Stochastic Galerkin Methods: Fundamentals and Algorithms

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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.