Solving Dirichlet Problems Numerically Using the Feynman–Kac Representation

Abstract

In this paper we study numerical solutions of the Dirichlet problem in high dimensions using the Feynman–Kac representation. What is involved are Monte-Carlo simulations of stochastic differential equations and algorithms to accurately determine exit times and process values at the boundary. It is assumed that the radius of curvature of the boundary is much larger than the square root of the step size. We find that the canonical reciprocal square root behavior of statistical errors as a function of the sample size holds regardless of the dimension of the space. In fact, the coefficient of the recriprocal square root actually seems to decrease with dimension. Additionally, acceptance ratios for finding the boundary become less sensitive to the step size in higher dimensions.

The walk on cubes method, wherein the model increments of Brownian motion are 3-point random variables, is of particular interest. Our motivation for this approach will be to use well established Runge–Kutta (RK) methods, although we do not use these familiar methods here where a Taylor series is fairly simple. Comparisons are made between this walk on cubes method, Milstein's walk on spheres, and a simpler 2-point method. Because of conditional expectations used in the formulations of the walk on spheres procedure, it is not amenable to RK constructions.

Our examples have hyperspherical domains in up to 64 dimensions.