Uncertainty Quantification for the BGK Model of the Boltzmann Equation Using Multilevel Variance Reduced Monte Carlo Methods

We propose a control variate multilevel Monte Carlo method for the kinetic Bhatnagar-Gross-Krook model of the Boltzmann equation subject to random inputs. The method combines a multilevel Monte Carlo technique with the computation of the optimal control variate multipliers derived from local or global variance minimization problems. Consistency and convergence analysis for the method equipped with a second-order positivity-preserving and asymptotic-preserving scheme in space and time is also performed. Various numerical examples confirm that the optimized multilevel Monte Carlo method outperforms the classical multilevel Monte Carlo method especially for problems with discontinuities.