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A Stochastic Particle Method with Random Weights for the Computation of Statistical Solutions of McKean-Vlasov Equations. Part I: Foundation of the Method and Empirical Evidence

Denis Talay 1 Olivier Vaillant 1
1 OMEGA - Probabilistic numerical methods
CRISAM - Inria Sophia Antipolis - Méditerranée , UHP - Université Henri Poincaré - Nancy 1, Université Nancy 2, CNRS - Centre National de la Recherche Scientifique : UMR7502
Abstract : We are interested in statistical solutions [28] of McKean-Vlasov equations. An example of motivation is the Navier-Stokes equation for the vorticity of a 2D incompressible fluid flow. We propose an original and efficient numerical method to compute moments of such solutions. It is a stochastic particle method with random weights. These weights are defined through nonparametric estimators of a regression function, and convey the uncertainty on the initial condition of the considered equation. In this first part of our work, we prove an existence and uniqueness result for a class of nonlinear stochastic differential equations (SDE), and we study the relation between these nonlinear SDEs and statistical solutions of the corresponding McKean-Vlasov equations. This result founds our stochastic particle method, for which we show results of numerical experiments obtained for the Burgers and the 2D Navier-Stokes equation.
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Submitted on : Tuesday, May 23, 2006 - 8:15:08 PM
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Denis Talay, Olivier Vaillant. A Stochastic Particle Method with Random Weights for the Computation of Statistical Solutions of McKean-Vlasov Equations. Part I: Foundation of the Method and Empirical Evidence. [Research Report] RR-4326, INRIA. 2001, pp.28. ⟨inria-00072261⟩

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