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

**Title:
Monte Carlo Methods for Partial Differential Equations:
Computing Permeability**

**Abstract:**

**We present a brief
overview of Monte Carlo methods for the solution of elliptic and parabolic partial differential
equations (PDEs). We begin with a review of the Feynman-Kac formula, and
its use in the probabilistic representation of the solutions of elliptic and
parabolic PDEs. We then consider some specific Monte Carlo methods used for
obtaining the solution of simple elliptic partial differential equations (PDEs)
as part of exterior boundary value problems that arise in electrostatics and
flow through porous media. These Monte Carlo methods use Feynman-Kac to
represent the solution of the elliptic PDE at a point as the expected value of
functionals of Brownian motion trajectories started at the point of interest. We
discuss the rapid solution of these equations, in complex exterior geometries,
using both the ``walk on spheres" and ``Greens function first-passage"
algorithms. We then concentrate on methods for quickly computing the isotropic
permeability using the ``unit capacitance" and ``penetration depth'' methods.
The first of these methods, requires computing a linear functional of the
solution to an exterior elliptic PDE. Both these methods for computing
permeability are simple, and provide accurate solutions in a few seconds on
laptop-scale computers. We then conclude with a brief look at other Monte
Carlo methods and problems that arise on related application areas.**

Home | Educational Background | Research Experience | Curriculum Vitae | Research Interests | Research Projects | Recent Papers | Courses | Abstracts of Talks