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Speaker: Jehanzeb H. Chaudhry, University of New Mexico

Title: Adjoint Based Analysis for Non-Standard Quantities of Interest

Abstract: Accurate estimation of the error in a quantity-of-interest (QoI) is a fundamental requirement for the robust application of numerical simulations in science and engineering. Adjoint based error estimates for QoIs modeled as bounded linear functionals (or linearized as such) have long been developed. However, there are often QoIs, which we refer to non-standard QoIs, which do not t into this framework. In this talk we examine two such QoIs: (i) time at which a functional of the solution achieves a threshold value (ii) statistical quantities of interest. Both these quantities of interest are often primary concerns in practical applications. An important example of (i) is the time at which a tsunami wave crosses a certain threshold height, while (ii) arises in quantifying uncertainty in dynamical systems due to uncertain initial conditions. In this talk we adapt adjoint-based a posteriori analysis to derive an accurate error estimate for such non-standard QoIs.

In addition to the error estimation, we also perform uncertainty quantification for the such QoIs. In particular, for (i) we consider the scenario of random differential equations, which leads to a probability distribution on the QoI. We derive an error estimate for the computed probability distribution function which decomposes the error into contributions due to sampling and discretization. For (ii) we model the probability density as a Normalizing Flow and use machine learning techniques to compute local sensitivity along with the error of the computed density.