Wednesday, July 14 2021
10:00am - 12:00pm
PhD Thesis Presentation
Stochastic Uncertainty Analysis for Data-Consistent Approaches to Inverse Problems

As predictive science increasingly relies upon the outputs of computational models, it is imperative that uncertainties impacting these models be quantified, and reduced, whenever possible. In the inverse uncertainty quantification (UQ) problem, the goal is to quantify and reduce the uncertainties of input parameters to a specified model by using observations and data from output quantities of interest. There are several factors complicating the formulation and solution of the inverse UQ problem. First, the uncertainties impacting either the input parameters or the output quantities of interest may be of different types. Generally speaking, uncertainties are categorized as either being aleatoric (i.e., irreducible) or epistemic (i.e., reducible). The formulation and solution of the inverse UQ problem significantly depends upon the assumptions of either aleatoric or epistemic uncertainties (or some combination of the two). Second, the mapping from model inputs to model outputs need not be invertible, which complicates the solution to the inverse UQ problem since sets of inputs may exist that map to a single output. Moreover, the mapping itself need not be deterministic if the modeled process is stochastic or if other sources of uncertainty impact model outputs but are not present in the initial model specification. This type of stochastic map also complicates the inverse UQ problems and its solutions. Finally, finite amounts of observable data on model outputs may additionally contribute to the uncertainty of the solution to the inverse UQ problems.

In this dissertation, we focus on addressing these complicating factors in the context of the data-consistent approach to inverse UQ problems. First, we provide a comparative analysis of the data-consistent approach to popular Bayesian approaches utilized in the UQ community. In this analysis, we explain how each approach naturally arises from defining a particular inverse UQ problem based on assumptions of either aleatoric or epistemic uncertainties. This serves to distinguish the types of problems solved by these approaches as well as to clarify the appropriate interpretations of their solutions. Furthermore, we explore how data-consistent and Bayesian methods may be modified to solve different inverse UQ problems based on alternative assumptions about their sources of uncertainty. Through this comparative analysis, we address both the issue involving various types of uncertainty and the complications that arise when using non-invertible maps. We then address situations where the map between the input and output spaces is stochastic. Specifically, we develop an extension to the theoretical and algorithmic framework of the data-consistent approach that solves the inverse UQ problem with stochastic maps, broadening the applicability of the data-consistent approach. Finally, we introduce theoretical and practical tools for analyzing the effects of finite observable data on the data-consistent approach to inverse problems. We thoroughly investigate two popular non-parametric methods used to estimate densities from finite data and explain how such estimates impact the data-consistent solution. Taken together, the results of this dissertation demonstrate the robustness and flexibility of the data-consistent approach in confronting the various challenges involved in solving inverse UQ problems.
Speaker:Tian Yu Yen
Location:See Email for Zoom link

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