Error bounds for POD expansions of parameterized transient temperatures

Accepted Manuscript Error bounds for POD expansions of parameterized transient temperatures M. Aza¨ ıez, F. Ben Belgacem, T. Chac´ on Rebollo PII: S0045-7825(16)30046-9 DOI: http://dx.doi.org/10.1016/j.cma.2016.02.016 Reference: CMA 10848 To appear in: Comput. Methods Appl. Mech. Engrg. Received date: 31 March 2015 Revised date: 13 October 2015 Accepted date: 12 February 2016 Please cite this article as: M. Aza¨ ıez, F.B. Belgacem, T.C. Rebollo, Error bounds for POD expansions of parameterized transient temperatures, Comput. Methods Appl. Mech. Engrg. (2016), http://dx.doi.org/10.1016/j.cma.2016.02.016 This is a PDF file of an unedited manuscript that has been accepted for publication. As a service to our customers we are providing this early version of the manuscript. The manuscript will undergo copyediting, typesetting, and review of the resulting proof before it is published in its final form. Please note that during the production process errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain. Error Bounds for POD expansions of parameterized transient temperatures. M. Aza¨ıez∗ , F. Ben Belgacem† , T. Chac´on Rebollo ‡ March 19, 2016 Abstract We focus on the convergence analysis of the POD expansion for the parameterized solution of transient heat equations. The parameter of interest is the conductivity coefficient. We prove that this expansion converges with exponential accuracy, uniformly if the conductivity coefficient remains within a compact set of positive numbers. This convergence result is independent of the regularity of the temperature with respect to the space and time variables. We present some numerical experiments to show that a reduced number of modes allows to represent with high accuracy the family of solutions corresponding to parameters that lie in the compact set under study. 1 Introduction Karhunen-Lo`eve’s expansion (KLE) provides a reliable procedure for computing a low dimensional representation of spatiotemporal signals (see [20, 10]) and is widely used in various communities. It is referred to as the principal components analysis (PCA) in statistics (see [22, 14, 15]), or called singular value decomposition (SVD) in linear algebra (see [11]). In computational mechanics where it is wide-spread, it bears the name of proper orthogonal decomposition (POD) (see [3, 13]). Since the last decades, it is fitted within the frame of reduced basis methods to perform the numerical simulations of parameter-depending dynamic systems, which allow substantial savings of computational costs and makes affordable the solution of problems that need a large amount of data. An important work may be found in the context of parameterized partial differential equations. We refer to (see [13, 9, 25, 26, 6, 5]) without being exhaustive We focus our contribution on the accuracy of the truncated KLE or POD for the transient temperature functions T, when parameterized by the conduction coefficient γ. This is a continuation work of [2], where the time was taken as the parameter while the space was the central variable. ∗I2M, IPB (UMR CNRS 5295), Universit´e de Bordeaux, 33607 Pessac (France) †Sorbonne University, Universit´e de technologie de Compi`egne, LMAC Laboratory of Applied Mathematics of Compi`egne - CS 60 319 - 60 203 Compi`egne cedex ‡I2M, IPB (UMR CNRS 5295), Universit´e de Bordeaux, 33607 Pessac (France) and Departamento EDAN & IMUS, Universidad de Sevilla, C/Tarfia, s/n, 41012 Sevilla (Spain) 1 The problem being bivariant, POD is strictly equivalent to decomposition by PGD and so the results obtained therefore apply for both cases The study we undertake here aims a bound of the approximation error that decays exponentially fast, in the number of the retained modes. This convergence is uniform in the conductivity coefficient provided it remains bounded away from zero. Let us point out the important fact that the smoothness of the temperature with respect to the space-time variables has no impact on the final estimates. The main theoretical tool is to apply the Courant-Weyl Theorem [19] (also known as Courant-Fisher min-max principle) to the POD operator whose kernel is the temperature field T=T(γ). An appropriate approximation of the temperature field by polynomials yields the desired bound on the truncating POD error approximation. The accuracy of such an approximation requires some smoothness results of the temperature T=T(γ). In spite of their role, the way these results may be derived is not the heart of our work. Hence, the simplicity option is preferred here ; we call hence for the Fourier analysis to obtain the desired regularity. In case the Fourier basis is not accessible, alternatives for the investigation of the regularity of Tupon γmay be available. The reader interested in is referred to the highly technical mathematics in [7, 17], where similar results can be found for elliptic value problems. We present afterwards some numerical simulations to check out the theoretical convergence rate of the POD expansion of the solution to the heat equation. We also confirm the dependence of the convergence rate with respect to the range of thermal diffusivities. The outlines are as follows. Section 2 recalls the POD or Karhunen-Lo`eve expansion for biparametric functions. Section 3 analyzes the velocity of convergence of the POD expansion when applied to the solutions of the heat equation. Finally, in Section 4 we present the numerical investigation. Notation — Let X⊂Rdbe a given Lipschitz domain. We denote by L2(X) the space of measurable and square integrable functions on X. The Sobolev spaces H1(X) contain all the function that belong to L2(X) together with all its first derivatives (see [1]). Then let Gbe a measure space and Ha Hilbert space. We denote by L2(G, H) the Bochner space of measurable and square integrable vector-valued functions from Gon H(cf. [8]). 2 2 Karhunen-Lo`eve decomposition The Karhunen-Lo`eve decomposition, also known as Proper Orthogonal decomposition (POD in the sequel) provides a technique to obtain low-dimensional approximations of parametric functions. For a rapid description of it, we consider G⊂Rdand Q ⊂ Rntwo bounded domains, dand nare integers ≥1. Gwill be the set of parameters and Qstands for the spatio-temporal domain. Let Tbe a given function in the Lebesgue space L2(G×Q) that we want to approximate in a low-dimensional variety. Define then the integral operator with kernel Texpressed as ϕ7→ B ϕ, (B ϕ)(z) = ZG T(γ, z)ϕ(γ)dγ. (1) Bbelongs to L=L(L2(G), L2(Q)), the space of bounded linear operators mapping L2(G) into L2(Q) and we have kBkL≤ kTkL2(G×Q). The adjoint operator B∗is defined from L2(Q) into L2(G) as v7→ B∗v, (B∗v)(γ) = ZQ T(γ, z)v(z)dz. (2) Let us then consider the POD operator A=B∗B, which is also an integral operator whose kernel K∈L2(G×G) is expressed by K(γ, µ) = ZQ T(γ, z)T(µ, z)dz. (3) The operator Ais linear, bounded, self-adjoint and compact. This results from the fact that the operator Bis compact by the Kolmogorov compactness criterion in L2(G) (cf. Muller [21], Chapter 2). Consequently, there exists a complete orthonormal basis of L2(G) formed by eigenvectors (ϕm)m≥0of A, associated to non-negative eigenvalues (λm)m≥0, that we assume to be ordered in decreasing value, such that lim m→∞ λm= 0 . Each non-zero eigenvalue has a finite multiplicity, and 0 is the only possible accumulation point of the spectrum. Moreover, setting σm= (λm)1/2, the sequence (vm)m≥0given by vm=1 σm Bϕm, is an orthogonal basis of L2(Q) , and B∗vm=σmϕm.(4) The terms of the sequence (σm)m≥0are the singular values of B. The positivity of the operator A makes the kernel Kbe a Mercer kernel. Using Mercer’s theorem yields the following decomposition (see [13, Chapter 3.8.2, Proposition 2]) K(γ, µ) = X m≥0 λmϕm(γ)ϕm(µ),in G×G. 3 This statement comes from the Hilbert-Schmidt theorem on the spectral decomposition of selfadjoint compact operators. This decomposition results in the POD expansion. Lemma 2.1 We have that T(γ, z) = X m≥0 σmϕm(γ)vm(z),in G×Q,(5) where the series is convergent in L2(G×Q). The POD expansion is optimal in the L2-norm (cf. [21], Chapter 2). Indeed, let us denote TMthe truncated POD expansion to the order M≥1. Then , let WMbe the subspace Span (w0,···, wM)⊂ L2(Q). The following holds, Lemma 2.2 ([13]) The following inequality holds kT−TMkL2(G×Q)≤inf SM∈L2(G,WM)kT−SMkL2(G×Q). 4 3 Analysis of parameterized transient temperatures We focus here on the homogeneous Dirichlet boundary value problem for the heat equation on a bounded domain Ω ⊂Rnand a time interval [0, b], ∂tT−γ∆T=fin Q, T= 0 in ∂Ω×(0, b), T(x, 0) = a(x) in Ω, (6) where γ > 0 is the thermal conductivity, and we set Q= Ω ×(0, b). If f∈L2(Q) and a∈L2(Ω) then problem (6) admits a unique solution T∈C([0, b], L2(Ω)) ∩L2(0, b;H1 0(Ω)). In particular, the following stability holds (see [18]) kT(t)kL2(Ω) ≤C(kakL2(Ω) +kfkL2(Q)),∀t∈(0, b). The constant Cdepends on band is independent of γ. We shall be concerned, in some places, with the thermal conductivity γranging in a finite interval Gstrongly contained in ]0,+∞[ that is G⊂G⊂]0,+∞[. Let us consider the POD expansion (5) of the temperature T=T(γ, z) as a function in γand z= (x, t). Then, we denote TM, the function given the truncated expansion TM(γ, z) = M X m=0 σmϕm(γ)vm(z).(7) Recall that the sequence TMconverges to Tin L2(G×Q) with a rate obviously connected with the asymptotics of the singular values. Our purpose is to prove that the rate of convergence is exponential. This requires some beforehand preparation. Let (eℓ=eℓ(x))ℓ≥0be the orthonormal Fourier basis of L2(Ω), formed by eigenfunctions of the Laplace operator, −∆eℓ=λℓeℓin Ω, eℓ= 0 on ∂Ω, where λℓ>0 is the eigenvalue associated to eℓ. The sequence (λℓ)ℓ≥0is ordered so that it is non-decreasing. We have that limk→∞ λℓ= +∞. For the calculation of the Fourier series of T, we need first the Fourier decompositions of a(·) and f(·, t), a(x) = X ℓ≥0 aℓeℓ(x), f(x, t) = X ℓ≥0 fℓ(t)eℓ(x). Their Fourier coefficients are respectively given by aℓ= (a, eℓ)L2(Ω), fℓ(t) = (f(·, t), eℓ)L2(Ω). The series are respectively convergent in L2(Ω) and L2(Q), and kak2 L2(Ω) =X ℓ≥0|aℓ|2,kfk2 L2(Q)=X ℓ≥0kfℓk2 L2(0,b). 5 Now, based on these Fourier decompositions, the solution of problem (6) can be expressed as T(γ, x, t) = X ℓ≥0 θℓ(γ, t)eℓ(x), where the coefficients (θk)k≥0are determined by θℓ(γ, t) = aℓe−γ λℓt+Zt 0 fℓ(s)e−γλℓ(t−s)ds. (8) This is set, we consider now the temperature field Tas a function mapping the segment G= [γm, γM] into L2(Q). We have that 0 < γm≤γM, and we denote |G|= (γM−γm) the length of G. Finally we introduce ρ∗=(√γm+√γM)2 γM−γm >1. The following important approximation result holds. Theorem 3.1 Assume that f∈L2(Q)and a∈L2(Ω). The truncated POD series expansion TM satisfies the error estimate kT−TMkL2(G×Q)≤Cρρ−M,∀ρ, 1< ρ < ρ∗,(9) where Cρ>0is a constant depending on ρ. As indicated above, the clue of the proof is the asymptotics of the singular values of the integral operator B, with kernel T. Their decreasing rate is tightly related to the smoothness of the temperature Twith respect to the conductivity γ. The proof is long and technical, we chose to expose it in several steps. We start by proving the preliminary statement Lemma 3.2 Let g∈L2(0, b) and λ > 0 be given, the function G:γ7→ Zt 0 g(s)e−γλ(t−s)ds, mapping ]0,+∞[ into L2(0, b) is analytic. Proof: To check it out, one could use results from the theory of parameter dependent integrals. We rather follow the direct procedure and show that γ7→ G(γ) is locally expressed as a convergent entire or power series. Let γ0>0 be fixed. On account of the analyticity of the exponential we derive that G(γ, t) = X n≥0 (γ−γ0)n n!Zt 0 g(s)[−λ(t−s)]ne−γ0λ(t−s)ds := X n≥0 (γ−γ0)n n!Gn(t), This series is absolutely convergent in L2(0, b). Indeed, the integral term being a convolution, then Young’s inequality can be used which implies that X n≥0 (γ−γ0)n n!kGnkL2(0,b)≤X n≥0 (γ−γ0)n n!kgkL2(0,b)k[−λt]ne−γ0λtkL1(0,∞) =1 λkgkL2(0,b)X n≥0 (γ−γ0)n (γ0)n. 6 The geometrical series is convergent for γsuch that |γ−γ0|< η provided that η < γ0. The function γ7→ G(γ) is then analytic in ]0,+∞[ with L2(0, b) as co-domain. The proof is complete. Then, we step forth towards the analyticity of the vector-valued function γ7→ T(γ), subject of the following Lemma 3.3 The function γ7→ T(γ), mapping ]0,+∞[into L2(Q), is analytic. Proof: According to (8), the temperature field Tis the sum of two contributions, one is due to the initial condition a(·) and the other is generated by the source f(·,·). The analyticity is here checked out for each of them. i. — Let us begin by the part generated by the initial condition T(γ, x, t) = X ℓ≥0 aℓe−γλℓteℓ(x). Obviously each of the terms in the series determines an analytic function from ]0,+∞[ into L2(Q). If the series converges uniformly on each interval [ǫ, +∞[, forall ǫ > 0, then the limit will be analytic in ]0,+∞[ (see [16, ]). To check this out, let us bound the residual sup γ≥ǫ  X ℓ≥L aℓe−γλℓteℓ   2 L2(Q)= sup γ≥ǫX ℓ≥L (aℓ)2Zb 0 e−2γλℓtdt = sup γ≥ǫX ℓ≥L (aℓ)21−e−2γλℓb 2γλℓ≤1 2ǫλ0X ℓ≥L (aℓ)2, for some L > 0. The series is hence uniformly convergent in ]0,+∞[ and the limit Tdetermines thus an analytic function in γ. ii. — The second and last step is to investigate the part arisen from the source f, provided by T(γ, x, t) = X ℓ≥0Zt 0 fℓ(s)e−γλℓ(t−s)dseℓ(x).(10) According Lemma 3.2, this is a series of analytical functions we shall cope its uniformly convergent in [ǫ, +∞[, forall ǫ > 0. Before doing so, it is convenient to denote by Fℓ(γ, t) the integral term in the infinite sum. Then, for a given Lwe have that sup γ≥ǫ  X ℓ≥L Fℓ(γ, ·)eℓ   2 L2(Q)= sup γ≥ǫX ℓ≥LkFℓ(γ, ·)k2 L2(0,b) ≤sup γ≥ǫX ℓ≥Lkfℓk2 L2(0,b)ke−γλℓtk2 L1(0,∞)≤1 (ǫλ0)2X ℓ≥Lkfℓk2 L2(0,b). The last infinite sum decays towards zero when Lgrows up to infinity. This indicates that the series (10) of analytic functions is uniformly convergent. As a result, the limit is also analytic. The proof is complete. 7 Remark 3.1 The analyticity is readily extended to γbelonging to the right half complex plan (ℜζ > 0). No changes are required in the proof; it is identical. Another preliminary tool required in our study is related to the polynomial approximation of regular vector-valued functions. We shall adapt a result by S. Bernstein (in 1912), stated for complex-valued functions, and improved since then in many works (see for instance [19]). For some ρ > 1, let the set Eρin the complex plan be defined as Eρ=ζ∈C;|ζ−1|+|ζ+ 1| ≤ ρ+ρ−1. Consider a function F:Eρ→Hwhere His a Hilbert space. For a given integer number M≥0 let FMbe the truncated Chebyshev polynomial series expansion of Fof degree Mwith coefficients in H. Although the shape of the polynomial FMwill be fixed later on, we do not detail the construction of FMas we do not need it. Following the proof as exposed in [19], we come up with Lemma 3.4 Assume that Fis analytic and bounded in Eρ. There holds that max ξ∈[−1,1] kF(ξ)−FM(ξ)kH≤Cρρ−M. Remark 3.2 The constant in the lemma may be fixed to Cρ=2 ρ−1kFkL∞(Eρ). It diverges as ρclose to unity. We now need to derive similar approximation estimates for analytic vector valued functions defined from Ginto L2(G, L2(Q)). Notice first that polynomials SMwith degree ≤Mmay be written under the following form SM(γ) = X 0≤m≤M Um(γ)wm,∀γ∈G. The symbol Umis for the polynomial obtained, by transforming to the interval G, the Chebyshev polynomial of degree m, defined in [−1,1]. The coefficients (wm)0≤m≤Mbelong of course to L2(Q). The following result holds Lemma 3.5 There exists a polynomial SMranging from Ginto L2(Q), with degree ≤M, such that: forall ρ(1 < ρ < ρ∗), max γ∈[γm,γM]kT(γ)−SM(γ)kL2(Q)≤Cρρ−M, Proof: We only give a sketch of the proof. Following Lemma 3.3 complemented by Remark 3.1, the vector-valued function γ7→ Tis analytic in ]0,+∞[. This implies that provided that ρ < ρ∗, the ellipse Eρ=nζ∈C;|ζ−γM|+|ζ−γm| ≤ |G| 2(ρ+ρ−1)o, 8 References [1] Adams, Robert A. and Fournier, John J. F., Sobolev Spaces, Academic Press. 2003 [2] Aza¨ıez, M. and Ben Belgacem, F., Karhunen-Lo`eve’s truncation error for bivariate functions, Computer Methods in Applied Mechanics and Engineering, Vol 290, pp 57-72, 2015 [3] Berkoz, G. and Holmes, P. and Lumley, J. L., The proper orthogonal decomposition in the analysis of turbulent flows, Annu. Rev. 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