Covariance Expressions for Multi-Fidelity Sampling with Multi-Output, Multi-Statistic Estimators: Application to Approximate Control Variates

TL;DR

This paper derives covariance formulas for multi-output, multi-statistic Monte Carlo estimators and demonstrates their effectiveness in reducing variance in multi-fidelity uncertainty quantification, with applications to control variates.

Contribution

It provides new covariance expressions for multi-output estimators and applies them to improve variance reduction in multi-fidelity Monte Carlo methods.

Findings

Multi-output estimators significantly reduce variance compared to single-output.

Optimal sample allocation enhances estimator efficiency.

Application to flight simulation demonstrates practical benefits.

Abstract

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.