MATHEMATICS COLLOQUIUM
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, 3:354:30 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 lastpassage Monte Carlo
algorithms and show
their utility in problems coming from materials science and biology.
These techniques exploit the fact that the firstpassage probability
function is
the Green's function for the Dirchlet problem of the Laplace equation.
Firstpassage algorithms allow the rapid simulation of diffusion using
analytic or simulationbased 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 firstpassage
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 SolcStockmayer 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 lastpassage distribution is
useful. Using these analogous lastpassage algorithms, we have solved
the test problem of computing the charge distribution on a circular
twodimensional 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. ChiOk Hwang of Florida State Unviersity,
and Dr. James Given of Angle, Inc.
