Efficient numerical solutions of Neumann problems in inhomogeneous media from their probabilistic representations and applications. Issue 2 (11th February 2020)

• Record Type: Journal Article
• Authors: Raghavan, Raghu; Brady, Martin L.
• Is Part Of: Engineering reports. Volume 2:Issue 2(2020)
• Page Start: n/a
• Page End: n/a
• Publication Date: 2020-02-11
• Subjects: Convection‐enhanced Delivery (CED) -- Numerical solutions PDE -- Neumann problems -- Probabilistic representations of Neumann problems; Engineering -- Periodicals; Computer science -- Periodicals; 620.005
• DOI: 10.1002/eng2.12108
• Languages: English
• ISSNs: 2577-8196

Abstract

Abstract : 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. Abstract : The Neumann problem, particularly for problems exterior to a bounding surface, is very useful in a number of situations. The probabilistic representation of the solution is usually expressed in terms of the concept of local time. In this article, we show that a suitable characterization of local time results in an efficient numerical method, in contrast to previous more direct characterizations. The method is examined numerically and illustrated in particular applications where it results in a considerable advance in modeling transport of (therapeutic) molecules directly into tissue.