Multilevel estimators aim at reducing the variance of Monte Carlo statistical estimators, by combining samples generated with simulators of different costs and accuracies. In particular, the recent work of Schaden and Ullmann (2020) on the multilevel best linear unbiased estimator (MLBLUE) introduces a framework unifying several multilevel and multifidelity techniques. The MLBLUE is reintroduced here using a variance minimization approach rather than the regression approach of Schaden and Ullmann. We then discuss possible extensions of the scalar MLBLUE to a multidimensional setting, i.e. from the expectation of scalar random variables to the expectation of random vectors. Several estimators of increasing complexity are proposed: a) multilevel estimators with scalar weights, b) with element-wise weights, c) with spectral weights and d) with general matrix weights. The computational cost of each method is discussed. We finally extend the MLBLUE to the estimation of second-order moments in the multidimensional case, i.e. to the estimation of covariance matrices. The multilevel estimators proposed are d) a multilevel estimator with scalar weights and e) with element-wise weights. In large-dimension applications such as data assimilation for geosciences, the latter estimator is computationnally unaffordable. As a remedy, we also propose f) a multilevel covariance matrix estimator with optimal multilevel localization, inspired by the optimal localization theory of M\'en\'etrier and Aulign\'e (2015). Some practical details on weighted MLMC estimators of covariance matrices are given in appendix.

翻译：多层级估计器旨在通过组合不同成本和精度的模拟器生成的样本，降低蒙特卡洛统计估计器的方差。特别地，Schaden 和 Ullmann (2020) 近期关于多层级最佳线性无偏估计量（MLBLUE）的工作引入了一个统一多种多层级与多保真度技术的框架。本文采用方差最小化方法而非 Schaden 和 Ullmann 的回归方法重新介绍 MLBLUE，并进一步探讨标量 MLBLUE 向多维场景（即从标量随机变量的期望扩展至随机向量的期望）的可能扩展。我们提出了复杂度递增的多种估计器：a) 带标量权重的多层级估计器，b) 带元素级权重的估计器，c) 带谱权重的估计器，以及 d) 带一般矩阵权重的估计器，并讨论了每种方法的计算成本。最后，我们将 MLBLUE 扩展至多维情形下的二阶矩估计，即协方差矩阵的估计。提出的多层级估计器包括 d) 带标量权重的多层级估计器与 e) 带元素级权重的估计器。在诸如地球科学数据同化等大维度应用中，后一种估计器因计算成本过高而难以实现。为此，我们受 Ménétrier 和 Auligné (2015) 最优局地化理论的启发，提出 f) 具有最优多层级局地化的多层级协方差矩阵估计器。附录中给出了加权多层蒙特卡洛（MLMC）协方差矩阵估计器的若干实用细节。