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Reduced basis techniques for stochastic problems

We report here on the recent application of a now classical general reduction technique, the Reduced-Basis approach initiated in [C. Prud'homme, D. Rovas, K. Veroy, Y. Maday, A. T. Patera, and G. Turinici. Reliable real-time solution of parametrized partial differential equations: Reduced-basis output bounds methods. Journal of Fluids Engineering, 124(1):7080, 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: [S. Boyaval, C. Le Bris, Y. Maday, N.C. Nguyen, and A.T. Patera. A reduced basis approach for variational problems with stochastic parameters: Application to heat conduction with variable Robin co-efficient. Computer Methods in Applied Mechanics and Engineering, 198(4144):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 [S. Boyaval and T. Lelièvre, A variance reduction method for parametrized stochastic differential equations using the reduced basis paradigm with T. Lelièvre, Commun. Math. Sci. 8, special Issue "Mathematical Issue on Complex Fluids" P. Zhang ed., to appear, 2010, ARXIV preprint arXiv:0906.3600], 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.