Model Reduction and Domain Decomposition Methods for Uncertainty Quantification

Abstract

This dissertation focuses on acceleration techniques for Uncertainty Quantification (UQ). The manuscript is divided into five chapters. Chapter 1 provides an introduction and a brief summary of Chapters 2, 3, and 4. Chapter 2 introduces a model reduction strategy that is used in the context of elasticity imaging to infer the presence of an inclusion embedded in a soft matrix, mimicking tumors in soft tissues. The method relies on Polynomial Chaos (PC) expansions to build a dictionary of surrogates models, where each surrogate is constructed using a different geometrical configuration of the potential inclusion. A model selection approach is used to discriminate against the different models and eventually select the most appropriate to estimate the likelihood that an inclusion is present in the domain. In Chapter 3, we use a Domain Decomposition (DD) approach to compute the Karhunen-Loeve (KL) modes of a random process through the use of local KL expansions at the subdomain level. Furthermore, we analyze the relationship between the local random variables associated to the local KL expansions and the global random variables associated to the global KL expansions. In Chapter 4, we take advantage of these local random variables and use DD techniques to reduce the computational cost of solving a Stochastic Elliptic Equation (SEE) via a Monte Carlo sampling method. The approach takes advantage of a lower stochastic dimension at the subdomain level to construct a PC expansion of a reduced linear system that is later used to compute samples of the solution. Thus, the approach consists of two main stages: 1) a preprocessing stage in which PC expansions of a condensed problem are computed and 2) a Monte Carlo sampling stage where samples of the solution are computed in order to solve the SEE. Finally, in Chapter 5 some brief concluding remarks are provided.