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：我们考察了蒙特卡罗（MC）和多层级蒙特卡罗（MLMC）方法在求解随机系数偏微分方程时的计算效率。这类方程出现在例如地下水流动建模中，其中未知参数常用随机场模型进行描述。我们采用循环嵌入过程对上述系数进行采样。针对高频振荡随机场情形下MLMC估计量的计算复杂度问题，我们设计并实现了一种集成于循环嵌入方法中的平滑技术。该技术使得MLMC首层最粗糙网格的选取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。通过数值实验证明，在具有实际意义的精度需求下，计算成本可降低5-10倍。