Solving Dirichlet problems numerically using the Feynman-Kac representation

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 \mathcal{O}(N-1/2) behavior of statistical errors as a function of the sample size N holds regardless of the dimension n of the space. In fact, the coefficient of N-1/2 seems to actually decrease with n. Additionally, acceptance ratios for finding the boundary become less sensitive to the time step size in higher dimensions. The walk on cubes method, wherein the model increments of Brownian motion are three-point random variables, is of particular interest. Comparisons are made between this walk-on-cubes method, Milstein's walk on spheres, and a simpler 2-point method. Our examples have hyperspherical domains up to n=64 dimensions