Speaker: Michael Mascagni
Title: New Monte Carlo Methods for Problems in Materials and Biology.
Affiliation: Florida State University.
Date: Friday, February 23, 2001.
Place and Time: Room 101 - Love Building, 3:35-4:30 pm.
Refreshments: Room 204 - Love Building, 3:00 pm.

Abstract. Probabilistic potential theory enables us to solve a large class of parabolic and elliptic partial differential equations using diffusion techniques. Here, we present new first- and last-passage Monte Carlo algorithms and show their utility in problems coming from materials science and biology. These techniques exploit the fact that the first-passage probability function is the Green's function for the Dirchlet problem of the Laplace equation. First-passage algorithms allow the rapid simulation of diffusion using analytic or simulation-based Green's functions in rather less complicated basic geometries. This permits consideration of more complicated real geometries made up as combinations of the simple geometries where Green's functions are available. This new method is the extension of the well known "Walk on Spheres" method. Harnessing these first-passage algorithms, we have developed the fastest algorithms known to compute:

(1) The fluid permeability in overlapping, nonoverlapping, and polydispersed spherical models of random porous media

(2) The Solc-Stockmayer model with zero potential, a model of ligand binding

(3) The mean trapping rate of a diffusing particle in a domain of nonoverlapping spherical traps

(4) The effective conductivity for perfectly insulating, nonoverlapping spherical inclusions in a matrix of conductivity

    In certain problems, such as that of computing the electrostatic charge distribution on a conductor, using the last-passage distribution is useful. Using these analogous last-passage algorithms, we have solved the test problem of computing the charge distribution on a circular two-dimensional disk in three dimensions.
    Our plans for the future involve adding more surface Green's functions to our present set of known Green's functions, and the application of these techniques to more realistic problems in materials and biology.
    This is joint work with Dr. Chi-Ok Hwang of Florida State Unviersity, and Dr. James Given of Angle, Inc.