Solution Bounds for Elliptic Partial Differential Equations via Feynman-Kac Representation

Abstract

We transform suitable smooth functions into hard bounds for the solution to boundary value and obstacle problems for elliptic partial differential equations based on the probabilistic Feynman-Kac representation. Unlike standard approximate solutions, hard solution bounds are intended to limit the location of the solution, possibly to a large extent, and, thus, have the potential to be very useful information. Our approach requires two main steps. First, the violation of sufficient conditions is quantified for the test function to be a hard bounding function. After extracting those violation terms from the Feynman-Kac representation, it remains to deal with a boundary value problem with constant input data. Although the probabilistic Feynman-Kac representation is employed, the resulting numerical method is deterministic without the need for sophisticated probabilistic numerical methods, such as sample paths generation of reflected diffusion processes. Throughout this article, we provide numerical examples to illustrate the effectiveness of the proposed method.