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We propose a control variate multilevel Monte Carlo method for the kineticBGK model of the Boltzmann equation subject to random inputs. The methodcombines a multilevel Monte Carlo technique with the computation of the optimalcontrol variate multipliers derived from local or global variance minimizationproblems. Consistency and convergence analysis for the method equipped with asecond-order positivity-preserving and asymptotic-preserving scheme in spaceand time is also performed. Various numerical examples confirm that theoptimized multilevel Monte Carlo method outperforms the classical multilevelMonte Carlo method especially for problems with discontinuities.

Uncertainty quantification for the BGK model of the Boltzmann equation using multilevel variance reduced Monte Carlo methods