Non-Stochastic Methods for Uncertainty Quantification

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Department of Mathematics

Synopsis:

The goal is to make students familiar with some non-stochastic methods for uncertainty quantification. Uncertainty is considered in parameters entering mathematical models. Consequently, the model output represented by a quantity of interest is also uncertain and this uncertainty is to be assessed.

Topics: Aleatoric and epistemic uncertainty. Differential equations with uncertain data. Various approaches to uncertainty quantification. The worst- and best-case scenario method.

Elements of fuzzy set theory (membership function, alpha-cut, Zadeh’s extension principle). Fuzzification, various definitions of membership functions, a connection to information gap theory by Y. Ben-Haim. An introduction to the Dempster-Shafer theory (DST), belief and plauzibility, Dempster’s rule of combination. Probabilistic interpretation of DST. Application to engineering problems with uncertain data and a non-trivial state problem. Tools for solving such problems – minimization algorithms, sensitivity analysis, finite element method.

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