Numerical Techniques for Stochastic Partial Differential Equations with Non-Gaussian Noise

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The goal of this project is to develop and analyze numerical methodsfor parabolic stochastic partial differential equations (SPDE's)driven by non-Gaussian noise. As a first step, work will focus on thestochastic heat equation in one dimension, with an additive noise termgiven by a Poisson random measure. The research will proceed alongtwo paths that will be developed in parallel: mathematical analysisand numerical experimentation. These techniques will be used in orderto study the application of both finite difference methods (includingthe explicit and fully implicit Euler methods and the Crank-Nicholsonmethod) and finite element methods. Once the analysis of the heatequation is complete, it will be generalized to quasi-linear secondorder SPDE's with non-Gaussian noise. After the analysis of theone-dimensional case is complete, equations in many spatial dimensionswill be considered.Stochastic partial differential equations (SPDE's) are used to modelmany phenomena in the natural and social sciences. SPDE's are appliedin a wide variety of applications in oceanography, neurophysiology,quantum physics, economics, and the physics of porous media (asapplied, for example, in the geology of oil extraction). The purposeof this research is to develop and analyze techniques for thesimulation of such equations where the randomness assumes the form ofa series of discrete "shocks" (as is often the case in scientificapplications). The development of the mathematics, computeralgorithms and software for the analysis and simulation SPDE's willhave a broad impact on the fields mentioned above, since simulation andnumerical solution are an important part of the application of any stochastic model in science and engineering. Most current models usingSPDE's make use of continuous, "Gaussian" noise. This is notnecessarily due to the suitability of such a specification, but ratherto its ease of application and implementation (and wider familiarityamong scientists and engineers). The study of SPDE's driven by noiseterms that are not Gaussian, and methods for their numerical solutionin particular, will make an important new tool available toresearchers in many disciplines.