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.

翻译：我们提出了新颖的蒙特卡洛(MC)与多层级蒙特卡洛(MLMC)方法，通过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闭合形式无偏地构造估计量的均方误差。这与传统MC和MLMC协方差估计方法形成对比，后者基于仅由上限定义的偏置均方误差（尤其在MLMC框架中）。最后，通过估算一维随机椭圆型偏微分方程（如泊松模型）的随机响应的协方差，展示了算法的数值结果。