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Reduced basis techniques for stochastic problems. (English) Zbl 1269.65005
Summary: 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, No. 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, No. 41-44, 3187–3206 (2009; Zbl 1230.80013)], 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.

65C30 Numerical solutions to stochastic differential and integral equations
35R60 PDEs with randomness, stochastic partial differential equations
60H25 Random operators and equations (aspects of stochastic analysis)
62F15 Bayesian inference
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