Methods for Analysis and Optimization of Stochastic Systems with Model Uncertainty and Related Monte Carlo Schemes

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Mathematical models are used in every area of science, engineering, and policy to design systems or to understand physical or social phenomena. In every instance, the issue of model error is important. In general, it is not possible for practical reasons (such as limited amounts of data, or the need to maintain computational feasibility) to work with a perfectly accurate model. Hence it is important to identify those aspects of the model that are uncertain, quantify their impact on predictions, and perhaps even account for this uncertainty while using the model, for example as an engineering tool. The models of interest in this project are probabilistic. In this setting we acknowledge that the system is random, and the model error is due to an imperfect understanding of the parameters that describe the probability distribution. To assess how mathematical predictions based on the model change as the model itself changes, one needs metrics to compare the outcome based on different distributions (e.g., the distribution that is used for "design," and an ideal but not available "true" distribution). The topic of this research is the development of the theory and application of such metrics. In contrast to prior work, here we focus on situations where the quantities of interest are tied to rare events, such as a catastrophic system failure. Graduate students participate in the research of the project.The main theme of this project is the use of divergences and metrics on probability measures to study model uncertainty, and optimization and control in the presence of model uncertainty. The probability measures are typically on high-dimensional or complicated spaces, and typically on a path space to model stochastic dynamics. An important aspect of the work is to establish useful qualitative properties, such as scaling limits and chain rule-type formulas. In contrast to prior work, the focus here is on situations where (a) one wishes to consider differing models that are not absolutely continuous, and (b) performance measures and quantities of interest are largely determined by rare events and tail properties. The main mathematical tools used are convex duality or variational formulas that relate the divergences to exponential integrals. To implement the theory, one needs to evaluate such exponential integrals, which for example may take the form of a moment-generating function with respect to the stationary distribution of some Markov process. The project also considers the design and analysis of Monte Carlo methods for this class of problems. Graduate students participate in the research of the project.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.