Comptes Rendus
Partial Differential Equations/Numerical Analysis
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.

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.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2010.11.016
Toni Lassila 1; Gianluigi Rozza 2

1 Department of Mathematics and Systems Analysis, Aalto University, P.O. Box 11100, 00076 Aalto, Finland
2 Modelling and Scientific Computing, Mathematics Institute of Computational Science and Engineering, École Polytechnique Fédérale de Lausanne, station 8, EPFL, CH-1015 Lausanne, Switzerland
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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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