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

本项目的目标是发展和分析非高斯噪声驱动的抛物型随机偏微分方程（SPDE）的数值方法。作为第一步，工作将集中在一维的随机热方程，与一个加性噪声termsgiven的泊松随机测度。 本研究将沿着两条平行发展的路径进行：数学分析和数值实验。这些技术将被用来研究有限差分方法（包括显式和全隐式欧拉方法以及Crank-Nicholson方法）和有限元方法的应用。一旦热方程的分析完成，它将被推广到具有非高斯噪声的准线性二阶SPDE。 在一维情况的分析完成后，将考虑许多空间维度的方程。随机偏微分方程（SPDE）被用来模拟自然科学和社会科学中的许多现象。SPDE广泛应用于海洋学、神经生理学、量子物理学、经济学和多孔介质物理学（如石油开采地质学）。 这项研究的目的是开发和分析技术，用于模拟这样的方程，其中随机性假定为一系列离散“冲击”的形式（这在科学应用中经常发生）。 由于模拟和数值求解是任何随机模型在科学和工程中应用的重要部分，因此，用于分析和模拟随机微分方程的数学、计算机算法和软件的发展将对上述领域产生广泛的影响。 目前大多数使用SPDE的模型使用连续的“高斯”噪声。 这并不一定是因为这样一个规范的适用性，而是因为它易于应用和实现（以及科学家和工程师之间的广泛熟悉）。非高斯噪声项驱动的随机微分方程的研究，特别是其数值求解方法，将为许多学科的研究人员提供一个重要的新工具。

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.