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
Offices: 498 Dirac Science Library/207A Love Building
Phone:   +1.850.644.3290
FAX:     +1.850.644.0058
e-mail:  mascagni@fsu.edu 

Title:    Monte Carlo Methods for Partial Differential Equations: A Personal Journey

Abstract:

We give a brief overview of the early history of the Monte Carlo method.  We then give a quick review of the Feynman-Kac equations; these allow one to represent the solution of linear elliptic and parabolic partial differential equations (PDEs) as expected values over stochastic processes.  The particular stochastic process for a given PDE is the solution to a stochastic differential equation defined via the elliptic operator in the PDE.   We then return to elliptic PDEs and discuss, in detail, several acceleration techniques that are widely applicable Monte Carlo methods.  We begin with the "walk on spheres" algorithm, followed by the the "Greens function first-passage" method, the "simulation-tabulation" method, "last passage" methods, the "walk on the boundary" method, and finally the "walk on subdomains" method.  These various Monte Carlo methods are presented within the context of various problems that arise in flow through porous media, electrostatics, and continuum biochemistry that were researched and published by Prof. Mascagni and his group over the years.

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