A Modified Walk-on-sphere Method for High Dimensional Fractional Poisson Equation

TL;DR

This paper introduces a modified walk-on-sphere method for solving high-dimensional fractional Poisson equations, utilizing probabilistic representations, efficient quadrature, and rejection sampling, with theoretical analysis and numerical validation.

Contribution

It presents a novel, efficient walk-on-sphere algorithm tailored for high-dimensional fractional Poisson equations, including error estimates and probabilistic analysis.

Findings

Number of walks increases with fractional order and distance from origin.

The method is effective for dimensions 2 to 10.

Numerical results confirm theoretical predictions and efficiency.

Abstract

We develop walk-on-sphere for fractional Poisson equations with Dirichilet boundary conditions in high dimensions. The walk-on-sphere method is based on probabilistic represen tation of the fractional Poisson equation. We propose effcient 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 effciency of the modified walk-on-sphere method.