Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations

Yvon Maday; Anthony T. Patera; G. Turinici

doi:10.1016/S1631-073X(02)02466-4

Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations
[Résultats globaux a priori pour l'approximation d'équations aux dérivées partielles coercives symétriques elliptiques dépendant d'un paramètre]

Yvon Maday ¹ ; Anthony T. Patera ² ; G. Turinici ³

¹ Laboratoire Jacques-Louis Lions, Université Paris VI, 4, place Jussieu, 75252 Paris cedex 05, France
² Department of Mechanical Engineering, M.I.T., 77 Mass. Ave., Cambridge, MA 02139, USA
³ INRIA Rocquencourt, BP 105, 78153 Le Chesnay cedex, France

Comptes Rendus. Mathématique, Volume 335 (2002) no. 3, pp. 289-294.

Résumé
Abstract

On considère des méthodes de bases réduites de type Lagrange pour des équations aux dérivées partielles coercives symétriques elliptiques et dépendant d'un paramètre. On montre que, pour une répartition logarithmiquement quasi uniforme des points d'échantillonage, l'approximation en base réduite converge de façon exponentielle vers la solution exacte uniformément par rapport au paramètre. De plus la convergence ne dépend que faiblement du rapport entre les coefficients de coercivité et de continuité de l'opérateur : ainsi une approximation de très basse dimension procure une solution très précise même dans le cas d'un large eventail de paramètres. Des test numériques (présentés ailleurs) corroborent ces prédictions numériques.

We consider “Lagrangian” reduced-basis methods for single-parameter symmetric coercive elliptic partial differential equations. We show that, for a logarithmic-(quasi-)uniform distribution of sample points, the reduced-basis approximation converges exponentially to the exact solution uniformly in parameter space. Furthermore, the convergence rate depends only weakly on the continuity–coercivity ratio of the operator: thus very low-dimensional approximations yield accurate solutions even for very wide parametric ranges. Numerical tests (reported elsewhere) corroborate the theoretical predictions.

Reçu le : 2002-05-15
Accepté le : 2002-06-03
Publié le : 2002-01-01

DOI : 10.1016/S1631-073X(02)02466-4

Affiliations des auteurs :

Yvon Maday ¹ ; Anthony T. Patera ² ; G. Turinici ³

¹ Laboratoire Jacques-Louis Lions, Université Paris VI, 4, place Jussieu, 75252 Paris cedex 05, France
² Department of Mechanical Engineering, M.I.T., 77 Mass. Ave., Cambridge, MA 02139, USA
³ INRIA Rocquencourt, BP 105, 78153 Le Chesnay cedex, France

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     author = {Yvon Maday and Anthony T. Patera and G. Turinici},
     title = {Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {289--294},
     publisher = {Elsevier},
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     language = {en},
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Yvon Maday; Anthony T. Patera; G. Turinici. Global a priori convergence theory for reduced-basis approximations of single-parameter symmetric coercive elliptic partial differential equations. Comptes Rendus. Mathématique, Volume 335 (2002) no. 3, pp. 289-294. doi : 10.1016/S1631-073X(02)02466-4. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/S1631-073X(02)02466-4/

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