AbstractsComputer Science

This thesis presents key contributions towards devising highly efficient stochastic reconstruction algorithms for solving large scale inverse problems, where a large data set is available and the underlying physical systems is complex, e.g., modeled by partial differential equations (PDEs). We begin by developing stochastic and deterministic dimensionality reduction methods to transform the original large dimensional data set into the one with much smaller dimensions for which the computations are more manageable. We then incorporate such methods in our efficient stochastic reconstruction algorithms. In the presence of corrupted or missing data, many of such dimensionality reduction methods cannot be efficiently used. To alleviate this issue, in the context of PDE inverse problems, we develop and mathematically justify new techniques for replacing (or filling) the corrupted (or missing) parts of the data set. Our data replacement/completion methods are motivated by theory in Sobolev spaces, regarding the properties of weak solutions along the domain boundary. All of the stochastic dimensionality reduction techniques can be reformulated as Monte-Carlo (MC) methods for estimating the trace of a symmetric positive semi-definite (SPSD) matrix. In the next part of the present thesis, we present some probabilistic analysis of such randomized trace estimators and prove various computable and informative conditions for the sample size required for such Monte-Carlo methods in order to achieve a prescribed probabilistic relative accuracy. Although computationally efficient, a major drawback of any (randomized) approximation algorithm is the introduction of “uncertainty” in the overall procedure, which could cast doubt on the credibility of the obtained results. The last part of this thesis consists of uncertainty quantification of stochastic steps of our approximation algorithms presented earlier. As a result, we present highly efficient variants of our original algorithms where the degree of uncertainty can easily be quantified and adjusted, if needed. The uncertainty quantification presented in the last part of the thesis is an application of our novel results regarding the maximal and minimal tail probabilities of non-negative linear combinations of gamma random variables which can be considered independently of the rest of this thesis.