Revisiting Kac's Method: A Monte Carlo Algorithm for Solving the Telegrapher's Equations

Title: Revisiting Kac's Method: A Monte Carlo Algorithm for Solving the Telegrapher's Equations

The solution of partial differential equations (PDEs) using Monte Carlo methods (MCMs) has been mainly restricted to elliptic and parabolic equations. This was due to the fact that probabilistic representations of the solutions of PDEs via mathematical tools, such as the Feynman-Kac formulae, were similarly restricted to elliptic and parabolic equations. A curious counter example was derived by Mark Kac, the Kac in Feynman-Kac, in which he produced a stochastic representation for the solution of the telegrapher's equation. The telegrapher's equation models a lossy wave equation, and was originally used to study the propagation of telegraph signals in cables placed in the ocean. Here, we use Kac's stochastic model to derive an MC algorithm for the numerical solution of the telegrapher's equation. Compared with the MCM recently proposed by Acebrón and Ribeiro, the Kac's model based method is able to handle two and higher dimensional problems, and has a more efficient algorithmic implementation. With numerical experiments, we have validated the accuracy and efficiency of the proposed algorithms, and their applicability to some higher dimensional problems.

This work was done in collaboration with my graduate student, Bolong Zhang, and my collaborator, Prof. Wenjian Yu from the Department of Computer Science at Tsinghua University in Beijing.