Simultaneous Empirical Interpolation and Reduced Basis method for non-linear problems

Abstract
journal: the Académie des sciences

In this paper, we focus on the reduced basis methodology in the context of non-linear non-affinely parametrized partial differential equations in which affine decomposition necessary for the reduced basis methodology are not obtained (K. 3847, K. 2003). To deal with this issue, it is now standard to apply the EIM methodology (Barrault 2004, A. 2007) 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 (A. 2007, Cécile Daversin 2013). We propose a simultaneous EIM Reduced basis algorithm, named SER, that 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 (A. 2007). To cite this article: C. Daversin, C. Prud’homme, C. R. Acad. Sci. Paris, Ser. I 340 (2015).

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Résumé Une méthode EIM Bases Réduites simultanée pour les équations aux dérivées partielles non-linéaires et non-affines.

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 et pour lesquelles la décomposition affine nécessaire à la méthodologie RB ne peut être obtenue (K. 3847, K. 2003). Pour traiter ce problème, il est à présent standard d’appliquer la méthodologie EIM (Barrault 2004, A. 2007) avant de déployer la méthodologie RB. Cependant le coût de calcul de cette approche est en général considérable car il requiert de nombreuses évaluations élément fini, la rendant très peu compétitive, pour construire l’approximation EIM des termes non-linéaires (A. 2007, Cécile Daversin 2013). Nous proposons l’algorithme SER qui construit simultanément l’approximation EIM et RB, fournit ainsi un gain de calcul considérable et requiert au minimum \(N+1\) résolutions élément fini 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 et finalement nous illustrons ses performances sur un benchmark proposé par (A. 2007).

Pour citer cet article : C. Daversin, C. Prud’homme, C. R. Acad. Sci. Paris, Ser. I 340 (2015).

journal: the Académie des sciences

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journal: the Académie des sciences

Introduction

\label{sec:introduction}

Reduced order modeling is more and more used in engineering problems due to efficient evaluation of quantities of interest. The Reduced Basis Method (see \cite{prud'homme:70,veroy03:_reduc,veroy03:_poster_error_bound_reduc_basis,MR2061274,Reviewquarteroni2011certified,rozza2007reduced}) has been especially designed for real-time and many-query contexts, and cover a large range of problems among which non-affinely parametrized Partial Differential Equations (PDE). A core enabler of this method is the so-called offline/online decomposition of the problem. This allows for computing costly parameter independent terms that depend solely on the finite element dimension. However such decomposition is not necessarily or readily available in particular for non-affine/non-linear problems. The Empirical Interpolation Method (EIM, see (Barrault 2004, A. 2007)) has been developed to recover this core ingredient and is used prior to the reduced basis methodology on industrial based applications, see e.g. (Cécile Daversin 2013). However the EIM building step can be costly when the terms are non-linear and requires many non-linear finite element solves. It is a deterring trait for this methodology which forbids its application to non-linear applications.

In this paper, we propose a Simultaneous EIM-RB (named SER) construction that requires only but a few finite element solves and which builds together the affine decomposition as well as the RB ingredient. To start we first give an overview of both Reduced Basis and Empirical Interpolation methods in non-affinely parametrized PDE context to recall the necessary notions. Based on these considerations, a second part makes an assessment of the proposed simultaneous approach discussing the changes to be made in the EIM offline step. The last part illustrates our method with preliminar results obtained on a benchmark introduced in (A. 2007).