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Abstract: In many scientific areas, deterministic models (e.g., differential equations) use numerical parameters. In real-world settings, however, such parameters might be uncertain or noisy. A more comprehensive model should therefore provide a statistical description of the quantity of interest. Underlying this numerical analysis problem is a fundamental question - if two "similar" functions push-forward the same measure, would the new resulting measures be close, and if so, in what sense? We will first show how the probability density function (PDF) of the quantity of interest can be approximated. We will then discuss an alternative viewpoint: through Optimal Transport theory, a Wasserstein-distance formulation of our problem yields a more robust theoretical framework.
Finally, we will use similar measure-theoretic tools to understand two seemingly unrelated problems - the study of high-dimensional zero-sets, and the analysis of statistical sampling algorithms.