Stochastic Galerkin Methods: Fundamentals and Algorithms

具有随机数据的偏微分方程组构成了不确定性量化(UQ)的主要技术，在该技术中，关于微分方程数据的可变性和缺乏信息的情况是由表现出显著相关性的随机过程来模拟的。随机Galerkin(SG)方法通常用于逼近具有随机数据的这类偏微分方程组的解，自20世纪90年代末以来，在其数学分析和计算实现方面取得了很大进展。本项目的第二阶段旨在对SG方法的三个方面做出贡献：(1)深入分析用多项式混沌展开及其推广来表示随机过程的数学基础，这是SG方法的基础。在第一阶段给出了卡梅隆-马丁定理的严格推广之后，项目的第二阶段将集中于关于广义多项式混沌展开式的逼近性质的定量结果，并将其与基于小波的展开式进行比较。(2)建立具有随机系数的椭圆型偏微分方程组的适定公式：当平稳扩散方程的随机系数不一致地远离零和无穷大时，需要加权的L2-空间来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)基于最近发展的偏微分方程模型中贝叶斯逆问题的数学框架，我们将开发和实现反问题中UQ的贝叶斯逆方法，该方法以SG近似为基本步骤。我们将把这一技术应用于地下水流动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.