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Recent research shows that Monte Carlo diffusion methods are often the most efficient algorithms for solving certain elliptic boundary value problems. In this paper, we extend this research by providing two efficient algorithms based on the concept of "last-passage diffusion." These algorithms are qualitatively compared with each other (and with the best… (More)

It is well known that there is no analytical expression for the electrical capacitance of a cube, even though it has been claimed that one can compute this capacitance numerically to high precision. However, there have been some disparities between reported numerical results of the capacitance of the unit cube. In this article, the ''walk on planes'' ͑WOP͒… (More)

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. Green's function first-passage (GFFP) methods solve such problems efficiently by generalizing the " walk on spheres " (WOS) method to allow… (More)

This study presents a Feynman–Kac path-integral implementation for solving the Dirichlet problem for Poisson's equation. The algorithm is a modified " walk on spheres " (WOS) that includes the Feynman–Kac path-integral contribution for the source term. In our approach, we use an h-conditioned Green's function instead of simulating Brownian trajectories in… (More)

A discrete random walk method on grids was proposed and used to solve the linearized Poisson– Boltzmann equation ͑LPBE͒ ͓R.

We describe two efficient methods of estimating the fluid permeability of standard models of porous media by using the statistics of continuous Brownian motion paths that initiate outside a sample and terminate on contacting the porous sample. The first method associates the ''penetration depth'' with a specific property of the Brownian paths, then uses the… (More)

The " Walk On Spheres " (WOS) algorithm and its relatives have long been used to solve a wide variety of boundary value problems [Ann. All WOS algorithms that require the construction of random walks that terminate, employ an-shell to ensure their termination in a finite number of steps. To remove the error related to this-shell, Green's function… (More)