### Michael Mascagni

Department of Computer Science

Florida State University

April 21, 2015

Computational Geometry Aspects of Monte Carlo Approaches to PDE Problems in Biology, Chemistry, and Materials

We will introduce at some Monte Carlo methods for solving problems in electrostatics. These rely on evaluating functionals of the first-passage time of Brownian motion on geometries defined by the system of interest. We will use the Walk on Spheres (WOS) algorithm to quickly evaluate these functionals, and we will show how a computational geometric computation dominates this computation in complexity. We then consider computing the capacitance of a complicated shape, and use this as our model problem to find an efficient serial and parallel implementation. The capacitance computation is prototypical of many Monte Carlo approaches to problems in biology, biochemistry, and materials science.