An approximate control variates approach to multifidelity distribution estimation

Forward simulation-based uncertainty quantification that studies the output distribution of quantities of interest (QoI) is a crucial component for computationally robust statistics and engineering. There is a large body of literature devoted to accurately assessing statistics of QoI, and in particular, multilevel or multifidelity approaches are known to be effective, leveraging cost-accuracy tradeoffs between a given ensemble of models. However, effective algorithms that can estimate the full distribution of outputs are still under active development. In this paper, we introduce a general multifidelity framework for estimating the cumulative distribution functions (CDFs) of vector-valued QoI associated with a high-fidelity model under a budget constraint. Given a family of appropriate control variates obtained from lower fidelity surrogates, our framework involves identifying the most cost-effective model subset and then using it to build an approximate control variates estimator for the target CDF. We instantiate the framework by constructing a family of control variates using intermediate linear approximators and rigorously analyze the corresponding algorithm. Our analysis reveals that the resulting CDF estimator is uniformly consistent and budget-asymptotically optimal, with only mild moment and regularity assumptions. The approach provides a robust multifidelity CDF estimator that is adaptive to the available budget, does not require \textit{a priori} knowledge of cross-model statistics or model hierarchy, and is applicable to general output dimensions. We demonstrate the efficiency and robustness of the approach using several test examples.