The goal of this project is to develop and analyze numerical methods for parabolic stochastic partial differential equations (SPDE's) driven by non-Gaussian noise. As a first step, work will focus on the stochastic heat equation in one dimension, with an additive noise term given by a Poisson random measure. The research will proceed along two paths that will be developed in parallel: mathematical analysis and numerical experimentation. These techniques will be used in order to study the application of both finite difference methods (including the explicit and fully implicit Euler methods and the Crank-Nicholson method) and finite element methods. Once the analysis of the heat equation is complete, it will be generalized to quasi-linear second order SPDE's with non-Gaussian noise. After the analysis of the one-dimensional case is complete, equations in many spatial dimensions will be considered.
Stochastic partial differential equations (SPDE's) are used to model many phenomena in the natural and social sciences. SPDE's are applied in a wide variety of applications in oceanography, neurophysiology, quantum physics, economics, and the physics of porous media (as applied, for example, in the geology of oil extraction). The purpose of this research is to develop and analyze techniques for the simulation of such equations where the randomness assumes the form of a series of discrete "shocks" (as is often the case in scientific applications). The development of the mathematics, computer algorithms and software for the analysis and simulation SPDE's will have a broad impact on the fields mentioned above, since simulation and numerical solution are an important part of the application of any stochastic model in science and engineering. Most current models using SPDE's make use of continuous, "Gaussian" noise. This is not necessarily due to the suitability of such a specification, but rather to its ease of application and implementation (and wider familiarity among scientists and engineers). The study of SPDE's driven by noise terms that are not Gaussian, and methods for their numerical solution in particular, will make an important new tool available to researchers in many disciplines.
