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基于蒙特卡罗的多输出均值、方差及Sobol主效应方差估计器在模型集合中的协方差表达式的结果。这些协方差可用于多保真不确定性量化策略中，该策略旨在通过一组低保真模型降低高保真蒙特卡罗估计器的方差。在近似控制变量法和多层级最佳线性无偏估计器等方案中，此类协方差表达式是必需的。尽管现有文献针对均值、方差等单输出情形给出了这些表达式，但我们的结果适用于任意采样策略下的多函数输出与多统计量。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，利用多输出可显著降低估计器方差。通过合成算例突出最优样本分配和先导样本估计的影响，并利用进入、下降与着陆的飞行弹道仿真展示多输出估计在实际应用中的可行性。