Monte Carlo Methods for Partial Differential Equations

Name:    Prof. Michael Mascagni         
Address: Department of Computer Science and
         School of Computational Science
         Florida State University
         Tallahassee, FL  32306-4530  USA
Offices: 498 Dirac Science Library/172 Love Building
Phone:   +1.850.644.3290
FAX:     +1.850.644.0098
e-mail:  mascagni@fsu.edu

Title: Using Simple SDEs (Stochastic Differential Equations) to Solve Complicated PDEs (Partial Differential Equations)

Abstract:

This talk begins with an overview of methods to solve PDEs based on the representation of point solutions of the PDEs as expected values of functionals of stochastic processes defined by the Feynman-Kac formula. The particular stochastic processes that arise in the Feynman-Kac formula are solutions to specific SDEs defined by the characteristics of the differential operator in the PDE. The Feynman-Kac formula is applicable to wide class of linear initial and initial-boundary value problems for elliptic and parabolic PDEs. We then concentrate our attention on elliptic boundary value problems that arise in applications in materials science and biochemistry. These problems are similar in that the PDEs to be solved are rather simple, and hence the associated SDEs that arise in the Feynman-Kac formula are likewise simple. However, the geometry of the problem is often complicated and amenable to several acceleration approaches particular to these simple SDEs. We will specifically describe the walk on spheres, Greens function first passage, last passage, walk on the boundary, and walk on subdomains methods in this context. These methods will be presented in the setting of several applications studied by the author and his research collaborators.