SIAM/ASA Journal on Uncertainty Quantification, Volume 9, Issue 2, Page 650-680, January 2021.
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.

中文翻译：

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使用多级方差归约蒙特卡洛方法对Boltzmann方程的BGK模型进行不确定性量化

SIAM / ASA不确定性量化杂志，第9卷，第2期，第650-680页，2021年1月。
我们为随机输入的Boltzmann方程的动力学Bhatnagar-Gross-Krook模型提出了控制变量多级蒙特卡洛方法。该方法将多级蒙特卡洛技术与从局部或全局方差最小化问题得出的最优控制变量乘数的计算相结合。还对时空上具有二阶正性保存和渐近保存方案的方法进行了一致性和收敛性分析。各种数值例子证实，优化的多级蒙特卡洛方法优于经典的多级蒙特卡洛方法，特别是对于具有不连续性的问题。