We propose an improvement to the reduced basis method for parametric partial differential equations. An assumption of affine parameterization leads to an efficient offline–online decomposition when the problem is solved for many different parametric configurations. We consider an advection–diffusion problem, where the diffusive term is non-affinely parameterized and treated with a two-level affine approximation given by the empirical interpolation method. The offline stage and a posteriori error estimation is performed using the coarse-level approximation, while the fine-level approximation is used to perform a correction iteration that reduces the actual error of the reduced basis approximation while keeping the same certified error bounds.
On propose une amélioration de la méthode des bases réduites pour des équations aux dérivées partielles paramétriques. On utilise l'hypothèse de paramétrisation affine pour obtenir un problème ayant des formes bilinéaires indépendantes des paramètres et des fonctions scalaires qui dépendent des paramètres. Ceci mène à une décomposition offline–online (qui est) plus efficace lorsque le problème est résolu pour différentes configurations des paramètres. Toutefois, dans le cas général, la condition de paramétrisation affine n'est pas satisfaite. On considère un problème d'advection–diffusion où la matrice des coefficients d'advection est paramétrisée de manière affine et où on traite le terme diffusif non-affine avec une approximation affine à deux niveaux obtenue avec une méthode d'interpolation empirique. La partie offline est effectuée en utilisant une approximation affine grossière alors qu'une approximation affine plus fine est utilisée pour accomplir une itération de correction.
Accepted:
Published online:
Toni Lassila 1; Gianluigi Rozza 2
@article{CRMATH_2011__349_1-2_61_0, author = {Toni Lassila and Gianluigi Rozza}, title = {Model reduction of semiaffinely parameterized partial differential equations by two-level affine approximation}, journal = {Comptes Rendus. Math\'ematique}, pages = {61--66}, publisher = {Elsevier}, volume = {349}, number = {1-2}, year = {2011}, doi = {10.1016/j.crma.2010.11.016}, language = {en}, }
TY - JOUR AU - Toni Lassila AU - Gianluigi Rozza TI - Model reduction of semiaffinely parameterized partial differential equations by two-level affine approximation JO - Comptes Rendus. Mathématique PY - 2011 SP - 61 EP - 66 VL - 349 IS - 1-2 PB - Elsevier DO - 10.1016/j.crma.2010.11.016 LA - en ID - CRMATH_2011__349_1-2_61_0 ER -
%0 Journal Article %A Toni Lassila %A Gianluigi Rozza %T Model reduction of semiaffinely parameterized partial differential equations by two-level affine approximation %J Comptes Rendus. Mathématique %D 2011 %P 61-66 %V 349 %N 1-2 %I Elsevier %R 10.1016/j.crma.2010.11.016 %G en %F CRMATH_2011__349_1-2_61_0
Toni Lassila; Gianluigi Rozza. Model reduction of semiaffinely parameterized partial differential equations by two-level affine approximation. Comptes Rendus. Mathématique, Volume 349 (2011) no. 1-2, pp. 61-66. doi : 10.1016/j.crma.2010.11.016. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2010.11.016/
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