Probability And Partial Differential Equations In Modern Applied Mathematics

The IMA Summer Program on 'Probability and Partial Differential Equations in Modern Applied Mathematics' took place July 21-August 1, 2003. The program was devoted to the role of probabilistic methods in modern applied mathematics from perspectives of both a tool for analysis and as a tool in modeling. 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. A probabilistic representation of solutions to partial differential equations that arise as deterministic models allows one to exploit the power of stochastic calculus and probabilistic limit theory in analysis, as well as 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 both non-Monte Carlo as well as Monte Carlo type numerical methods. There is also a growing recognition of a role in the inclusion of stochastic terms in the modeling of complex flows, and the addition of such terms has led to interesting new mathematical problems at the interface of probability, dynamical systems, numerical analysis, and partial differential equations. This volume consists of original contributions by researchers with a common interest in the problems, but with diverse mathematical expertise and perspective. The volume will be useful to researchers and graduate students who are interested in probabilistic methods, dynamical systems approaches and numerical analysis for mathematical modeling in engineering and sciences.