Spatial filter averaging approach of probabilistic method to linear second-order partial differential equations of the parabolic type

Abstract

A backward-in-time probabilistic method with spatial filter averaging is presented to solve linear second-order partial differential equations of the parabolic type. An advantage of this methodology is that while forward methods are subject to region with loss of density of particles and hence loss of spatial resolution of the solution, the solution given by backward methods is given on any desired grid. However, traditional backward time probabilistic method using Monte Carlo averaging are computationally expensive. We prove a convergence result and present several examples. The method leads to important improvement in computational efficiency and is expected to perform well to solve high dimensional problems where a solution is needed on a large grid.