Efficient numerical solutions of Neumann problems in inhomogeneous media from their probabilistic representations and applications

The Neumann problem, particularly for problems exterior to a bounding surface, is very useful in a number of situations. Those that are described by elliptic and parabolic partial differential equations are immediately amenable to probabilistic representations for the solutions, the numerical implementation of which has several advantages including parallelism and for obtaining solutions at specific desired points or subregions. Although these representations have been known for a long time to mathematicians, the numerical implementation of these seems to have been little used. The purpose of this note is to point out that a suitable characterization of the mathematical representation results in an efficient scheme and to give some numerical examples, including from our own work involving detailed modeling of drug transport in tissue.