Monte Carlo methods for the solution of nonlinear partial differential equations

Abstract

Stochastic models for the solution of nonlinear partial differential equations are discussed. They consist of a discretized version of these equations and Monte Carlo techniques. The Markov transitions are based on a priori estimates of the solution. To improve the efficiency of stochastic smoothers a Monte Carlo multigrid method is presented. The numerical results presented show the convergence of these methods. Some directions for the parallelization of the Monte Carlo algorithms presented are outlined. The techniques introduced make possible the extension of Monte Carlo methods to nonlinear problems, offering a new approach with an analytic potential for a wide range of problems in computational physics.