# UNCERTAINTY QUANTIFICATION IN SCIENTIFIC MODELS

January 2014

Dissertation

## Degree Name

Doctor of Philosophy (PhD)

Mathematics

Dongbin Xiu

Suchuan Dong

Greg Buzzard

Jianlin Xia

## Abstract

Uncertainties widely exist in physical, finance, and many other areas. Some uncertainties are determined by the nature of the research subject, such as random variable and stochastic process. However, in many problems uncertainty is a result of lack of knowledge and may not be modeled as random variables/processes because of the lack of probability information. This is often referred to as epistemic uncertainty, and the traditional probabilistic approaches cannot be readily employed. First two parts of this work study epistemic uncertainties in the forward problems. A method to compute upper and lower bounds for the quantity of interest of problems whose uncertain inputs are of epistemic type is presented. Relative entropy is an important measure to study the distance between multiple probabilities. Its properties have motivated many important existing inequalities for quantifying epistemic uncertainties. Based on these works, we extend the inequalities to a large family of functions, the integrable functions, which play an important role in engineering and research. To be more specific, we provide upper and lower bounds for the statistics such as statistical moments of the quantities of our interest under the existence of epistemic uncertainty. We present the theoretical derivation of the bounds, along with numerical examples to illustrate their computations. Based on derived analytical lower and upper bounds, a procedure to compute numerical bounds of when the underlying system is subject to epistemic uncertainty is discussed. In particular, we consider the case where the uncertain inputs to the system take the form of parameters, physical and/or hyperparameters, and with unknown probability distributions. Our goal is to compute the lower and upper bounds of the statistical moments of quantity-of-interest of the system response. We discuss exclusively the numerical algorithms for computing such bounds. More importantly, we established the properties of such numerical bounds and analyzed their accuracy compared to the analytical bounds.

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