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: firstname.lastname@example.org
Title: Stochastic Methods for Elliptic Problems: Applications to Materials and Biology
We present a brief
overview of stochastic methods for the solution of elliptic problems that arise
as wither partial differential
equations (PDEs) or integral equations (IEs). For
simplicity, 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):
with the boundary conditions
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, pW(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 IEs. We conclude with some new applications to the biophysics of proteins and to the visualization of integral curves of stochastic vector fields.
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