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:     +1.732.601.4470
e-mail:  mascagni@fsu.edu

Title:    First- and Last-Passage Random Walk Algorithms

Abstract:

We present two complimentary Monte Carlo methods for computing the charge density on a conductor when that conductor is at potential V₀ with respect to infinity.  The first method extends our previous first-passage algorithm for calculating the capacitance of an arbitrarily shaped conducting object.  The capacitance of an object can be probabilistically calculated finding the probability that a Brownian walker starting at infinity hits the region before returning to infinity.  It is a standard result from probabilistic potential theory that the absorption locations of the first-passage capacitance calculation give the charge distribution on the conducting object. The second Monte Carlo method utilizes the last-passage concept. This last-passage method stems from the consideration of the isomorphism between the electrostatic potential and the probability of going to infinity without touching a conducting object at a certain distance from the conductor. This is analogous to "adjoint" methods, where time-reversed walks are used to solve the adjoint problem.  We demonstrate our methods for computing an analytically known charge density on a two dimensional thin disk in three dimensions.  In addition we solve for the charge distribution on the unit cube using an edge distribution concept.  Computing the capacitance of the unit cube is considered the most important open problem in electrostatics, and we not only have the most accurate calculation, but the edge distribution computation provides results never before computed.

This is joint work with Dr. Chi-Ok Hwang of Inha University and Dr. James Given of the Naval Research Lab.

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