Random walk on ellipsoids method for solving elliptic and parabolic equations

Abstract

Abstract A Random Walk on Ellipsoids (RWE) algorithm is developed for solving a general class of elliptic equations involving second- and zero-order derivatives. Starting with elliptic equations with constant coefficients, we derive an integral equation which relates the solution in the center of an ellipsoid with the integral of the solution over an ellipsoid defined by the structure of the coefficients of the original differential equation. This integral relation is extended to parabolic equations where a first passage time distribution and survival probability are given in explicit forms. We suggest an efficient simulation method which implements the RWE algorithm by introducing a notion of a separation sphere. We prove that the logarithmic behavior of the mean number of steps for the RWS method remains true for the RWE algorithm. Finally we show how the developed RWE algorithm can be applied to solve elliptic and parabolic equations with variable coefficients. A series of supporting computer simulations are given.