Resumen de Reduced Basis Techniques for Stochastic Problems

Sébastien Boyaval, Claude Le Bris, Tony Lelièvre, Yvon Maday , Ngoc-Cuong Nguyen, Anthony T. Patera

• We report here on the recent application of a now classical general reduction technique, the Reduced-Basis (RB) approach initiated by C. Prud�homme et al. in J. Fluids Eng. 124(1), 70�80, 2002, to the specific context of differential equations with random coefficients. After an elementary presentation of the approach, we review two contributions of the authors: in Comput. Methods Appl. Mech. Eng. 198(41�44), 3187�3206, 2009, which presents the application of the RB approach for the discretization of a simple second order elliptic equation supplied with a random boundary condition, and in Commun. Math. Sci., 2009, which uses a RB type approach to reduce the variance in the Monte-Carlo simulation of a stochastic differential equation. We conclude the review with some general comments and also discuss possible tracks for further research in the direction