A modified walk‐on‐sphere method for high dimensional fractional Poisson equation

Abstract

AbstractWe develop walk‐on‐sphere method for fractional Poisson equations with Dirichilet boundary conditions in high dimensions. The walk‐on‐sphere method is based on probabilistic representation of the fractional Poisson equation. We propose efficient quadrature rules to evaluate integral representation in the ball and apply rejection sampling method to drawing from the computed probabilities in general domains. Moreover, we provide an estimate of the number of walks in the mean value for the method when the domain is a ball. We show that the number of walks is increasing in the fractional order and the distance of the starting point to the origin. We also give the relationship between the Green function of fractional Laplace equation and that of the classical Laplace equation. Numerical results for problems in 2–10 dimensions verify our theory and the efficiency of the modified walk‐on‐sphere method.