We consider the computational efficiency of Monte Carlo (MC) and Multilevel Monte Carlo (MLMC) methods applied to partial differential equations with random coefficients. These arise, for example, in groundwater flow modelling, where a commonly used model for the unknown parameter is a random field. We make use of the circulant embedding procedure for sampling from the aforementioned coefficient. To improve the computational complexity of the MLMC estimator in the case of highly oscillatory random fields, we devise and implement a smoothing technique integrated into the circulant embedding method. This allows to choose the coarsest mesh on the first level of MLMC independently of the correlation length of the covariance function of the random field, leading to considerable savings in computational cost. We illustrate this with numerical experiments, where we see a saving of factor 5-10 in computational cost for accuracies of practical interest.

翻译：我们研究了蒙特卡罗(MC)和多层蒙特卡罗(MLMC)方法在含随机系数偏微分方程中的计算效率问题。此类方程出现在例如地下水流建模中，其中未知参数的常用模型为随机场。我们采用循环嵌入过程对前述系数进行采样。为改善高度振荡随机场情况下MLMC估计量的计算复杂度，我们设计并实现了一种集成于循环嵌入方法中的光滑技术。该技术使得MLMC第一层最粗糙网格的选择可独立于随机场协方差函数的相关长度，从而显著降低计算成本。数值实验表明，在实际感兴趣的精度要求下，计算成本可降低5至10倍。