonian does not increase along solution trajecto-

ries. This property is also reflected in a dissipative Hamiltonian structure on

the semi-discretized level, cf. the upcoming equation (6.24).

For the semi-discretization in space, we use the Galerkin FEM with mesh

size h=1

N+1 ,N∈N, analogously as in the preceding two sections. The

198

6.3. Nonlinear Reaction–Diffusion Equation

resulting semi-discretized system is given by

Eh˙xh(t) = −R1,hxh(t)−vh(xh(t)) for all t∈I,

where xh:I→RN+1 contains the coefficients corresponding to the FEM ansatz

functions. Moreover, the coefficient matrices Eh, R1,h ∈RN+1,N+1 are given by

Eh:=h







4 1 0 ··· 0

1 4 .......

0......1 0

....1 4 1

0··· 0 1 2







, R1,h :=d







2−1 0 ··· 0

−1 2 .......

0......−1 0

....−1 2 −1

0··· 0−1 1





(6.18)

and the nonlinearity vh:RN+1 →RN+1 is defined via

[vh(xh)]i:=ζ*ψi, v 



N+1

j=1

[xh]jψj, β

+L2(Ω)

=ζZ

Ω(xh)

ψi(ξ) exp −β

1 + PN+1

k=1 [xh]kψk(ξ)!N+1

j=1

[xh]jψj(ξ) dξ

=ζ

N+1

j=1

[xh]jZ

Ω(xh)

ψi(ξ)ψj(ξ) exp −β

1 + PN+1

k=1 [xh]kψk(ξ)!dξ

(6.19)

for i= 1, . . . , N + 1, with ψ1, . . . , ψN+1 denoting the usual hat functions with

ψi(jh) = δij for i= 1, . . . , N + 1 and j= 0, . . . , N + 1. Here, b

Ω(xh)⊆Ω

denotes the subset of the spatial domain where the reaction is active, i.e., the

mapping b

Ω: RN+1 → P(Ω) is defined via

Ω(xh):=



ξ∈Ω

N+1

j=1

[xh]jψj(ξ)>−1



,(6.20)

cf. (6.17). Since the evaluation of vhas defined in (6.19) still involves an

integral which depends on the unknown xh, we propose to approximate this

integral by means of the composite trapezoidal rule. As quadrature points, we

use the FEM grid points, while setting the weights of all grid points which lie

outside of b

Ω(xh)to zero. The resulting system reads

Eh˙xh(t) = −R1,hxh(t)−˜vh(xh(t)) for all t∈I,(6.21)

199

6. Numerical Examples

where ˜vh:RN+1 →RN+1 is defined via

[˜vh(xh)]i:=ζ

N+1

j=1

[xh]j

N+1

k=1

ˆωk(xh)ψi(kh)ψj(kh) exp −β

1 + PN+1

`=1 [xh]`ψ`(kh)!

=ζ

N+1

j=1

[xh]j

N+1

k=1

ˆωk(xh)δikδjk exp −β

1 + PN+1

`=1 [xh]`δk` !

=ζ[xh]iˆωi(xh) exp −β

1+[xh]i!,if [xh]i6=−1,

[˜vh(xh)]i:= 0,otherwise

(6.22)

for i= 1, . . . , N + 1 and the weights ˆω1,...,ˆωN+1 :RN+1 →R≥0are given by

ˆω1(xh):=









h, if ([xh]1,[xh]2)∈(R>−1)2,

21 + [xh]1+1

[xh]1−[xh]2,if ([xh]1,[xh]2)∈R>−1×R≤−1,

0,otherwise,

ˆωN+1(xh):=









2,if ([xh]N,[xh]N+1)∈(R>−1)2,

[xh]N+1+1

[xh]N+1−[xh]N,if ([xh]N,[xh]N+1)∈R≤−1×R>−1,

0,otherwise,

ˆωk(xh):=











h, if ([xh]k−1,[xh]k,[xh]k+1)∈(R>−1)3,

z1,k(xh),if ([xh]k−1,[xh]k,[xh]k+1)∈R≤−1×R>−1×R≤−1,

z2,k(xh),if ([xh]k−1,[xh]k,[xh]k+1)∈(R>−1)2×R≤−1,

z3,k(xh),if ([xh]k−1,[xh]k,[xh]k+1)∈R≤−1×(R>−1)2,

0,otherwise,

z1,k(xh):=h

2([xh]k+ 1) 1

[xh]k−[xh]k−1

[xh]k−[xh]k+1 !,

z2,k(xh):=h

2 1 + [xh]k+ 1

[xh]k−[xh]k+1 !,

z3,k(xh):=h

2 1 + [xh]k+ 1

[xh]k−[xh]k−1!

(6.23)

for k= 2, . . . , N, see appendix D.2 for the corresponding derivation. Based on

(6.22) we may write (6.21) equivalently as

Eh˙xh(t) = −(R1,h +R2,h(xh(t)))xh(t)for all t∈I,(6.24)

where R2,h :RN+1 →RN+1,N+1 is pointwise diagonal and defined via

[R2,h(xh)]i,i :=





ζˆωi(xh) exp −β

1+[xh]i,if [xh]i6=−1,

0,otherwise (6.25)

200

6.3. Nonlinear Reaction–Diffusion Equation

0 0.5 1

100

200

−1

x(t, ξ)

0 0.5 1

−1

t= 0 t= 60 t= 120 t= 180 t= 240

Figure 6.13.: Nonlinear reaction–diffusion equation: pseudocolor plot of the FOM solution

(left) and some selected snapshots (right).

for i= 1, . . . , N +1. In particular, based on (6.18) and (6.25), we note that Eh

is symmetric and positive definite, R1,h is symmetric and positive semi-definite,

and R2,h is pointwise symmetric and positive semi-definite. Consequently, the

semi-discretized system (6.24) has a dissipative Hamiltonian structure with

Hamiltonian Hh:RN+1 →Rdefined via Hh(xh):=1

2x>

hEhxh. Especially, as

in the preceding two sections, the Hamiltonian is a quadratic function and

hence we may apply the implicit midpoint rule for the time discretization to

ensure a dissipation inequality on the time-discrete level.

For the following numerical experiments, we choose the PDE parameters as

d= 5 ·10−5,ζ= 1, and β= 1, the final time as tend = 250, and the initial

value as

x0(ξ) = 









0,if ξ∈[0,0.1],

−exp(1

1−10ξ)

exp(1

1−10ξ)+exp(1

10ξ−2),if ξ∈(0.1,0.2),

−1,otherwise

for all ξ∈Ω, cf. Figure 6.13, right. Furthermore, we divide the spatial domain

into N+ 1 = 1000 equidistant intervals, which corresponds to a mesh size of

h= 10−3, and for the time discretization we use a step size of 0.2. Figure 6.13

depicts the corresponding numerical solution by means of a pseudocolor plot

and some selected snapshots. We observe that the initial wave profile is trans-

ported to the right with an approximately constant wave speed. There is a

slight change in the shape of the wave at the beginning of the time interval,

which may be seen for instance by comparing the initial snapshot with the

one corresponding to t= 60. However, afterwards the shape remains nearly

constant.

As in the previous sections, we employ the residual minimization approach

from section 3.1 for the mode determination based on the FOM snapshot data.

201

6. Numerical Examples

In particular, we use W=Ehdefined in (6.18) as spatial weighting matrix and

we consider an approximation based on r= 1 mode with dimension dφ=N+1.

Furthermore, we use a discretized version of the constant extrapolation shift

operator defined in Definition 3.3.8 and restrict it to functions which vanish at

the left boundary. Due to this restriction, the resulting transformation family

may also be regarded as a combination of the zero padding shift operator

for non-negative shift values and the constant extrapolation shift operator for

negative shift values. This choice reflects the homogeneous Dirichlet condition

on the left boundary and the homogeneous Neumann condition on the right

boundary in (6.16). For the discretization, we use cubic splines such that

the resulting family of discretized shift operators Tc,h :R→ L(RN+1)may be

specified as

Tc,h(η)φ:=T1(q(η))φ+T2(q(η))a(φ)−ζ(η)b(φ) + ζ(η)2c(φ)−ζ(η)3d(φ)

with q(η):=jη

hk,ζ(η):=ηmod h,

T1(q):=









0∈RN+1,N+1,if q≥0,



0 0

01min(N+1,−q)

∈RN+1,N+1,otherwise,

T2(q):=













0 0

Imax(N+1−q,0) 0

∈RN+1,N+1,if q≥0,



0Imax(N+1+q,0)

0 0 

∈RN+1,N+1,otherwise.

(6.26)

Here, a(φ), b(φ), c(φ), d(φ)∈RN+1 are vectors containing the spline coefficients

corresponding to φ, see appendix D.1.3 for their definition and a derivation of

Tc,h from Tc.

In contrast to the test cases considered in the previous two sections, the wave

propagation observed in Figure 6.13 is not caused by an advection term in the

corresponding PDE, but instead originates from an interplay between reaction

and diffusion. Consequently, the wave speed may not be simply read from the

PDE in (6.16), which prevents us from choosing the offline shift values in the

same way as in sections 6.1 and 6.2. Instead, we determine them by finding

the minimum of the spatial derivative of each FOM snapshot to identify the

position of the reaction front depicted in Figure 6.13, right. To this end, we

use a simple forward finite difference scheme for obtaining an approximation

of the first derivative of each snapshot.

As in the previous sections, we use the variable projection approach from sec-

tion 3.1.2 for the actual computation of the modes and solve the corresponding

optimization problem by means of the GRANSO software with default settings.

For the starting values, we use similarly as in section 6.1 the first left singular

202

6.3. Nonlinear Reaction–Diffusion Equation

vector of the transformed snapshot matrix

hTc,h(−p(t1))xh(t1)··· Tc,h(−p(tq))xh(tq)i.

This seems to be a natural way of aligning the FOM snapshots, cf. Figure 6.13.

Based on these starting values, the GRANSO solver terminates after 184 iter-

ations and the resulting relative offline error is about 2.7·10−3.

We proceed to construct a ROM of the form (4.1)–(4.2) based on the de-

termined mode φ1and using the derivative of Tc,h provided in (D.14) in ap-

pendix D.1.3. As in the previous two sections, we slightly differ from the

setting in section 4.1 and use the weighted E−1

h-norm for the residual mini-

mization, cf. section 5.2. Especially, we note that although Theorem 5.2.1 is

stated for a FOM with constant coefficient matrices, the theorem and its proof

straightforwardly extend to the case of a state-dependent dissipation matrix.

Consequently, we may also apply this theorem here, which yields that the

ROM is dissipative Hamiltonian. The resulting ROM with r= 1 has the form

"Mα(p)αN(p)

αN(p)α2Mp(p)#"˙α

˙p#="Aα(p)

αAp(p)#α+"vα(α, p)

αvp(α, p)#,(6.27)

where Mα,N, and Mpare obtained as in (4.2), but the occurring inner prod-

ucts need to be replaced by weighted inner products with weighting matrix

Eh. Furthermore, Aαand Apare as specified in (4.37) with F=−R1,h and

vα, vp:R×R→Rare defined as

vα(α, p):=−φ>

1Tc,h(p)>˜vh(Tc,h(p)φ1α),

vp(α, p):=−φ>

1T0

c,h(p)>˜vh(Tc,h(p)φ1α).

As discussed in section 4.3, the evaluation of the ROM still scales with the

dimension of the FOM, since Mα,N,Mp,Aα,Ap,vα, and vpinvolve prod-

ucts of high-dimensional matrices and vectors. Moreover, since the FOM is

nonlinear, this is also reflected in vαand vp, whose evaluations involve the

FOM nonlinearity −˜vh, and this is a second reason which prevents an efficient

evaluation of the ROM. As outlined in sections 4.3.1 and 4.3.2, we treat these

two issues differently. In particular, the path-dependent coefficients Mα,N,

Mp,Aα, and Apare treated as in section 6.2, i.e., we sample them and con-

struct corresponding piecewise linear interpolants. For this purpose, we use a

sampling range of [0,1] and an equidistant grid whose grid size coincides with

the one used for the FEM discretization.

To approximate the functions vαand vp, we follow the approach outlined

in section 4.3.2. We note that the framework in section 4.3.2 is presented for

a PDE as FOM and in particular requires the evaluated nonlinearity to be

a function defined on the spatial domain Ω. Instead, here we have formally

derived the ROM (6.27) based on the finite-dimensional FOM (6.21). Never-

theless, since (6.21) is obtained via a spatial semi-discretization of (6.16) based

203

6. Numerical Examples

on piecewise linear FEM basis functions, the entries of all occurring vectors

of dimension N+ 1 correspond to evaluations of continuous functions at the

FEM grid points. This allows us to still apply the hyperreduction framework

from section 4.3.2 by interpolating between the grid points if necessary.

As the first step of the approach from section 4.3.2, we approximate the FOM

nonlinearity −˜vhbased on a linear combination of transformed EIM ansatz

vectors or modes. These are obtained based on snapshot data of the FOM

nonlinearity, see Figure 6.14 for a depiction of the corresponding snapshots

and note that −˜vhcorresponds to the term −v(·,1) on the infinite-dimensional

level, cf. (6.16) with ζ= 1 and β= 1. In particular, we apply the residual

minimization approach from section 3.1 with one mode and using the same

paths and transformation operators as for the snapshots of the state. Also the

starting value is determined similarly as for the state snapshots, i.e., we choose

the first left singular vector of the transformed snapshot matrix

−hTc,h(−p(t1))˜vh(xh(t1)) ··· Tc,h(−p(tq))˜vh(xh(tq))i.

In fact, this starting value appears to be already very close to a local minimum

point as the GRANSO optimization solver does not perform any iteration, but

instead immediately terminates, since the stationarity tolerance is already met.

The resulting relative approximation error is around 0.026.

Based on the determined EIM mode ψ1,1, we proceed to approximate the

ROM (6.27) as outlined in section 4.3.2. In particular, we replace vα(α, p)

and vp(α, p)in (6.27) by ˆ

Aα(p)βand ˆ

Ap(p)β, respectively, where ˆ

Aαand ˆ

are as specified in (4.48) and their efficient offline/online decomposition works

analogously as for Mα,N,Mp,Aα, and Ap. Here, β:I→Ris given by the

solution of the linear equation system

AEIM(p(t))β(t) = b(α(t), p(t)) (4.47c)

for all t∈I. As stated in (4.46), the definitions of AEIM and binvolve the

EIM collocation point p1,1(p). For the latter we use the update rule (4.52),

i.e., p1,1(p) = p1,1(0) + p. Furthermore, for determining p1,1(0) we apply the

Q-DEIM algorithm introduced in [83] to the EIM mode ψ1,1. We note that the

algorithm in [83] actually outputs a value b

p1,1(0) ∈ {1, . . . , N + 1}, whereas

p1,1(0) as introduced in section 4.3.2 is formally an element of Ω. However,

since the entries of ψ1,1∈RN+1 correspond to the FEM grid points, we may

relate b

p1,1(0) and p1,1(0) via p1,1(0) = hb

p1,1(0). Moreover, since pis a scalar-

valued function for the problem at hand, we may follow Remark 4.3.18 to infer

that AEIM is constant with respect to pand especially given by

AEIM(p) = [ψ1,1]bp1,1(0) .

Formally, this is only valid as long as p1,1(p) = p1,1(0) + pis in Ω, cf. Re-

mark 4.3.18, but in our numerical experiments this assumption turns out to

204

6.3. Nonlinear Reaction–Diffusion Equation

0 0.5 1

100

200

0.02

0.04

0.06

−v(x(t, ξ),1)

0 0.5 1

0.02

0.04

0.06

t= 0 t= 60 t= 120 t= 180 t= 240

Figure 6.14.: Nonlinear reaction–diffusion equation: pseudocolor plot of the FOM nonlin-

earity (left) and some selected snapshots (right).

be always satisfied. Especially, since AEIM is constant with respect to p, it may

be precomputed in the offline phase.

To also derive an efficient way of evaluating the right-hand side bin (4.47c),

we first observe that the ith component of ˜vh(xh)defined via (6.22)–(6.23)

only depends on the three components [xh]i−1,[xh]i, and [xh]i+1, respectively

for i= 2, . . . , N. Thus, we may write [˜vh(xh)]ias

[˜vh(xh)]i= ˆvh([xh]i−1,[xh]i,[xh]i+1),(6.28)

for i= 2, . . . , N, where ˆvh:R3→Ris defined via

ˆvh(η1, η2, η3):=





0,if η2=−1,

ζη2ˆω(η1, η2, η3) exp −β

1+η2,otherwise,

ˆω(η1, η2, η3):=











h, if (η1, η2, η3)∈(R>−1)3,

z1(η1, η2, η3),if (η1, η2, η3)∈R≤−1×R>−1×R≤−1,

z2(η1, η2, η3),if (η1, η2, η3)∈(R>−1)2×R≤−1,

z3(η1, η2, η3),if (η1, η2, η3)∈R≤−1×(R>−1)2,

0,otherwise,

z1(η1, η2, η3):=h

2(η2+ 1) 1

η2−η1

η2−η3!,

z2(η1, η2, η3):=h

2 1 + η2+ 1

η2−η3!, z3(η1, η2, η3):=h

2 1 + η2+ 1

η2−η1!,

cf. (6.22)–(6.23). Moreover, using (6.28) and similar arguments as in Re-

mark 4.3.18, we infer that bis constant with respect to pand given by

b(α, p) = ˆvhα[φ1]bp1,1(0)−1, α [φ1]bp1,1(0) , α [φ1]bp1,1(0)+1.

205

6. Numerical Examples

Here, we have assumed that p1,1(p) = p1,1(0) + pis in [2h, Nh](Ωand

note that this assumption indeed turns out to be satisfied in our numerical

experiments. In particular, bmay be efficiently evaluated in the online phase,

since ˆvhdoes not scale with the FOM dimension in contrast to ˜vh.

For comparing the ROM (6.27) and its hyperreduced counterpart, we use as

in sections 6.1 and 6.2 the implicit midpoint rule for time integration and the

same parameter values which have been used for generating the FOM snapshot

data in the offline phase. In particular, the evaluation of the non-hyperreduced

ROM takes approximately four seconds, whereas the hyperreduced ROM re-

quires only a computation time of about 2.5seconds. This corresponds to a

relative speed-up of roughly 1.6, while the error of the hyperreduced ROM is

around 5.6·10−3and thus only slightly larger than the 5·10−3obtained by the

non-hyperreduced ROM. Moreover, both ROMs are significantly faster than

the corresponding FOM, which requires a computation time of approximately

five minutes. We note that, in contrast to the test cases considered in sec-

tions 6.1 and 6.2, the reaction–diffusion FOM (6.21) is nonlinear and hence

its evaluation involves solving a high-dimensional nonlinear equation system

in each time step.

We also compare the ROMs based on transformed modes with a ROM based

on a classical approach using POD and DEIM, cf. section 2.5. In particular,

a ROM based on 40 POD and 60 DEIM modes yields a relative online error

of around 6.1·10−3, which is thus comparable to the error obtained via the

hyperreduced ROM based on transformed modes. However, the evaluation of

the POD/DEIM-based ROM takes about 9.6seconds, which corresponds to

almost four times the computation time required for the hyperreduced ROM

based on transformed modes. Another advantage of the latter ROM is that it

allows for more flexibility of choosing the time step size, similarly as reported

in sections 6.1 and 6.2. For example, the ROMs based on transformed modes

yield a relative error of roughly one per cent when increasing the time step

size from 0.2to 50. The fact that this large increase of the time step size is

only accompanied by a rather small increase of the error, may be explained

by the observation that the path pis nearly a linear function of time and

the amplitude αis almost constant, see also the corresponding discussion for

the wave equation test case in section 6.1. Also the POD/DEIM-based ROM

allows to significantly increase the time step size without leading to a significant

increase of the error. However, this effect is less pronounced than for the ROMs

based on transformed modes. For instance, the POD/DEIM-based ROM yields

a relative error of more than 0.1when increasing the time step size to 10.

Remark 6.3.1 (Choice of the number of DEIM modes for the POD-based

ROM).The number of DEIM modes used for the hyperreduction of the POD-

based ROM has been chosen such that the relative error between the hyper-

reduced and non-hyperreduced ROM is smaller than 10−5. Consequently, this

error is negligible in comparison to the error between the non-hyperreduced

ROM and the full-order model. However, in our numerical experiments we

206

6.3. Nonlinear Reaction–Diffusion Equation

have observed that decreasing the number of DEIM modes may lead to a

decreasing error with respect to the FOM, even though the corresponding er-

ror with respect to the non-hyperreduced ROM increases at the same time.

For instance, a POD/DEIM-based reduced-order model with 40 POD and 40

DEIM modes yields a relative error of 2.8·10−3in comparison to the FOM.

This is significantly smaller than the 6.1·10−3obtained using 40 POD and 60

DEIM modes, even though the latter corresponds to a better approximation

of the non-hyperreduced ROM. Thus, it appears that the performance of the

POD/DEIM-based reduced-order model could be improved by exploiting this

counterintuitive behavior. However, since it is in general not clear how the

number of DEIM modes may be chosen to minimize the error with respect to

the FOM, we omit further numerical experiments in this direction. ¨

We close this section by briefly discussing the compliance with the power

balance associated with the considered reaction–diffusion test case. As pointed

out after (6.25), the FOM (6.21) allows for a dissipative Hamiltonian repre-

sentation of the form (6.24). Consequently, an energy-consistent ROM should

have the property that the associated Hamiltonian does not increase over time.

Indeed, the ROM (6.27) may be formulated as a dissipative Hamiltonian sys-

tem, which follows from the fact that we have used the weighted E−1

h-norm for

the residual minimization and from the considerations in section 5.2. Thus,

the ROM is energy-consistent in the sense that the associated Hamiltonian is

non-increasing with respect to time. However, as illustrated in sections 6.1

and 6.2, this property is not necessarily preserved after discretization in time,

at least not when using the implicit midpoint rule. Nevertheless, in our nu-

merical experiments we have not observed any increase of the Hamiltonian

associated with the ROM (6.27), even when using the implicit midpoint rule

and significantly increasing the time step size. Accordingly, we omit a cor-

responding numerical comparison between the implicit midpoint rule and the

discrete gradient pair approach from appendix C.2.

207

7. Conclusion

We close this thesis by providing a summary in section 7.1 and by addressing

some future research directions in section 7.2.

7.1. Summary

In this thesis we consider a specific nonlinear model order reduction (MOR)

approach which is suitable for an effective reduction of transport-dominated

systems. To this end, we approximate the state of the full-order model (FOM)

by a linear combination of transformed modes, where the transformation op-

erators are parametrized by time-dependent path variables. In the examples

presented in this thesis, we mostly consider translation or shift operators on

one-dimensional spatial domains and in this case the paths correspond to

the respective shift amounts. As a consequence, this class of approximation

ansatzes may result in very low-dimensional and accurate reduced-order mod-

els even for transport-dominated problems which are challenging for classical

methods based on linear approximation ansatzes. Moreover, the new model

reduction framework allows to preserve important system properties such as

stability and passivity by exploiting a port-Hamiltonian (pH) representation

of the full-order model.

The determination of suitable modes based on snapshot data of the FOM

is subject of chapter 3. For this purpose, we propose to solve an optimiza-

tion problem such that the approximation error between the snapshot data

and the corresponding approximation based on transformed modes is mini-

mized. Especially, we assume the paths to be given or determined in a pre-

processing step and present two different optimization approaches. The first

one is based on directly solving the full optimization problem in terms of

the modes and the corresponding amplitudes. The second approach uses the

variable projection method, which results in a reduced problem where the opti-

mization parameters consist only of the modes. Furthermore, for the treatment

of parameter-dependent problems, we extend the classical POD-greedy algo-

rithm to nonlinear approximation approaches based on transformed modes. In

particular, the resulting algorithm determines one mode after another based

on adaptively chosen parameter samples and uses the reduced optimization

approach obtained via variable projection. While the techniques mentioned

so far are presented for a general class of transformation operators, we also

discuss some special classes of shift operators for one-dimensional problems

with different boundary conditions. In addition, we compare the proposed

209

7. Conclusion

optimization framework with some related techniques from the literature.

In chapter 4 we demonstrate how to construct a reduced-order model (ROM)

using a nonlinear approximation ansatz based on transformed modes. To this

end, we assume the modes to be given and derive a ROM by minimizing the

residual which is obtained by substituting the approximation ansatz into the

full-order model. The state of the resulting ROM consists of the amplitudes

and the paths. Moreover, the ROM is in general nonlinear and involves a

state-dependent mass matrix, which is due to the nonlinearity of the used ap-

proximation ansatz. In general, it is not guaranteed that this mass matrix is

nonsingular and thus we present a corresponding regularization approach to

ensure its invertibility. Furthermore, we discuss the solvability of the ROM and

present an a posteriori error bound for a special class of linear full-order mod-

els. In addition, we compare a special case of the presented model reduction

approach with a framework based on symmetry reduction and demonstrate a

relation between the corresponding optimization problems. Finally, since the

evaluation of the ROM scales in general with the dimension of the FOM, we

present a hyperreduction approach, which is based on an additional approxi-

mation of the ROM to ensure an efficient evaluation in the online phase. In

particular, for the case that the FOM is nonlinear, we present a technique which

is based on ideas of the (discrete) empirical interpolation method ((D)EIM)

and uses an approximation of the FOM nonlinearity via a linear combination

of transformed ansatz functions.

Chapter 5 is devoted to structure-preserving model order reduction for port-

Hamiltonian systems in a general finite-dimensional setting. We consider dif-

ferent classes of pH systems for the FOM including linear, nonlinear, time-

invariant, and time-varying systems. Moreover, we discuss not only linear

approximation ansatzes but also two classes of nonlinear ansatzes, which in

particular include the class of approximation ansatzes considered in the previ-

ous chapters of this thesis. Furthermore, we demonstrate that in many cases

structure-preserving MOR may be achieved via a residual minimization ap-

proach by using a suitable weighted norm for the residual, especially if the

corresponding FOM is linear. The weighted norm used for the residual mini-

mization may be motivated by a residual-based error bound, where the error

is measured via another weighted norm which is related to the Hamiltonian.

Besides, we also discuss the stability of the ROMs and provide corresponding

sufficient conditions.

Finally, in chapter 6 we demonstrate the application of some of the methods

presented in chapters 3 to 5 by means of three numerical test cases. First,

we consider a linear wave equation with periodic boundary conditions and

obtain a corresponding reduced-order model based on two shifted modes. In

particular, the numerical experiments reveal that the time step size may be

considerably increased for this ROM without significantly increasing the er-

ror. Furthermore, we demonstrate that the Hamiltonian of the ROM is a

conserved quantity even after time discretization, when using a discrete gra-

210

7.2. Outlook

dient pair approach as outlined in appendix C. The second test case involves

an advection–diffusion equation with mixed Robin–Neumann boundary con-

ditions, which lead to incoming and outgoing waves at the boundaries. We

demonstrate the performance of the greedy algorithm presented in chapter 3

and obtain a parameter-dependent ROM which yields a decent approximation

quality over the considered range of parameter values. For the last test case,

we consider a nonlinear reaction–diffusion equation whose solution features a

traveling reaction front. Here, we apply the hyperreduction approach from

chapter 4 and obtain a ROM which may be efficiently evaluated and especially

outperforms a corresponding ROM based on POD and DEIM.

7.2. Outlook

The research fields of nonlinear and structure-preserving model order reduction

methods currently experience a significant increase in research efforts and there

are still many open questions to be addressed in the future. In the following

we mention some of them with a special focus on those which are the most