Local and Optimal Transport Perspectives on Uncertainty Propagation

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 computational 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, using spectral and local methods. We will then discuss an alternative viewpoint: through optimal transport theory, a Wasserstein-distance formulation of our problem yields a much simpler and widely applicable theory.