Multi-fidelity Monte Carlo (MFMC) is a variance reduction method that leverages a multi-fidelity ensemble of models of varying cost and accuracy levels. Constructing an MFMC estimator with optimal variance requires knowledge of the correlation coefficients between the different fidelity models which are not usually known in practice. The correlations are typically estimated using offline pilot samples and the sample correlation formula, after which the MFMC method proceeds as if the estimated correlations are the true correlations. Computational cost often restricts the number of pilot samples used leading to poor correlation estimates and suboptimal estimators. Leveraging the MFMC problem setting and probabilistic information about the sample covariance matrix, we present a method to improve standard sample-based correlation estimates in the presence of limited pilot samples. We define a novel discrepancy function quantifying the estimator suboptimality which in turn facilitates selecting a correlation estimator minimizing the worst-case expected discrepancy, where the expectation is taken with respect to the pilot sampling variability. Through a simple bivariate Gaussian example and a multi-fidelity modeling application from a NASA Entry, Descent, and Landing (EDL) problem, we show that this method produces better MFMC estimators than the standard sample covariance under small pilot sample sizes and limited total budgets.

翻译：多保真蒙特卡洛（MFMC）是一种方差缩减方法，通过利用具有不同计算成本与精度水平的多保真模型集成来实现估计量的方差降低。构建具有最优方差的MFMC估计量需要预知不同保真度模型间的相关系数，然而实际中这些系数通常未知。目前普遍采用离线试点采样和样本相关系数公式估计相关性，随后MFMC方法将估计值视为真实相关系数继续执行。受计算成本限制，可用的试点采样数量往往有限，导致相关性估计不准确及估计量次优。本文利用MFMC问题的特殊性及样本协方差矩阵的概率信息，提出一种在有限试点采样条件下改进标准样本相关性估计的方法。我们定义了一个新的差异函数来量化估计量的次优性，该函数可辅助选择使最坏情况期望差异最小化的相关性估计量——其中期望值针对试点采样的变异性计算。通过二元高斯分布算例以及美国国家航空航天局（NASA）进入-下降-着陆（EDL）问题的多保真建模应用，我们证明：在试点采样量小且总预算有限的情况下，该方法能生成比标准样本协方差更优的MFMC估计量。