Covariance estimation using h-statistics in Monte Carlo and Multilevel Monte Carlo methods

We present novel Monte Carlo (MC) and multilevel Monte Carlo (MLMC) methods for determining the unbiased covariance of random variables using h-statistics. The advantage of this procedure lies in the unbiased construction of the estimator's mean square error in a closed form. This is in contrast to the conventional MC and MLMC covariance estimators, which are based on biased mean square errors defined solely by upper bounds, particularly within the MLMC. Finally, the numerical results of the algorithms are demonstrated by estimating the covariance of the stochastic response of a simple 1D stochastic elliptic PDE such as Poisson's model.