Uncertainty quantification in scientific models

Xin Qi, Purdue University

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. Besides the uncertainties in forward problems, quantification of uncertainties in inverse problems is also discussed: Bayesian posterior estimation and model discrepancy. After a posterior is well studied for a selected prior, we proposed a method of estimating the posterior when a new prior is selected. . The method is based on the initial choice of prior distribution and a surrogate of the forward model under the initial prior. If a new prior is selected, instead of another complete circle for computing posterior distribution, we first study the relation between the two priors and then approximate the new forward model using the previous forward model surrogate. We also present an error analysis for the difference between our estimate and the true posterior distribution. This efficient numerical strategy is based on stochastic collocation methods and generalized polynomial chaos (gPC) to construct a polynomial approximation of the forward solution over the support of the initial prior distribution. The gPC strategy is also applied in model discrepancy to bring in more structure to the forward model. The problem is solved via an optimization procedure. The gPC based algorithms not only reduce computational cost but also bring more useful structure to the forward model.

Degree

Ph.D.

Advisors

Xiu, Purdue University.

Subject Area

Mathematics

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