Covariance Expressions for Multifidelity Sampling with Multioutput, Multistatistic Estimators: Application to Approximate Control Variates

SIAM/ASA Journal on Uncertainty Quantification, Volume 12, Issue 3, Page 1005-1049, September 2024.
Abstract.We provide a collection of results on covariance expressions between Monte Carlo–based multioutput mean, variance, and Sobol main effect variance estimators from an ensemble of models. These covariances can be used within multifidelity 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 such as the approximate control variate and multilevel 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 multioutput estimation in practical applications.