Monte Carlo Methods for the Telegrapher's Equation (Based on Mark Kac's Probabilistic Representation)

Name:    Prof. Michael Mascagni                 
Address: Department of Computer Science and
         Department of Mathematics and
	 Department of Scientific Computing and
         Graduate Program in Molecular Biophysics	
         Florida State University
         Tallahassee, FL  32306-4530  USA
         AND
         Information Technology Laboratory
         Applied and Computational Mathematics Division
         100 Bureau Drive M/S 8910
         National Institute of Standards and Technology (NIST)
         Gaithersburg, MD  20899-8910  USA

Offices: 498 Dirac Science Library/207A Love Building (FSU) Building 225/Room B154 (NIST)
Phone:   +1.850.644.3290 (FSU) +1.301.975.2051 (NIST)
FAX:     +1.850.644.0058
e-mail:  mascagni@fsu.edu (FSU) mascagni@nist.gov (NIST)

Title: Monte Carlo Methods for the Telegrapher's Equation (Based on Mark Kac's Probabilistic Representation)

Monte Carlo methods for PDEs are mainly restricted to elliptic and parabolic equations using probabilistic representations based on the Feynman-Kac formulas. However, Mark Kac came up with an ingenious probabilistic representation for the solution to the Telegrapher's equation. The Telegrapher's equation is a transitional equation in the sense that one limit yields the heat equation (parabolic), and a different limit yields the wave equation (hyperbolic). We show how to compute numerical solutions to the Telegrapher's equation using an algorithm based on Kac's representation. We show that this provides a finite variance Monte Carlo estimator, and that the technique seems to work in arbitrary dimensions. This provides the first Monte Carlo methods we are aware of for hyperbolic PDEs.

This is joint work with Prof. Wenjian Yu in CS at Tsinghua University, and Bolong Zhang, who is currently my doctoral student.