Monte Carlo Solution of Dirichlet Problem for Semi-linear Equation

Abstract

In the present work Dirichlet boundary value problem (BVR) for semi-linear elliptic equation is considered. Assuming the existence of solution BVR obtained a probabilistic representation of the solution as a mathematical expectation of some random variable. In accordance with a probabilistic representation on the trajectories of the branching random process were constructed unbiased estimator of the solution. An unbiased estimator of the solution has finite variance, based on the trajectories of a branching process with a finite average number of branching and easily simulated. Some numerical experiments are performed.