The Simulation–Tabulation Method for Classical Diffusion Monte Carlo

Many important classes of problems in materials science and biotechnology require the solution of the Laplace or Poisson equation in disordered two-phase domains in which the phase interface is extensive and convoluted. Greenʹs function first-passage (GFFP) methods solve such problems efficiently by generalizing the “walk on spheres” (WOS) method to allow first-passage (FP) domains to be not just spheres but a wide variety of geometrical shapes. (In particular, this solves the difficulty of slow convergence with WOS by allowing FP domains that contain patches of the phase interface.) Previous studies accomplished this by using geometries for which the Greenʹs function was available in quasi-analytic form. Here, we extend these studies by using the simulation–tabulation (ST) method. We simulate and then tabulate surface Greenʹs functions that cannot be obtained analytically. The ST method is applied to the Solc–Stockmayer model with zero potential, to the mean trapping rate of a diffusing particle in a domain of nonoverlapping spherical traps, and to the effective conductivity for perfectly insulating, nonoverlapping spherical inclusions in a matrix of finite conductivity. In all cases, this class of algorithms provides the most efficient methods known to solve these problems to high accuracy.