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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.

MSC:
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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rbMIT
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