Stochastic Methods in Electrostatics

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:     +1.850.644.0098
e-mail:  mascagni@fsu.edu

Title: Stochastic Methods in Electrostatics: Applications to Biological and Physical Science

Abstract:

We present an overview of stochastic methods for the solution of elliptic partial differential equations (PDEs). In particular, we consider the solution of linear and nonlinear problems that arise in electrostatics computations in various applications. We discuss the "Walk On Spheres" (WOS), "Greens Function First-Passage" (GFFP), and "Simulation-Tabulation" (S-T) Monte Carlo methods for the computation of capacitance, charge density, and related problems in materials science and biophysics. In addition, we introduce the "Walk on the Boundary" method for rapid calculation of capacitance. We then present generalizations that permit direct computation of charge density. Finally, we consider the problem of the electrostatics of large molecules in aqueous solution. An implicit model of the solvent leads to consideration of the Poisson-Boltzmann equation (PBE) as a continuum electrostatic model. We present new Monte Carlo methods for the solution of the linear PBE based on WOS, GFFP, and other methods. In particular, we solve an elliptic PDE system with the Poisson equation inside the molecule of interest, the linear PBE outside, and matching Neumann boundary conditions on the molecular surface. We demonstrate the intrinsic advantages of these methods on an electrostatic internal energy computation with the use of new, efficient Monte Carlo approaches to the boundary conditions.

This work is joint with Dr. Nikolai Simonov of Florida State University and the Siberian Branch of the Russian Academy of Sciences.