HAL - INRIA :: [inria-00311463, version 1] A Reduced Basis Approach for Variational Problems with Stochastic Parameters: Application to Heat Conduction with Variable Robin Coefficient

inria-00311463, version 1

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

Sébastien Boyaval ()^a^, 1, 2, Claude Le Bris ()^a^, 1, 2, Yvon Maday^b^, 3, Ngoc Cuong Nguyen^c^, 4, Anthony T. Patera ()^c^, 4

N° RR-6617 (2008)

Résumé : 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.

a – Ecole Nationale des Ponts et Chaussées
b – Université Pierre et Marie Curie - Paris VI
c – Massachussetts Institute of Technology (MIT)
1 : MICMAC (INRIA Paris - Rocquencourt)
Ecole des Ponts ParisTech – INRIA
2 : Centre d'Enseignement et de Recherche en Mathématiques, Informatique et Calcul Scientifique (CERMICS)
INRIA – Ecole des Ponts ParisTech
3 : Laboratoire Jacques-Louis Lions (LJLL)
CNRS : UMR7598 – Université Paris VI - Pierre et Marie Curie
4 : Department of Mechanical Engineering (MIT-MECHE)
Massachussetts Institute of Technology (MIT)

inria-00311463, version 1
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Contributeur : Sébastien Boyaval <>
Soumis le : Lundi 18 Août 2008, 11:44:59
Dernière modification le : Lundi 18 Août 2008, 15:40:58

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%% inria-00311463, version 1
%% http://hal.inria.fr/inria-00311463
@techreport{boyaval:inria-00311463,
    hal_id = {inria-00311463},
    url = {http://hal.inria.fr/inria-00311463},
    title = {{A Reduced Basis Approach for Variational Problems with Stochastic Parameters: Application to Heat Conduction with Variable Robin Coefficient}},
    author = {Boyaval, S{\'e}bastien and Le Bris, Claude and Maday, Yvon and Cuong Nguyen, Ngoc and Patera, T., Anthony},
    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-Lo{\`e}ve 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 = {Stochastic Partial Differential Equations; Karhunen-Lo{\`e}ve; Monte Carlo; Parameterized Partial Differential Equations; Order Reduction Methods; Reduced Basis Method; A posteriori estimation},
    language = {Anglais},
    affiliation = {MICMAC - INRIA Paris - Rocquencourt , Centre d'Enseignement et de Recherche en Math{\'e}matiques, Informatique et Calcul Scientifique - CERMICS , Laboratoire Jacques-Louis Lions - LJLL , Department of Mechanical Engineering - MIT-MECHE},
    pages = {50},
    type = {Rapport de recherche},
    institution = {INRIA},
    number = {RR-6617},
    year = {2008},
    pdf = {http://hal.inria.fr/inria-00311463/PDF/SPDE\_final.pdf},
}

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