We investigate the use of multilevel Monte Carlo (MLMC) methods for estimating the expectation of discretized random fields. Specifically, we consider a setting in which the input and output vectors of numerical simulators have inconsistent dimensions across the multilevel hierarchy. This requires the introduction of grid transfer operators borrowed from multigrid methods. By adapting mathematical tools from multigrid methods, we perform a theoretical spectral analysis of the MLMC estimator of the expectation of discretized random fields, in the specific case of linear, symmetric and circulant simulators. We then propose filtered MLMC (F-MLMC) estimators based on a filtering mechanism similar to the smoothing process of multigrid methods, and we show that the filtering operators improve the estimation of both the small- and large-scale components of the variance, resulting in a reduction of the total variance of the estimator. Next, the conclusions of the spectral analysis are experimentally verified with a one-dimensional illustration. Finally, the proposed F-MLMC estimator is applied to the problem of estimating the discretized variance field of a diffusion-based covariance operator, which amounts to estimating the expectation of a discretized random field. The numerical experiments support the conclusions of the theoretical analysis even with non-linear simulators, and demonstrate the improvements brought by the F-MLMC estimator compared to both a crude MC and an unfiltered MLMC estimator.

翻译：本文研究了多级蒙特卡洛（MLMC）方法在估计离散化随机场期望中的应用。具体而言，我们考虑数值模拟器的输入和输出向量在多级层级间维度不一致的情形。这需要引入从多重网格方法借鉴的网格传递算子。通过适配多重网格方法中的数学工具，我们对离散化随机场期望的MLMC估计量进行了理论谱分析，该分析针对线性、对称且循环的模拟器这一特定情况。随后，我们提出了一种基于滤波机制的滤波多级蒙特卡洛（F-MLLC）估计量，该机制类似于多重网格方法中的平滑过程。我们证明了滤波算子能够改善方差的小尺度与大尺度分量的估计，从而降低估计量的总方差。接下来，通过一维算例对谱分析的结论进行了实验验证。最后，将所提出的F-MLMC估计量应用于估计基于扩散的协方差算子的离散化方差场问题，该问题等价于估计一个离散化随机场的期望。数值实验支持了理论分析的结论，即使在非线性模拟器情况下亦然，并证明了与原始蒙特卡洛估计量以及未滤波的MLMC估计量相比，F-MLMC估计量所带来的改进。