Stochastic Galerkin Methods: Fundamentals and Algorithms

随机数据的偏微分方程是不确定性量化的一种主要方法，它利用具有显著相关性的随机过程来模拟微分方程数据的变异性和信息缺失。随机Galerkin（SG）方法被广泛地应用于随机数据下的偏微分方程的求解，自20世纪90年代末以来，SG方法在数值分析和计算实现方面取得了很大的进展。本项目的第二阶段主要在三个方面对SG方法进行了研究：（1）深入分析了SG方法的数学基础，即用多项式混沌展开及其推广来表示随机过程。在第一阶段中给出了严格的Cameron-Martin定理的推广之后，该项目的第二阶段将专注于广义多项式混沌展开的近似性质的定量结果，并将其与基于小波的展开进行比较。(2)建立具有随机系数的椭圆型偏微分方程的适定公式：当一个定态扩散方程的随机系数不是一致有界的远离零和无穷大时，加权L2-空间是必要的，以获得适定的变分公式。我们将研究这些问题及其对计算技术的影响。(3)我们将开发和实施一个贝叶斯反演方法UQ在反问题的基础上，最近开发的数学框架贝叶斯反演偏微分方程模型，其中使用SG近似作为一个基本步骤。我们将把这种技术应用于地下水流和地球物理勘探中的两个反问题的案例研究。

PDEs with random data constitute a primary technique of uncertainty quantification (UQ), in whichvariability and lack of information on the data of a differential equation are modeled by stochasticprocesses displaying significant correlations. Stochastic Galerkin (SG) methods are routinely used toapproximate the solutions of such PDEs with random data, and much progress in their mathematicalanalysis and computational implementation has been made since the late 1990s. The second phaseof this project intends to make a contribution in three aspects of SG methods:(1) A deeper analysis of the mathematical foundations underlying the representation of stochasticprocesses by polynomial chaos expansions and their generalizations, which underly the SG method. After giving a rigorous generalization of the Cameron-Martin theorem in the first phase, the second phase of the project will focus on quantitative results on the approximation properties of generalized polynomial chaos expansions as well as compare these with wavelet-based expansions.(2) Establishing well-posed formulations of elliptic PDEs with random coefficients: When the randomcoefficient of a stationary diffusion equation is not uniformly bounded away from zero and infinity, weighted L2-spaces are necessary to obtain well-posed variational formulations. We will investigate these and their implications on computational techniques.(3) We will develop and implement a Bayesian inversion methodology for UQ in inverse problems based on recently developed mathematical frameworks for Bayesian inversion in PDE models, which use SG approximation as a basic step. We will apply this techniqe to two case studies of inverse problems in groundwater flow and geophysical exploration.