We provide a collection of results on covariance expressions between Monte Carlo based multi-output mean, variance, and Sobol main effect variance estimators from an ensemble of models. These covariances can be used within multi-fidelity uncertainty quantification strategies that seek to reduce the estimator variance of high-fidelity Monte Carlo estimators with an ensemble of low-fidelity models. Such covariance expressions are required within approaches like the approximate control variate and multi-level best linear unbiased estimator. While the literature provides these expressions for some single-output cases such as mean and variance, our results are relevant to both multiple function outputs and multiple statistics across any sampling strategy. Following the description of these results, we use them within an approximate control variate scheme to show that leveraging multiple outputs can dramatically reduce estimator variance compared to single-output approaches. Synthetic examples are used to highlight the effects of optimal sample allocation and pilot sample estimation. A flight-trajectory simulation of entry, descent, and landing is used to demonstrate multi-output estimation in practical applications.

翻译：我们提供了一系列关于蒙特卡洛方法中多输出均值、方差及Sobol主效应方差估计器之间协方差表达式的研究结果，这些估计器源自一个模型集合。这些协方差可用于多保真度不确定性量化策略中，旨在通过低保真度模型集合来降低高保真度蒙特卡洛估计器的估计方差。近似控制变量法和多层级最佳线性无偏估计器等方法均需要此类协方差表达式。尽管现有文献已针对某些单输出情况（如均值和方差）提供了相关表达式，但我们的研究结果适用于任意抽样策略下的多函数输出与多统计量场景。在阐述这些结果后，我们将其应用于近似控制变量框架，证明相较于单输出方法，利用多输出能显著降低估计方差。通过合成算例重点分析了最优样本分配与先导样本估计的影响，并采用进入-下降-着陆阶段的飞行轨迹仿真来展示多输出估计在实际工程中的应用。