Comptes Rendus
no. 9
Numerical Analysis
An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations
[Une méthode d' « interpolation empirique » : application à la discrétisation efficace par base réduite d'équations aux dérivées partielles.]

Présenté par : Olivier Pironneau

Maxime Barrault1 ; Yvon Maday2 ; Ngoc Cuong Nguyen3 ; Anthony T. Patera4
1 CERMICS – ENPC, cité Descartes, Champs sur Marne, 77455 Marne la Vallée cedex 2, France
2 Laboratoire J.-L. Lions, université Pierre et Marie Curie, B.C. 187, 75242 Paris cedex 05, France
3 National University of Singapore, 10 Kent Ridge Crescent, Singapore 117576
4 Massachusetts Institute of Technology, Department of Mechanical Engineering, Room 3-264, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA
Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 667-672.

Nous présentons dans cette Note une méthode rapide de base réduite pour la résolution d'équations aux dérivées partielles ayant une dépendance non affine en ses paramètres. L'approche propose de remplacer le calcul des fonctionelles non affines par un développement en base réduite annexe qui conduit à une évaluation en ligne effectivement affine. Les points essentiels de cette approche sont (i) un bon système de base réduite annexe, (ii) une méthode stable et peu coûteuse d'interpolation dans cette base, et (iii) un estimateur a posteriori pertinent pour quantifier les nouvelles erreurs introduites. Des résultats théoriques et numériques viennent anticiper puis confirmer le bon comportement de cette technique.

We present an efficient reduced-basis discretization procedure for partial differential equations with nonaffine parameter dependence. The method replaces nonaffine coefficient functions with a collateral reduced-basis expansion which then permits an (effectively affine) offline–online computational decomposition. The essential components of the approach are (i) a good collateral reduced-basis approximation space, (ii) a stable and inexpensive interpolation procedure, and (iii) an effective a posteriori estimator to quantify the newly introduced errors. Theoretical and numerical results respectively anticipate and confirm the good behavior of the technique.

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DOI : 10.1016/j.crma.2004.08.006
Maxime Barrault 1 ; Yvon Maday 2 ; Ngoc Cuong Nguyen 3 ; Anthony T. Patera 4

1 CERMICS – ENPC, cité Descartes, Champs sur Marne, 77455 Marne la Vallée cedex 2, France
2 Laboratoire J.-L. Lions, université Pierre et Marie Curie, B.C. 187, 75242 Paris cedex 05, France
3 National University of Singapore, 10 Kent Ridge Crescent, Singapore 117576
4 Massachusetts Institute of Technology, Department of Mechanical Engineering, Room 3-264, 77 Massachusetts Avenue, Cambridge, MA 02139-4307, USA 
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     title = {An {\textquoteleft}empirical interpolation{\textquoteright} method: application to efficient reduced-basis discretization of partial differential equations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {667--672},
     publisher = {Elsevier},
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Maxime Barrault; Yvon Maday; Ngoc Cuong Nguyen; Anthony T. Patera. An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus. Mathématique, Volume 339 (2004) no. 9, pp. 667-672. doi : 10.1016/j.crma.2004.08.006. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2004.08.006/

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