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Multilevel estimators aim at reducing the variance of Monte Carlo statisticalestimators, by combining samples generated with simulators of different costsand accuracies. In particular, the recent work of Schaden and Ullmann (2020) onthe multilevel best linear unbiased estimator (MLBLUE) introduces a frameworkunifying several multilevel and multifidelity techniques. The MLBLUE isreintroduced here using a variance minimization approach rather than theregression approach of Schaden and Ullmann. We then discuss possible extensionsof the scalar MLBLUE to a multidimensional setting, i.e. from the expectationof scalar random variables to the expectation of random vectors. Severalestimators of increasing complexity are proposed: a) multilevel estimators withscalar weights, b) with element-wise weights, c) with spectral weights and d)with general matrix weights. The computational cost of each method isdiscussed. We finally extend the MLBLUE to the estimation of second-ordermoments in the multidimensional case, i.e. to the estimation of covariancematrices. The multilevel estimators proposed are d) a multilevel estimator withscalar weights and e) with element-wise weights. In large-dimensionapplications such as data assimilation for geosciences, the latter estimator iscomputationnally unaffordable. As a remedy, we also propose f) a multilevelcovariance matrix estimator with optimal multilevel localization, inspired bythe optimal localization theory of M\'en\'etrier and Aulign\'e (2015). Somepractical details on weighted MLMC estimators of covariance matrices are givenin appendix.

Multivariate extensions of the Multilevel Best Linear Unbiased Estimator for ensemble-variational data assimilation