The main topics of the mini-symposium include model uncertainty, robust uncertainty quantification & optimization, and their implications in predictive modeling guarantees and rare-event analysis. We aim at bringing together closely related but possibly disparate communities in applied mathematics, applied probability, information theory, operations research, optimization and economics, to foster interdisciplinary discussions and collaborations. Speakers will demonstrate recent mathematical and conceptual developments of related UQ methods, and also their applications ranging from engineering design of materials to econometrics and risk analysis.
14:00
Predictive Probabilistic Graphical Models for Energy Materials
Eric Hall | RWTH Aachen University | Germany
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Eric Hall | RWTH Aachen University | Germany
We discuss a new class of Probabilistic Graphical Models (PGMs) in energy research including models for the design of nanoporous metamaterials for supercapacitors and energy storage devices. Since their introduction, PGMs have proved to be a fundamental mathematical concept for modeling uncertainty and causal relationships in Artificial Intelligence. Here the proposed PGMs involve multi-scale physicochemical mechanisms and must incorporate data, e.g. from electronic structure calculations or observations/experiments. More specifically, the hierarchical structure of PGMs allows us to bring together both statistical and multi-scale modeling in a systematic way by providing a framework for informing physical models with domain knowledge including available data, which here are typically sparse or incomplete, along with expert opinion and correlations and causal relationships between model components. The process of building such models necessarily involves numerous sources of uncertainty, arising from different components of the PGM, corresponding to either data or modeling errors. In this direction, we also introduce information-based Uncertainty Quantification (UQ) methods capable of assessing and improving the predictive ability of our proposed computational models.
14:30
Large deviation properties of the empirical measure of a dynamical system subject to small random perturbations
Guo-Jhen Wu | KTH Stockholm | Sweden
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Guo-Jhen Wu | KTH Stockholm | Sweden
Among the many interesting results proved by Freidlin and Wentzell in the 70's and 80's concerning small random perturbations of dynamical systems, one of particular note is the formulation and proof of a large deviation principle for the invariant measure of a small noise diffusion. Additionally, it was shown that the rate function of the invariant measure can be formulated in terms of quasipotentials which are quantities used to measure the difficulty of a transition from one point to another.
In this talk, we present some large deviation type estimates for a quantity closely related to the invariant measure, which is the empirical measure over a time interval whose length grows as the noise decreases to zero. Our interest in proving large deviations estimates with vanishing noise and accompanying with growing time interval is due to the fact that one might hope it is easier to extract information in this double limit, analogous to the simplified approximation to the invariant measure just mentioned. As will be discussed, the large deviations estimates for the first and second moments of an integration with respect to the empirical measure can again be expressed by quasipotentials. This information is useful for, among other things, analysis and design of Monte Carlo methods in the small noise limit, when the Monte Carlo method will have the greatest difficulty.
15:00
Likelihood ratio methods for estimating sensitivity and linear response in stochastic dynamics
Petr Plechac | University of Delaware | United States
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Petr Plechac | University of Delaware | United States
We discuss schemes for computing the sensitivity or linear response of steady-state averages of stochastic dynamics. The schemes are based on Girsanov's change-of-measure theory and apply reweighting of trajectories by factors derived from a linearization of the Girsanov weights. We investigate both the discretization error and the finite time approximation error. The designed numerical schemes are shown to be of bounded variance with respect to the integration time, which is a desirable feature for long time simulations. We also show how the discretization error can be improved to second order accuracy in the timestep by modifying the weight process in an appropriate way. (joint work with T. Wang and G. Stoltz).
15:30
Uncertainty quantification for non-absolutely continuous perturbations of probability measures
Yixiang Mao | Harvard University | United States
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Yixiang Mao | Harvard University | United States
Uncertainty quantification is an important subject in evaluating model robustness under perturbation. In a number of prior works it has been shown that the variational representation of relative entropy can be used to produce useful bounds for expected values under model form uncertainty. However, when the perturbed distribution is not absolutely continuous with respect to the original one, this bound turns out to be useless. In this talk, I will define a new class of divergences that generalize relative entropy, show some of its important properties, and use it to derive useful bounds for uncertainty quantification under non-absolutely continuous perturbation. Moreover, comparison with using optimal transport cost will be presented. This is based on joint work with Paul Dupuis.