Off-campus UMass Amherst users: To download campus access dissertations, please use the following link to log into our proxy server with your UMass Amherst user name and password.

Non-UMass Amherst users: Please talk to your librarian about requesting this dissertation through interlibrary loan.

Dissertations that have an embargo placed on them will not be available to anyone until the embargo expires.

Author ORCID Identifier

0000-0001-9330-4983

Document Type

Open Access Dissertation

Degree Name

Doctor of Philosophy (PhD)

Degree Program

Mathematics

Year Degree Awarded

2019

Month Degree Awarded

September

First Advisor

Markos A. Katsoulakis

Second Advisor

Luc Rey-Bellet

Subject Categories

Applied Mathematics | Materials Chemistry | Statistics and Probability

Abstract

In this thesis, we focus on Uncertainty Quantification and Sensitivity Analysis, which can provide performance guarantees for predictive models built with both aleatoric and epistemic uncertainties, as well as data, and identify which components in a model have the most influence on predictions of our quantities of interest.

In the first part (Chapter 2), we propose non-parametric methods for both local and global sensitivity analysis of chemical reaction models with correlated parameter dependencies. The developed mathematical and statistical tools are applied to a benchmark Langmuir competitive adsorption model on a close packed platinum surface, whose parameters, estimated from quantum-scale computations, are correlated and are limited in size (small data). The proposed mathematical methodology employs gradient-based methods to compute sensitivity indices. We observe that ranking influential parameters depend critically on whether or not correlations between parameters are taken into account. The impact of uncertainty in the correlation and the necessity of the proposed non-parametric perspective are demonstrated.

In the second part (Chapter 3-4), we develop new information-based uncertainty quantification and sensitivity analysis methods for Probabilistic Graphical Models. Probabilistic graphical models are an important class of methods for probabilistic modeling and inference, probabilistic machine learning, and probabilistic artificial intelligence. Its hierarchical structure allows us to bring together in a systematic way statistical and multi-scale physical modeling, different types of data, incorporating expert knowledge, correlations, and causal relationships. However, due to multi-scale modeling, learning from sparse data, and mechanisms without full knowledge, many predictive models will necessarily have diverse sources of uncertainty at different scales. The new model-form uncertainty quantification indices we developed can handle both parametric and non-parametric probabilistic graphical models, as well as small and large model/parameter perturbations in a single, unified mathematical framework and provide an envelope of model predictions for our quantities of interest. Moreover, we propose a model-form Sensitivity Index, which allows us to rank the impact of each component of the probabilistic graphical model, and provide a systematic methodology to close the experiment - model - simulation - prediction loop and improve the computational model iteratively based on our new uncertainty quantification and sensitivity analysis methods. To illustrate our ideas, we explore a physicochemical application on the Oxygen Reduction Reaction (ORR) in Chapter 4, whose optimization was identified as a key to the performance of fuel cells.

In the last part (Chapter 5), we complete our discussion for the uncertainty quantification and sensitivity analysis methods on probabilistic graphical models by introducing a new sensitivity analysis method for the case where we know the real model sits in a certain parametric family. Note that the uncertainty indices above may be too pessimistic (as they are inherently non-parametric) when studying uncertainty/sensitivity questions for models confined within a given parametric family. Therefore, we develop a method using likelihood ratio and fisher information matrix, which can capture correlations and causal dependencies in the graphical models, and we show it can provide us more accurate results for the parametric probabilistic graphical models.

Share

COinS