[Une méthode d' « interpolation empirique » : application à la discrétisation efficace par base réduite d'équations aux dérivées partielles.]
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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@article{CRMATH_2004__339_9_667_0,
author = {Barrault, Maxime and Maday, Yvon and Nguyen, Ngoc Cuong and Patera, Anthony T.},
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},
volume = {339},
number = {9},
year = {2004},
doi = {10.1016/j.crma.2004.08.006},
language = {en},
url = {http://www.numdam.org/articles/10.1016/j.crma.2004.08.006/}
}
TY - JOUR AU - Barrault, Maxime AU - Maday, Yvon AU - Nguyen, Ngoc Cuong AU - Patera, Anthony T. TI - An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations JO - Comptes Rendus. Mathématique PY - 2004 SP - 667 EP - 672 VL - 339 IS - 9 PB - Elsevier UR - http://www.numdam.org/articles/10.1016/j.crma.2004.08.006/ DO - 10.1016/j.crma.2004.08.006 LA - en ID - CRMATH_2004__339_9_667_0 ER -
%0 Journal Article %A Barrault, Maxime %A Maday, Yvon %A Nguyen, Ngoc Cuong %A Patera, Anthony T. %T An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations %J Comptes Rendus. Mathématique %D 2004 %P 667-672 %V 339 %N 9 %I Elsevier %U http://www.numdam.org/articles/10.1016/j.crma.2004.08.006/ %R 10.1016/j.crma.2004.08.006 %G en %F CRMATH_2004__339_9_667_0
Barrault, Maxime; Maday, Yvon; Nguyen, Ngoc Cuong; Patera, Anthony T. An ‘empirical interpolation’ method: application to efficient reduced-basis discretization of partial differential equations. Comptes Rendus. Mathématique, Tome 339 (2004) no. 9, pp. 667-672. doi : 10.1016/j.crma.2004.08.006. http://www.numdam.org/articles/10.1016/j.crma.2004.08.006/
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