IMA 2003 Summer Program:

Probability and Partial Differential Equations in Modern Applied Mathematics

July 21-August 1, 2003

Organizers:

Edward C. Waymire (chair)
Mathematics Department
Oregon State University
waymire@math.orst.edu

Jinqiao Duan
Department of Applied Mathematics
Illinois Institute of Technology
duan@iit.edu

This summer program is devoted to the role of probabilistic methods in modern applied mathematics from perspectives of both a tool for analysis and in modeling. Researchers involved in contemporary problems involving dispersion and flow, e.g. fluid flow, cash flow, genetic migration, flow of internet data packets, etc., were selected as speakers and to lead discussion groups. There is a growing recognition in the applied mathematics research community that stochastic methods are playing an increasingly prominent role in the formulation and analysis of diverse problems of contemporary interest in the sciences and engineering. In organizing this workshop an explicit effort is being made to bring together researchers with a common interest in the problems, but with diverse mathematical expertise and perspective. In particular, many are researchers who might not ordinarily interact with one another.

The Navier-Stokes equations, Euler equations, Burgers equation etc. arise naturally in the context of the basic physics of fluid motions. As such these equations play a fundamental role in the science and engineering of all varieties of fluid flow, from oceanic circulation to the flow of water beneath the earth's surface. The nonlinearity of the equations makes explicit solutions possible only for the simplest of flows. The development of a more complete understanding of these equations from the laminar to the turbulent regime ranks among the most important outstanding problems of modern mathematical physics.

A probabilistic representation of solutions allows one to exploit the power of stochastic calculus and probabilistic limit theory in the analysis of deterministic problems, as well as to offer new perspectives on the phenomena for modeling purposes. In addition such approaches can be effective in sorting out multiple scale structure and in the development of Monte Carlo type numerical methods. Among the relevant probability topics are multitype branching random walks, Brownian motion and stochastic calculus, Kolmogorov cascade theory, the vortex method and multiscale methods. In addition to Navier-Stokes equations, other deterministic linear and nonlinear partial differential equations amenable to probabilistic methods of analysis will be of interest in this workshop, e.g. variations on Black-Scholes options equations, contaminant transport, reaction-diffusion, Schrodinger equation, etc.

There is a growing recognition of a role for the inclusion of stochastic terms in the modeling of complex flows. The addition of such terms has led to interesting new mathematical problems at the interface of probability, numerical analysis, and partial differential equaitons. During the last decade, significant progress has been made towards building a comprehensive theory of random dynamical systems, statistical cascades, stochastic flows, stochastic pde's, etc. Some core questions in the modeling, analysis and simulation of complex flows under uncertainty are: exploring appropriate ways to take stochastic effects into account; understanding the impact of randomness on the evolution of complex systems; and designing efficient numerical algorithms to simulate random phenomena.

There is also a need for new ways in which to incorporate the impact of probability, statistics, pde's and numerical analysis in the training of present and future PhD students in the mathematical sciences. The engagement of graduate students is an important feature of this summer program. A poster session will be included in each workshop.

The first workshop will be devoted to the role of probability as an analysis tool as described above. The second workshop will emphasize stochastic methods in connection with modeling of various dispersion and flow scenarios. Thus the combined focus of the two workshops is broadly on probabilistic methods in the modeling and analysis of dispersion and flow, but from two different perspectives.

Workshop 1: July 21-July 25, 2003: Probabilistic Methods in Analysis of PDE's.
Organizers: Edward C. Waymire, Mina E. Ossiander, Enrique A. Thomann, Ronald B. Guenther, Larry Chen (Mathematics, Oregon State University), Rabi N. Bhattacharya (Mathematics, Indiana University)

Workshop 2: July 28-August 1, 2003: Stochastic Models and their Analysis
Organizers: Jinqiao Duan (Illinois Institute of Technology), Bjorn Schmalfuss (Merseburg), Peter E. Kloeden (Frankfurt), Salah Mohammed (Southern Illinois University), Sri Namachchivaya (UIUC)