A Study of Multiscale Kinetic Models with Uncertainties

Abstract

There have been many challenges in studying multiscale kinetic models with uncertainties. Quantifying uncertainties in the models such as arisen from collision kernels, initial or boundary data, forcing terms, and developing efficient computational methods have become an important task in industrial applications. In this article, we will report some of the recent progress on multiscale kinetic models with random inputs. We will discuss from the aspects of both mathematical theory, by using the hypocoercivity of kinetic operators, and numerical computation. Two categories of intrusive and non-intrusive methods will be addressed, in particular the stochastic Galerkin method for problems with relatively low-dimensional random inputs and stochastic collocation method in a multi-fidelity setting for more challenging, higher-dimensional problems. Numerical experiments for several kinetic models such as the Boltzmann, bipolar Boltzmann-Poisson, linear transport and epidemic kinetic equations will be shown as examples.