UC San Diego

This dissertation deals with mathematical modeling of complex distributed systems whose parameters are heterogeneous and heavily under-specified by data. Such problems are ubiquitous in every field of science and engineering, where one or more deterministic models exist to describe a given phenomenon but only a limited number of measurements of a model's parameters and its state variables are available. The main focus of this dissertation is on parameter identification (PI) and uncertainty quantification (UQ). The first part of this dissertation deals with development and numerical implementation of an algorithm to compute accurately and efficiently Green's functions, which are often used in both PI and UQ analyses of linear systems with piece-wise continuous parameters. The second part of this dissertation explores the propagation of parametric uncertainty through a modeling process, in which quantities of interest are described by nonlinear elliptic and parabolic partial differential equations. We demonstrate that the variance of uncertain parameters (a measure of their uncertainty) strongly affects the regularity of a system's stochastic response, restricting the use of modern probabilistic UQ methods (e.g, polynomial chaos expansions and stochastic collocation methods) to low distributed parameters with low noise-to- signal ratios. High ratios adversely affect the stability and scalability of such methods. The third part of this dissertation deals with this issue by developing a multi- level Monte Carlo algorithm that outperforms direct Monte Carlo and allows for systematic treatment of different sources of bias in the computed estimators. In the final part of this dissertation we explore two PI strategies based on a Bayesian framework. The first strategy is to sample a posterior distribution using a generalized hybrid Monte Carlo (gHMC) method. We develop acceleration schemes for improving the efficiency of gHMC, and use them to estimate parameters in reactive transport systems with sparse concentration measurements. The second strategy is to compute the maximum a posteriori estimator of the configuration of spatially distributed, piece-wise continuous parameters by using a linearized functional minimization algorithm. Total variation regularization (TV) is employed as a prior on the parameter distribution, which allows one to capture large-scale features of system behavior from sparse measurements of both system parameters and transient system states