Some Continuous Monte Carlo Methods for the Dirichlet Problem

Abstract

Monte Carlo techniques are introduced, using stochastic models which are Markov processes. This material includes the $N$-dimensional Spherical, General Spherical, and General Dirichlet Domain processes. These processes are proved to converge with probability 1, and thus to yield direct statistical estimates of the solution to the $N$-dimensional Dirichlet problem. The results are obtained without requiring any further restrictions on the boundary or the function defined on the boundary, in addition to those required for the existence and uniqueness of the solution to the Dirichlet problem. A detailed study is made for the $N$-dimensional Spherical process; this includes a study of the order of the average number of steps required for convergence. Asymptotic confidence intervals are obtained. When computing effort is measured in terms of the order of the average number of steps required for convergence, the often-made conjecture that the computing effort of a Monte Carlo procedure should be a linear function of the dimensionality of the problem is shown to be true for the cases considered. Comments are included regarding the application of these processes on digital computers, and truncation methods are suggested.