A Reduced Basis Approach for Variational Problems with Stochastic Parameters: Application to Heat Conduction with Variable Robin Coefficient

Sébastien Boyaval 1, 2 Claude Le Bris 1, 2 Yvon Maday 3 Ngoc Cuong Nguyen 4 Anthony Patera 4
1 MICMAC - Methods and engineering of multiscale computing from atom to continuum
Inria Paris-Rocquencourt, ENPC - École des Ponts ParisTech
2 CERMICS - Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique
3 LJLL - Laboratoire Jacques-Louis Lions
4 MIT-MECHE - Department of Mechanical Engineering
Abstract : In this work, a Reduced Basis (RB) approach is used to solve a large number of Boundary Value Problems (BVPs) parametrized by a stochastic input — expressed as a Karhunen-Loève expansion — in order to compute outputs that are smooth functionals of the random solution fields. The RB method proposed here for variational problems parametrized by stochastic coefficients bears many similarities to the RB approach developed previously for deterministic systems. However, the stochastic framework requires the development of new a posteriori estimates for “statistical” outputs — such as the first two moments of integrals of the random solution fields; these error bounds, in turn, permit efficient sampling of the input stochastic parameters and fast reliable computation of the outputs in particular in the many-query context.
Keywords : Order Reduction Methods Reduced Basis Method A posteriori estimation Parameterized Partial Differential Equations Stochastic Partial Differential Equations Karhunen-Loève Monte Carlo
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Rapport
[Research Report] RR-6617, INRIA. 2008, pp.50
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Sébastien Boyaval, Claude Le Bris, Yvon Maday, Ngoc Cuong Nguyen, Anthony Patera. A Reduced Basis Approach for Variational Problems with Stochastic Parameters: Application to Heat Conduction with Variable Robin Coefficient. [Research Report] RR-6617, INRIA. 2008, pp.50. 〈inria-00311463〉

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