Abstract:
In many scientific areas, a deterministic model (e.g., a differential equation) is equipped with parameters. In practice, these parameters might be uncertain or noisy, and so an honest model should account for these uncertainties and provide a statistical description of the quantity of interest. In this talk, I will first show how the probability density function (PDF) can be approximated, and show applications to nonlinear optics and fluid dynamics. However, we will see that the analysis of PDF approximation is challenging, and can only be applied to a limited number of use cases. Perhaps surprisingly, an alternative framing of this task as a measure approximation task, and measuring the error with the Wasserstein metric, allows us to use tools from optimal transport theory to solve a very abstract and fundamental question - If two "similar" functions push-forward the same measure, are the new resulting measures close, and if so, in what sense? |