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Reduced Basis Techniques for Stochastic Problems

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Abstract

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.

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Authors and Affiliations

  1. CERMICS, Ecole des ponts ParisTech, Université Paris-Est, 6-8 avenue Blaise Pascal, Cité Descartes, 77455, Marne la Vallée Cedex 2, France

    S. Boyaval, C. Le Bris & T. Lelièvre

  2. MICMAC project team, INRIA, Domaine de Voluceau, Rocquencourt, 78153, Le Chesnay Cedex, France

    S. Boyaval, C. Le Bris & T. Lelièvre

  3. UMR 7598, Laboratoire Jacques-Louis Lions, UPMC Univ Paris 06, 75005, Paris, France

    Y. Maday

  4. Division of Applied Mathematics, Brown University, Providence, USA

    Y. Maday

  5. Department of Mechanical Engineering, Massachusetts Institute of Technology, Cambridge, MA, 02139, USA

    N. C. Nguyen & A. T. Patera

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  1. S. Boyaval
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  2. C. Le Bris
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Boyaval, S., Le Bris, C., Lelièvre, T. et al. Reduced Basis Techniques for Stochastic Problems. Arch Computat Methods Eng 17, 435–454 (2010). https://doi.org/10.1007/s11831-010-9056-z

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