Name: Prof. Michael Mascagni Address: Department of Computer Science and School of Computational Science Florida State University Tallahassee, FL 323064530 USA Offices: 498 Dirac Science Library/172 Love Building Phone: +1.850.644.3290 FAX: +1.850.644.0098 email: mascagni@fsu.edu
Title: Stochastic Methods for Elliptic Problems: Applications to Materials and Biology
Abstract:
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):

(1) 
with the boundary conditions

(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 firstpassage 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 ``FeynmanKac" methods as well as some Markovchain 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.
Home  Educational Background  Research Experience  Curriculum Vitae  Research Interests  Research Projects  Recent Papers  Courses  Abstracts of Talks