A new class of grid-free Monte Carlo algorithms for elliptic boundary value problems

Abstract

In this paper we consider the following mathematical model: an elliptic boundary value problem, where the partial differential equation contains advection, diffusion, and deposition parts. A Monte Carlo (MC) method to solve this equation uses a local integral representation by the Green's function and a random process called "Walks on Balls" (WOB). A new class of grid free MC algorithms for solving the above elliptic boundary value problem is suggested and studied. We prove that the integral transformation kernel can be taken as a transition density function in the Markov chain in the case when the deposition part is equal to zero. An acceptance-rejection (AR) and an inverse-transformation methods are used to sample the next point in the Markov chain. An estimate for the efficiency of the AR method is obtained. © Springer-Verlag Berlin Heidelberg 2003.