Analysis of the Challenges in Developing Sample-Based Multi-fidelity Estimators for Non-deterministic Models

Multifidelity (MF) Uncertainty Quantification (UQ) seeks to leverage and fuse information from a collection of models to achieve greater statistical accuracy with respect to a single-fidelity counterpart, while maintaining an efficient use of computational resources. Despite many recent advancements in MF UQ, several challenges remain and these often limit its practical impact in certain application areas. In this manuscript, we focus on the challenges introduced by non-deterministic models to sampling MF UQ estimators. Non-deterministic models produce different responses for the same inputs, which means their outputs are effectively noisy. MF UQ becomes complicated by this noise since many state-of-the-art approaches rely on statistics, e.g., the correlation among models, to optimally fuse information and allocate computational resources. We demonstrate how the statistics of the quantities of interest, which impact the design, effectiveness, and use of existing MF UQ techniques, change as functions of the noise. With this in hand, we extend the unifying Approximate Control Variate framework to account for non-determinism, providing for the first time a rigorous means of comparing the effect of non-determinism on different multifidelity estimators and analyzing their performance with respect to one another. Numerical examples are presented throughout the manuscript to illustrate and discuss the consequences of the presented theoretical results.