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

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.

Mots clés: STOCHASTIC PARTICLE SYSTEM; MCKEAN-VLASOV EQUATIONS; STATISTICAL SOLUTIONS

Domaines: Autre [cs.OH]

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⟩