We present novel Monte Carlo (MC) and multilevel Monte Carlo (MLMC) methods to determine 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 conventional MC and MLMC covariance estimators, which are based on biased mean square errors defined solely by upper bounds, particularly within the MLMC. 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.

翻译：我们提出了新型蒙特卡洛（MC）与多层蒙特卡洛（MLMC）方法，利用h统计量确定随机变量的无偏协方差。该方法的优势在于能够以封闭形式无偏构建估计量的均方误差。这与传统MC和MLMC协方差估计方法形成对比——后者基于仅由上界定义的有偏均方误差，尤其在MLMC框架中尤为突出。通过估计一维随机椭圆型偏微分方程（如泊松模型）随机响应的协方差，数值实验结果展示了该算法的有效性。