Partial differential equations/Numerical analysis
Simultaneous empirical interpolation and reduced basis method for non-linear problems
Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1105-1109.

In this paper, we focus on the reduced basis methodology in the context of non-linear non-affinely parameterized partial differential equations in which affine decomposition necessary for the reduced basis methodology are not obtained [9,8]. To deal with this issue, it is now standard to apply the EIM methodology [1,3] before deploying the Reduced Basis (RB) methodology. However, the computational cost is generally huge as it requires many finite element solves, hence making it inefficient, to build the EIM approximation of the non-linear terms [3,2]. We propose a simultaneous EIM reduced basis algorithm, named SER, which provides a huge computational gain and requires as little as N+1 finite-element solves where N is the dimension of the RB approximation. The paper is organized as follows: we first review the EIM and RB methodologies applied to non-linear problems and identify the main issue, then we present SER and some variants and finally illustrates its performances in a benchmark proposed in [3].

Dans ce papier, nous nous intéressons à la méthodologie bases réduites (RB) dans le contexte d'équations aux dérivées partielles paramétrisées non linéaires et non affines, pour lesquelles la décomposition affine nécessaire à la méthodologie RB ne peut être obtenue [9,8]. Pour traiter ce problème, il est à présent standard d'appliquer la méthodologie EIM [1,3] avant de déployer la méthodologie RB. Cependant, le coût de calcul de cette approche est en général considérable, car il requiert de nombreuses évaluations éléments finis pour construire l'approximation EIM des termes non linéaires [3,2], ce qui la rend très peu compétitive. Nous proposons l'algorithme SER, qui construit simultanément l'approximation EIM et la méthodologie RB, fournissant ainsi un gain de calcul considérable, et qui requiert au minimum N+1 résolutions éléments finis, où N est la dimension de l'approximation RB. Le papier est organisé comme suit : tout d'abord, nous passons en revue les méthodes EIM et RB appliquées aux problèmes non linéaires et identifions la difficulté principale, puis nous présentons SER et quelques variantes ; finalement, nous illustrons ses performances sur un benchmark proposé par [3].

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Published online:
DOI: 10.1016/j.crma.2015.08.003
Daversin, Cécile 1; Prud'homme, Christophe 1

1 IRMA (UMR 7501), 7, rue René-Descartes, 67084 Strasbourg cedex, France
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Daversin, Cécile; Prud'homme, Christophe. Simultaneous empirical interpolation and reduced basis method for non-linear problems. Comptes Rendus. Mathématique, Volume 353 (2015) no. 12, pp. 1105-1109. doi : 10.1016/j.crma.2015.08.003. http://www.numdam.org/articles/10.1016/j.crma.2015.08.003/

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