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)与多级蒙特卡罗(MLMC)方法应用于随机系数偏微分方程时的计算效率。此类问题常见于地下水流建模等领域，其中未知参数通常采用随机场模型进行描述。我们利用循环嵌入方法对前述随机系数进行采样。为改善MLMC估计器在处理高频振荡随机场时的计算复杂度，我们设计并实现了一种与循环嵌入方法相结合的平滑技术。该技术使得MLMC第一层的最粗网格选择不再受限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，在达到实际应用所需精度时，该方法可节省5-10倍的计算开销。