Error bounds in high-order Sobolev norms for POD expansions of parameterized transient temperaturas Estimations d'erreur d'ordre élevé pour la décomposition POD appliquée à l'équation de la chaleur parametrisée

C. R. Acad. Sci. Paris, Ser. I 355 (2017) 432–438 Contents lists available at ScienceDirect C. R. Acad. Sci. Paris, Ser. I www.sciencedirect.com Partial differential equations/Numerical analysis Error bounds in high-order Sobolev norms for POD expansions of parameterized transient temperatures Estimations d’erreur d’ordre élevé pour la décomposition POD appliquée à l’équation de la chaleur parametrisée Mejdi Azaïez a, Faker Ben Belgacemb, Tomás Chacón Rebollo c, Macarena Gómez Mármol d, Isabel Sánchez Muñoz e aI2M, IPB (UMR CNRS 5295), Université de Bordeaux, 33607 Pessac, France bSorbonne University, Université de technologie de Compiègne, LMAC Laboratory of Applied Mathematics of Compiègne, CS 60319, 60203 Compiègne cedex, France cDepartamento de Ecuaciones Diferenciales y Análisis Numérico & Instituto de Matemáticas IMUS, Universidad de Sevilla, 41080 Sevilla, Spain dDepartamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, 41080 Sevilla, Spain eDepartamento de Matemática Aplicada I, Universidad de Sevilla, 41013 Sevilla, Spain a r t i c l e i n f o a b s t r a c t Article history: Received 29 July 2016 Accepted after revision 3 March 2017 Available online 22 March 2017 Presented by Olivier Pironneau In this work, we analyze the convergence of the POD expansion for the solution to the heat conduction parameterized with respect to the thermal conductivity coefficient. We obtain error bounds for the POD approximation in high-order norms in space that assure an exponential rate of convergence, uniformly with respect to the parameter whenever it remains within a compact set of positive numbers. We present some numerical tests that confirm this theoretical accuracy. ©2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. r é s u m é On considère le problème de la conduction thermique paramétrée par rapport au coefficient de conductivité thermique, et on s’intéresse à la décomposition de sa solution par la méthode POD. Nous analysons la convergence de la solution tensorielle. Nous obtenons des bornes d’erreur pour l’approximation POD dans les normes de Sobolev d’ordre élevé, qui assurent un taux exponentiel de convergence, uniformément par rapport au paramètre si celui-ci reste dans un ensemble compact de nombres positifs. Enfin, nous présentons quelques tests numériques qui confirment nos résultats théoriques. ©2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. E-mail addresses: [email protected] (M. Azaïez), [email protected] (F. Ben Belgacem), [email protected] (T.ChacónRebollo), [email protected] (M. Gómez Mármol), [email protected] (I. Sánchez Muñoz). http://dx.doi.org/10.1016/j.crma.2017.03.002 1631-073X/©2017 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. M. Azaïez et al. / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 432–438 433 Version française abrégée La POD est une technique très connue pour son efficacité dans la construction d’approximations tensorielles de données, particulièrement quand celles-ci sont les solutions de problèmes physiques linéaires. (On peut consulter à ce sujet [5,8, 10,11].) Dans [1], on démontre que la décomposition par POD de la solution de l’équation de la chaleur paramétrée par rapport à la diffusivité thermique converge en norme L2en espace-temps avec un taux exponentiel. Dans cette, note nous apportons une extension de ce résultat pour des normes de Sobolev d’ordre élevé. Nous considérons la solution T=T(γ, z) de l’équation de la chaleur (5), où γ∈G =[γmin, γmax] ⊂(0, +∞)est la diffusivité thermique et z=(x, t)appartient au cylindre Q= ×(0, τ).  ⊂Rnest un domaine borné. Si le domaine est de classe Cs, nous prouvons que, pour des données initiales et aux limites compatibles, Test une fonction analytique de Gà valeurs dans H:= L2((0, τ), Vs()), où l’espace Vs() est défini par (7) (Proposition 5.1). Pour démontrer l’analyticité, on développe Tcomme série de Fourier des fonctions propres du laplacien sur et on démontre que la convergence est uniforme par rapport à γ∈G. Ce résultat est basé sur l’équivalence dans Vs() de la norme  ·Vs() définie par (8) et la norme usuelle dans Hs() (Lemme 5.1). L’analyticité de Tcomme fonction de γest la clé pour démontrer la convergence exponentielle du développement POD tronqué, car celui-ci possède la propriété d’être la meilleure approximation en norme L2(G, H)de Tpar rapport à tous les sous-espaces de Hà dimension donnée (4). L’erreur en norme L2(G, H)entre Tet la série POD tronquée à Mtermes, TM, est donc bornée par celle de l’approximation de Tpar des polynômes de Chebyshev de degré Mde la variable γ. Les estimations classiques d’erreur d’interpolation de fonctions analytiques (Lemme 5.3) donnent alors le taux exponentiel de convergence (Théorème 5.2): T−TML2(G×Q)≤Cρρ−M,∀ρ:1<ρ<ρ∗,(1) où Cρ>0est une constante qui dépend de ρet ρ∗=(√σ+1)2 σ−1,with σ=γmax γmin . Nous présentons quelques tests numériques pour l’équation de la chaleur 2D confirmant le taux théorique de convergence exponentielle (1), indépendamment de s, en normes L2((0, τ), Vs()), pour s =0, 1, 2(voirFig. 1). 1. Introduction The Proper Orthogonal Decomposition (POD) is a method for deriving low-rank approximations of parametric functions, which has been successfully used in different fields (see [5,8,10,11]). We focus this work on quantifying the accuracy of the truncated POD expansion in approximating the solution to the parameterized heat equation with respect to the thermal conductivity coefficient. This is a continuation of work [1], where it is proved that the POD expansion converges in space–time L2norm with an exponential rate, with respect to the number of retained modes, and uniformly in the parameter if it remains within a compact set of positive numbers. In the present work, we extend this convergence result to higher-order norms in space, equivalent to the Hs()-norm for any s ≥0, provided that initial and source data are smooth enough and compatible. We also present some numerical tests for the 2D heat equation for s =0, 1 and 2, and the obtained results confirm the theoretical expectations. In this context, where the parameter domain is finite dimensional, the POD approximation error is equivalent to the Kolmogorov width of the solution manifold. This quantity is used as a benchmark for the accuracy of the approximations obtained by reduced-order methods. For example, for the reduced-basis method generated by the greedy algorithm, convergence results are established in [3] and [4], which show convergence rates similar to the Kolmogorov width of the solution manifold. Basic references on the application of reduced basis method to parameterized PDEs can be found in [7] and [12]. In this work, the POD approximation error is estimated by an appropriate approximation of the heat solution by Chebyshev polynomials in the parametric variable. This approximation requires the analytic dependence of this function (with values in high-order Sobolev spaces) with respect to the thermal diffusivity coefficient, that we prove by means of a Fourier analysis. This analyticity property also is at the basis of the proof in [4] of the rate of convergence of the approximation of the solution to parameterized elliptic equations with infinitely many parameters, and also to the parameterized heat equation. However, there are several differences between both results (for heat equation): the analysis within [4] apply to the estimation of the n-width (in mean quadratic error measured in low-order parabolic norms) of the variety of solutions to the parameterized heat equation. The n-width is bounded by means of the error of the Taylor and Legendre polynomial expansion. Here we estimate the rate of convergence of the POD expansion in mean quadratic error measured in high-order parabolic norms (that also yield the n-width, but with respect to this norm). We bound the error by means of the error of Chebyshev’s polynomial expansion. The convergence rate turns out to be somewhat better (see Remark 2). The paper is structured as follows: in Section 2we introduce the POD method that we apply in Section 3to obtain the POD approximation of solutions to the heat equation. Then we analyze the convergence rate of this approximation, firstly in space–time L2norm in Section 4and later in higher-order norms in space in Section 5. Finally, we present numerical tests in Section 6. 434 M. Azaïez et al. / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 432–438 2. The POD expansion on Hilbert spaces Let G ⊂Rdbe a bounded domain, with d ≥1an integer, and let Hbe a Hilbert space endowed with the inner product (·, ·)H. Assume that Tis a given function in the Lebesgue space L2(G, H), that we want approximate in a low-dimensional variety. The POD method consists in choosing an orthonormal set of Mfunctions in H, BM={vi}M i=1, such that VM= span{v1, ..., vM}has the following best-approximation property: VMminimizes the mean distance square between Tand any subspace of Hof dimension M, WM. That is, VMis the subspace where the infimum δM(T)2 H:= inf dim(WM)=M G min w∈WMw−T(μ)2 Hdμ,(2) is attained. The components of the POD basis, BM, are characterized from the eigenvectors of the POD operator A :L2(G) → L2(G) such that Aϕ(μ) = G (T(μ), T(δ))Hϕ(δ) dδ. If we denote TMthe truncated POD series expansion of T: TM(μ)= M  m≥1 (T(μ), vm)Hvm(3) then TMconverges to Tin L2(G, H)(see [11] for details). Note that, in terms of this approximation, the property (2) implies δM(T)H=T−TML2(G,H)≤T−SML2(G,H),(4) for all SM∈L2(G, WM)with WMany subspace of Hof dimension M. 3. POD expansion for the heat equation We apply the POD method to the solution to the heat equation parameterized with respect to the thermal conductivity coefficient. Let  ⊂Rnbe a bounded domain, (0, τ)a time interval and Q= ×(0, τ). We consider the following boundary value problem for the heat equation: ⎧ ⎨ ⎩ ∂tT−γT=fin Q T=0in∂×(0,τ) T(x,0)=T0(x)in , (5) where γ>0is the thermal conductivity coefficient. Is well known that if f∈L2(Q)and T0∈L2() this problem admits a unique solution T∈L2(0, τ; H1 0()) and ∂tT∈L2(0, τ; H−1()). We consider the POD expansion of the function T=T(γ, z) with z=(x, t)and its POD approximation (3), TM. Our purpose is to analyze the rate of the convergence of TMto T, considering different norms. 4. Convergence analysis in L2norm Using the notation in the section above, we set H=L2(Q)and so, L2(G, H) =L2(G ×Q). The analysis of convergence of TMto Tin L2(G ×Q)norm is realized in [1]. This work proves that the rate of convergence is exponential with respect to the number of the modes and this convergence is uniform with respect to the diffusion coefficient, whenever it remains in a compact set of positive numbers. More concretely, the following approximation result holds. Theorem 4.1. Assume that G =[γmin, γmax] ⊂(0, +∞), f∈L2(Q)and T0∈L2(). Then the truncated POD series expansion TM (given by (3) with H=L2(Q)) of the solution Tto problem (5) satisfies the following error estimate T−TML2(G×Q)≤Cρρ−M,∀ρ:1<ρ<ρ∗, where Cρ>0is a constant depending on ρand ρ∗=(√σ+1)2 σ−1,with σ=γmax γmin .(6) M. Azaïez et al. / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 432–438 435 5. Convergence analysis in higher-order norm We aim the extension of the convergence result stated in Theorem 4.1 to higher-order Sobolev norms in space. For that, we introduce the following space: for any integer k ≥1 and s =2k −1or s =2k, Vs() =v∈Hs() :v=v=...=k−1v=0on∂.(7) For a unified notation, we denote V0() =L2(). We also define vVs() =∇k−1vL2() if s=2k−1 kvL2() if s=2k(8) Lemma 5.1. Vs() is a closed subspace of Hs(). If is of class Cs, then  ·Vs() is a norm equivalent to the usual norm in Hs(). Proof. We use the regularity of the weak solution for the problem: find v, solution to −v =fin with v =0on ∂ stated in [2], p. 181, and reads if is of class Csand f∈Hs−2(), then v ∈Hs() and vHs() ≤CfHs−2().(9) For k =1: V1() =H1 0(), the  ·V1() norm and the H1() norm are equivalent. Next, V2() =H2() ∩H1 0(), where the result holds, applying (9) with s =2. Let us suppose that the result holds for any k ≥2. We prove it now for k +1: let v ∈Vs(). Then f=v ∈Hs−2() and vverifies (9). Note that nf=n+1vfor 0 ≤n ≤k −1, thus f=f=...= k−1f=0on∂ and f∈Vs−2(). We now use the induction hypothesis: for s =2k +1, s −2 =2k −1 and fverifies fHs−2() ≤C∇k−1fL2() =C∇kvL2() =CvVs(). Also, for s =2k +2, s −2 =2kand fverifies fHs−2() ≤ CkfL2() =Ck+1vL2() =CvVs(). Combining these estimates with (9), we obtain vHs() ≤CvVs(),∀v∈Vs(). 2 We study the convergence of the POD expansion in the space H:= L2((0, τ), Vs()). The following result holds. Theorem 5.2. Assume that is of class Cs, for some integer s ≥1. Let G =[γmin, γmax] ⊂(0, +∞), f∈L2((0, τ), Vs−1()) and T0∈Vs−1(). Then the truncated POD series expansion TM(given by (3) with H=H) of the solution Tto the problem (5) satisfies the following error estimate T−TML2(G,H)≤Cρρ−M,∀ρ:1<ρ<ρ∗,(10) where Cρ>0is a constant depending on ρand ρ∗given in (6). Remark 1. Note that when s =1in Theorem 5.2, under the same hypotheses about the data as in Theorem 4.1, the estimate of the POD error obtained in (10) applies to L2(G, L2(0, τ; H1 0())) instead of L2(G ×Q), as in Theorem 4.1. The proof of Theorem 5.2 follows from the analyticity of the function Tupon γ, which we prove below. Proposition 5.1. The function γ∈(0, +∞) → T(γ) ∈His analytic. Proof. We consider the Dirichlet eigenvalue problem for the Laplace operator: −en=αnenin , en=0on∂. The spectrum of this operator is a non-decreasing sequence of positive numbers, {αn}n≥1such that lim n→∞αn=+∞. Also, the sequence {en}n≥1, with ena eigenfunction associated with αn, is an orthonormal basis of L2(). The sequence {en}n≥1is also an orthogonal basis of Vs(), with (en,em)Vs() =αs nδnm,en2 Vs() =αs n.(11) We consider the Fourier decomposition of the data of problem (5) T0and f: T0(x)= n≥1 anen(x), f(x,t)= n≥1 fn(t)en(x), with an=(T0, en)L2() and fn(t) =(f(·, t), en)L2(). Due to the assumed regularity of T0and f, these series are convergent in Vs−1() and L2((0, τ), Vs−1()), respectively, and from (11) it holds 436 M. Azaïez et al. / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 432–438 T02 Vs−1() = n≥1|an|2αs−1 n,(12) f2 L2((0,τ),Vs−1()) = n≥1fn2 L2(0,τ)αs−1 n.(13) As a consequence, the solution to problem (5) can be expressed as T(γ,x,t)= n≥1 Tn(γ,t)en(x), with Tn(γ,t)=ane−γα nt+ t  0 fn(s)e−γα n(t−s)ds.(14) We analyze the convergence of the series (14). For that, we check out the two terms coming from the two sums in Tn. On the one hand,  n≥1ane−γα nten2 H= n≥1|an|2e−γα nt2 L2(0,τ)en2 Vs(). Using the fact that e−γα nt2 L2(0,τ)= τ  0 e−2γα ntdt=1 2γα n (1 −e−2γα nτ) <1 2γα n and (11), we have  n≥1ane−γα nten2 H≤1 2γ n≥1|an|2αs−1 n=1 2γT02 Vs−1(),(15) taking into account (12). By another hand, denoting Fn(γ, t) = t  0 fn(s) e−γα n(t−s)ds,  n≥1Fn(γ)en2 H= n≥1Fn(γ)2 L2(0,τ)en2 Vs(). As |Fn(γ, t)|2≤fn2 L2(0,τ) t  0 e−2γα n(t−s)ds =1 2γα nfn2 L2(0,τ)(1 −e−2γα nt), then Fn(γ)2 L2(0,τ)≤τ 2γα nfn2 L2(0,τ). Thus,  n≥1Fn(γ)en2 H≤τ 2γ n≥1fn2 L2(0,τ)αs−1 n=τ 2γf2 L2((0,τ),Vs−1()),(16) taking into account (11) and (13). From the estimates (15) and (16) we deduce that the series (14) is uniformly convergent in any compact sub-sets of (0, +∞). Also, each of the terms in the series is an analytic function in (0, +∞)(see [1] for details), thus the limit is also an analytic function in (0, +∞).2 To prove Theorem 5.2, we use the following preliminary result (cf. [9]). Lemma 5.3. Let Eρ={z∈Csuch that |z−1| +|z+1| ≤ρ+ρ−1}for some ρ>1. Let F:Eρ→Hbe an analytic and bounded function in Eρ, with Ha Hilbert space. For a given integer number M≥0, let FMbe the truncated Chebyshev polynomial series expansion of Fof degree Mwith coefficients in H. Then it holds max z∈[−1,1]F(z)−FM(z)H≤Cρρ−M,(17) with Cρ=2 ρ−1FL∞(Eρ). Lemma 5.4. There exists a polynomial with degree smaller than or equal to M, SM:G →Hsuch that: max γ∈GT(γ)−SM(γ)H≤Cρρ−M,∀ρ:1<ρ<ρ∗,(18) with ρ∗given in (6). M. Azaïez et al. / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 432–438 437 Fig. 1. POD error in L2(G, H)norm with H=L2(Q), H=L2(0, 1; V1(0, 1)) and H=L2(0, 1; V2(0, 1)). x-Axis: number of modes. y-Axis: error in L2(G, H) norm, in logarithmic coordinates. Proof. We consider the affine transformation gthat transforms [−1, 1]into Gand the function ˜ T(˜ γ) =T(g(˜ γ)). From Proposition 5.1, Tis analytic in (0, +∞), and hence ˜ Tis analytic in a ellipse ˜ Eρ, with ρ<ρ∗. So, we can apply Lemma 5.3 and obtain (17) for the function ˜ Tin H: max ˜ γ∈[−1,1]˜ T(˜ γ)−˜ SM(˜ γ)H≤Cρρ−M, where ˜ SMis the truncated Chebyshev polynomial series expansion of ˜ Tof degree M. Now, we construct the polynomial SM(γ) =˜ SM(g−1(γ)) and it holds max γ∈GT(γ)−SM(γ)H=max ˜ γ∈[−1,1]˜ T(˜ γ)−˜ TM(˜ γ)H. Then, (18) follows from the error estimate above. 2 Now we are in a position to conclude about the proof of Theorem 5.2. Proof of Theorem 5.2.Let SMbe the polynomial constructed in Lemma 5.4. From the best-approximation property of TM, (4), it holds T−TML2(G,H)≤T−SML2(G,H)≤|G|1/2max γ∈GT(γ)−SM(γ)H. Finally, from (18) it follows (10).2 Remark 2. Note that the convergence rate (6) established by Theorem 5.2 is determined by the Chebyshev polynomial SMderived in Lemma 5.4. This approximation in estimating the POD error improves the convergence rate obtained in [4] (estimates (3.175) and (3.185)): ρ∗=σ+1 σ−1, obtained by comparison with the truncated Taylor expansion. 6. Numerical tests We carry out a numerical investigation in order to illustrate the convergence rates predicted by Theorem 5.2. We consider the problem (5) in the domain Q=(0, 1)2×(0, 1)with data T0(x, y) =x(x −1)y(y −1)and f(x, y, t) =x(x −1)y(y −1)t. We set the thermal conductivities interval to G =[1, 100]. The heat problem is discretized by a semi-implicit Euler in time and P3 Finite Element method in space with time step t=10−2, mesh size h =0.28, and 22801 degrees of freedom. The calculation of the correlation matrix corresponding to the kernel of operator A, K(γi, γj) =(T(γi), T(γj))H, is performed using the trapezoid quadrature formula. We have used the LAPACK package to compute the eigenvalues and eigenvectors of matrix K. All computations have been performed with the free software FreeFem++ (cf. [6]). In Fig. 1, we display the 438 M. Azaïez et al. / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 432–438 POD error T−TML2(G,H)(in logarithmic coordinates) in terms of the number of modes in the approximation TMfor the solution Tto the problem proposed. We have considered H=L2(0, 1; Vs(0, 1)) for s =0, 1 and 2. In all cases, we obtain an exponential convergence rate, with approximately the same slope in logarithmic coordinates, according to the theoretical estimate (10), where ρ∗does not depend on s. Acknowledgements The research of T. Chacón, M. Gómez and I. Sánchez was partially supported by the Spanish Ministerio de Economía y Competitividad – EU FEDER program grant MTM2015-64577-C2-1-R. T. Chacón his deeply grateful to the LaBeX program of the ‘Université de Bordeaux’, whose funding allowed him to start this collaborative research during several stays in 2014 and 2015. References [1] M. Azaïez, F. Ben Belgacem, T. Chacón Rebollo, Error bounds for POD expansions of parameterized transient temperatures, Comput. Methods Appl. Mech. Eng. 305 (2016) 501–511. [2] H. Brézis, Análisis Funcional. Teoría y aplicaciones, Alianza Editorial, Madrid, Spain, 1984. [3] A. Buffa, Y. Maday, A.T. Patera, C. Prud homme, G. 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