An Approximate Control Variates Approach to Multifidelity Distribution Estimation

Abstract

Forward simulation–based uncertainty quantification that studies the distribution of quantities of interest (QoI) is crucial for computationally robust engineering design and prediction. A large body of literature is devoted to accurately assessing QoI statistics. In particular, multilevel or multifidelity approaches are known to be effective, leveraging cost-accuracy trade-offs within a given ensemble of models. However, effective algorithms that can estimate the full distribution of QoIs are still under active development. In this paper, we introduce a general multifidelity framework for estimating the cumulative distribution function (CDF) of a vector-valued QoI associated with a high-fidelity model under a budget constraint. Given a family of control variates obtained from lower-fidelity surrogates, our framework involves identifying the most cost-effective model subset under a weighted 2 error metric and then using it to build an approximate control variates estimator for the target CDF. We instantiate the framework by constructing control variates using linear regression and rigorously analyze the corresponding algorithm. Our analysis reveals that the resulting CDF estimator is uniformly consistent and asymptotically optimal under appropriate criteria as the budget tends to infinity, with only mild moment and regularity assumptions on the joint distribution of QoIs. The approach provides a robust multifidelity CDF estimator that is adaptive to the available budget, does not require a priori knowledge of cross-model statistics or model hierarchy, and applies to multiple dimensions. We demonstrate the efficiency and robustness of the approach using test examples of parametric PDEs and stochastic differential equations including both academic instances and more challenging engineering problems.