Stochastic Methods for Elliptic PDEs

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 for Elliptic Partial Differential Equations: Avoiding Complicated Deterministic Structures in Applications

Abstract:

We present a brief overview of stochastic methods for the solution of elliptic partial differential equations (PDEs). We focus our attention on the solution of the Dirichlet problem for the Laplace equation in a domain W, with boundary, ¶W, and with given boundary data f(x):

Df(x) = 0, x Î W,

(1)

with the boundary conditions

f(x) = f(x), x Î ¶W.

(2)

The value of f(·) can be efficiently estimated at a point x by averaging the boundary value of where Brownian walkers started at x first strike the boundary, ¶W. In addition, p_W(x,y), the first-passage probability of Brownian walkers starting at x striking the boundary of W first at the point y is equal to the boundary Green's function for W. This fact provides the mathematical basis for a wide variety of fast algorithms to exploit these stochastic ideas for solving PDEs.

We develop these ideas to provide fast algorithms to solve a variety of problems involving just the Laplace equation. In each case, the problem has a considerable complication that slows or completely inhibits solution through more standard deterministic approaches. In problems involving the computation of material properties where the Laplace equation must be solved in the presence of extremely complicated boundaries we will provide examples where we compute effective electrical and transport properties. In addition, we discuss the effectiveness of these methods on two, related, problems in electrostatics that are difficult via deterministic methods due to their singular nature. These concrete examples show that in many cases where the stochastic method can avoid the construction of a discrete object whose dimensionality and/or size impedes the deterministic method, the ``best" method of solution is stochastic. We provide examples in many different application areas involving variants of the stochastic methods that include the above ``Feynman-Kac" methods as well as some Markov-chain methods for solving related integral equations. We also show some new applications of these ideas to the visualization of integral curves of stochastic vector fields.