Comptes Rendus
no. 4
Partial differential equations/Numerical analysis
Error bounds in high-order Sobolev norms for POD expansions of parameterized transient temperatures

Presented by: Olivier Pironneau

Mejdi Azaïez1; Faker Ben Belgacem2; Tomás Chacón Rebollo3; Macarena Gómez Mármol4; Isabel Sánchez Muñoz5
1 I2M, IPB (UMR CNRS 5295), Université de Bordeaux, 33607 Pessac, France
2 Sorbonne University, Université de technologie de Compiègne, LMAC Laboratory of Applied Mathematics of Compiègne, CS 60319, 60203 Compiègne cedex, France
3 Departamento de Ecuaciones Diferenciales y Análisis Numérico & Instituto de Matemáticas IMUS, Universidad de Sevilla, 41080 Sevilla, Spain
4 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, 41080 Sevilla, Spain
5 Departamento de Matemática Aplicada I, Universidad de Sevilla, 41013 Sevilla, Spain
Comptes Rendus. Mathématique, Volume 355 (2017) no. 4, pp. 432-438.

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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2017.03.002

Mejdi Azaïez 1; Faker Ben Belgacem 2; Tomás Chacón Rebollo 3; Macarena Gómez Mármol 4; Isabel Sánchez Muñoz 5

1 I2M, IPB (UMR CNRS 5295), Université de Bordeaux, 33607 Pessac, France
2 Sorbonne University, Université de technologie de Compiègne, LMAC Laboratory of Applied Mathematics of Compiègne, CS 60319, 60203 Compiègne cedex, France
3 Departamento de Ecuaciones Diferenciales y Análisis Numérico & Instituto de Matemáticas IMUS, Universidad de Sevilla, 41080 Sevilla, Spain
4 Departamento de Ecuaciones Diferenciales y Análisis Numérico, Universidad de Sevilla, 41080 Sevilla, Spain
5 Departamento de Matemática Aplicada I, Universidad de Sevilla, 41013 Sevilla, Spain 
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     title = {Error bounds in high-order {Sobolev} norms for {POD} expansions of parameterized transient temperatures},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {432--438},
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Mejdi Azaïez; Faker Ben Belgacem; Tomás Chacón Rebollo; Macarena Gómez Mármol; Isabel Sánchez Muñoz. Error bounds in high-order Sobolev norms for POD expansions of parameterized transient temperatures. Comptes Rendus. Mathématique, Volume 355 (2017) no. 4, pp. 432-438. doi : 10.1016/j.crma.2017.03.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2017.03.002/

[1] M. Azaïez; F. Ben Belgacem; T. Chacón Rebollo Error bounds for POD expansions of parameterized transient temperatures, Comput. Methods Appl. Mech. Eng., Volume 305 (2016), pp. 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. Turicini A priori convergence of the greedy algorithm for the parametrized reduced basis, Math. Model. Numer. Anal., Volume 46 (2012), pp. 595-603

[4] A. Cohen; R. Devore Approximation of high-dimensional parametric PDEs, Acta Numer. (2015), pp. 1-159

[5] L. Cordier; M. Bergmann Proper Orthogonal Decomposition: An Overview. Lecture Series 2002-04 and 2003-04 on Post-Processing of Experimental and Numerical Data, Von Karman Institute for Fluid Dynamics, Rhode-Saint-Genèse, Belgium, 2003

[6] F. Hecht New development in FreeFem++, J. Numer. Math., Volume 20 (2012) no. 3–4, pp. 251-265 http://www.freefem.org

[7] J.S. Hesthaven; G. Rozza; B. Stamm Certified Reduced Basis Methods for Parametrized Partial Differential Equations, Springer Briefs Math., 2016

[8] P. Holmes; J.L. Lumley; G. Berkooz Coherent Structures, Dynamical Systems and Symmetry, Cambridge Monogr. Mech., Cambridge University Press, Cambridge, UK, 1996

[9] G. Little; J.B. Reade Eigenvalues of analytic kernels, SIAM J. Math. Anal., Volume 15 (1984), pp. 133-136

[10] M.M. Loève Probability Theory, Van Nostrand, Princeton, NJ, 1988

[11] M. Muller On the POD Method. An Abstract Investigation with Applications to Reduced-Order Modeling and Suboptimal Control, Georg-August Universität, Göttingen, 2008 (PhD Thesis)

[12] A. Quarteroni; A. Manzoni; F. Negri Reduced Basis Methods for Partial Differential Equations, Springer, 2016

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