SIAM Journal on Scientific Computing, Volume 42, Issue 4, Page B1067-B1091, January 2020.
We propose a novel $hp$-multilevel Monte Carlo method for the quantification of uncertainties in the compressible Navier--Stokes equations, using the discontinuous Galerkin method as deterministic solver. The multilevel approach exploits hierarchies of uniformly refined meshes while simultaneously increasing the polynomial degree of the ansatz space. It allows for a very large range of resolutions in the physical space and thus an efficient decrease of the statistical error. We prove that the overall complexity of the $hp$-multilevel Monte Carlo method to compute the mean field with prescribed accuracy is, in the best case, of quadratic order with respect to the accuracy. We also propose a novel and simple approach to estimate a lower confidence bound for the optimal number of samples per level, which helps to prevent overestimating these quantities. The method is in particular designed for application on queue-based computing systems, where it is desirable to compute a large number of samples during one iteration without overestimating the optimal number of samples. Our theoretical results are verified by numerical experiments for the two-dimensional compressible Navier--Stokes equations. In particular we consider a cavity flow problem from computational acoustics, demonstrating that the method is suitable to handle complex engineering problems.

中文翻译：

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可压缩的Navier-Stokes方程不确定性量化的$ hp $-多层蒙特卡洛方法

SIAM科学计算杂志，第42卷，第4期，第B1067-B1091页，2020年1月。
我们提出了一种新颖的$ hp $多级蒙特卡洛方法，用于将可压缩的Navier-Stokes方程中的不确定性量化，使用不连续的Galerkin方法作为确定性求解器。多层方法利用均匀精炼的网格的层次结构，同时增加ansatz空间的多项式度。它允许在物理空间中实现很大范围的分辨率，从而有效地减少了统计误差。我们证明，在最佳情况下，以规定的精度计算$ hp $-多层蒙特卡罗方法的总体复杂度在最佳情况下为二次方。我们还提出了一种新颖而简单的方法来估算每个级别的最佳样本数的较低置信区间，这有助于防止高估这些数量。该方法特别地设计用于基于队列的计算系统，其中期望在一次迭代期间计算大量样本而不会高估样本的最佳数量。二维可压缩Navier-Stokes方程的数值实验验证了我们的理论结果。特别是，我们从计算声学角度考虑了腔体流动问题，表明该方法适合处理复杂的工程问题。二维可压缩Navier-Stokes方程的数值实验验证了我们的理论结果。特别是，我们从计算声学角度考虑了腔体流动问题，表明该方法适合处理复杂的工程问题。二维可压缩Navier-Stokes方程的数值实验验证了我们的理论结果。特别是，我们从计算声学角度考虑了腔体流动问题，表明该方法适合处理复杂的工程问题。