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Two of the most significant challenges in uncertainty quantification pertainto the high computational cost for simulating complex physical models and thehigh dimension of the random inputs. In applications of practical interest,both of these problems are encountered, and standard methods either fail or arenot feasible. To overcome the current limitations, we present a generalizedformulation of a Bayesian multi-fidelity Monte-Carlo (BMFMC) framework that canexploit lower-fidelity model versions in a small data regime. The goal of ouranalysis is an efficient and accurate estimation of the complete probabilisticresponse for high-fidelity models. BMFMC circumvents the curse ofdimensionality by learning the relationship between the outputs of a referencehigh-fidelity model and potentially several lower-fidelity models. While thecontinuous formulation is mathematically exact and independent of thelow-fidelity model's accuracy, we address challenges associated with the smalldata regime (i.e., only a small number of 50 to 300 high-fidelity model runscan be performed). Specifically, we complement the formulation with a set ofinformative input features at no extra cost. Despite the inaccurate and noisyinformation that some low-fidelity models provide, we demonstrate that accurateand certifiable estimates for the quantities of interest can be obtained foruncertainty quantification problems in high stochastic dimensions, withsignificantly fewer high-fidelity model runs than state-of-the-art methods foruncertainty quantification. We illustrate our approach by applying it tochallenging numerical examples such as Navier-Stokes flow simulations andfluid-structure interaction problems.

A Generalized Probabilistic Learning Approach for Multi-Fidelity Uncertainty Propagation in Complex Physical Simulations