Uncertainty Quantification for Kinetic Equations

Kinetic equations contain uncertainties in their collision kernels or scattering coefficients, initial or boundary data, forcing terms, geometry, etc. Quantifying the uncertainties in kinetic models have important engineering and industrial applications. In this article we survey recent efforts in the study of kinetic equations with random inputs, including their mathematical properties such as regularity and long-time behavior in the random space, construction of efficient stochastic Galerkin methods, and handling of multiple scales by stochastic asymptotic-preserving schemes. The examples used to illustrate the main ideas include the random linear and nonlinear Boltzmann equations, linear transport equation and the Vlasov-Poisson-Fokker-Planck equations.