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 the numerical simulators have inconsistent dimensions across the multilevel hierarchy. This requires the introduction of grid transfer operators borrowed from multigrid methods. Starting from a simple 1D illustration, we demonstrate numerically that the resulting MLMC estimator deteriorates the estimation of high-frequency components of the discretized expectation field compared to a Monte Carlo (MC) estimator. By adapting mathematical tools initially developed for 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. This analysis provides a spectral decomposition of the variance into contributions associated with each scale component of the discretized field. We then propose improved MLMC estimators using a filtering mechanism similar to the smoothing process of multigrid methods. 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. These improvements are quantified for the specific class of simulators considered in our spectral analysis. The resulting filtered MLMC (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 proposed F-MLMC estimator compared to both a crude MC and an unfiltered MLMC estimator.

翻译：本文研究了使用多层级蒙特卡洛（MLMC）方法估计离散随机场期望的问题。具体而言，我们考虑一种设定，其中数值模拟器的输入和输出向量在多层级层级结构中具有不一致的维度。这需要引入从多重网格方法中借鉴的网格转移算子。通过一个简单的一维示例，我们数值验证了与蒙特卡洛（MC）估计器相比，所得到的MLMC估计器会恶化离散期望场高频分量的估计。通过改进最初为多重网格方法开发的数学工具，我们在线性、对称和循环模拟器的特定情况下，对离散随机场期望的MLMC估计器进行了理论谱分析。该分析将方差分解为与离散场各尺度分量相关的贡献。随后，我们提出了改进的MLMC估计器，该估计器采用类似于多重网格方法平滑过程的滤波机制。滤波算子改善了方差中小尺度和大尺度分量的估计，从而降低了估计器的总方差。这些改进在我们谱分析中考虑的特定模拟器类别中得到了量化。所得到的滤波多层级蒙特卡洛（F-MLMC）估计器被应用于估计基于扩散的协方差算子的离散方差场问题，即离散随机场期望的估计。数值实验支持了理论分析的结论，即使对于非线性模拟器也是如此，并展示了所提出的F-MLMC估计器相比原始MC和未滤波的MLMC估计器所带来的改进。