Recent advances in computational power and numerical algorithms have enabled revolutionary prediction of complex multi-scale multi-physics. However, because of their significant computational cost, it remains challenging to use these new high-fidelity (high-accuracy) for uncertainty quantification (UQ), which requires repeated evaluation of a model. Addressing this core challenge requires utilizing multiple simulation models and experiments of varying cost and accuracy. This talk will provide an overview of the multi-fidelity (MF) strategies for combining limited highfidelity data with a greater amount of lower-fidelity data to substantially increase the accuracy of uncertainty estimates for a limited computational budget. Focus will be given to multi-fidelity quadrature methods that leverage the correlation between different models, arising from varying numerical discretizations and/or idealized physics, to reduce the cost of computing statistical estimators of uncertainty. Initial discussion will contrast MF methods that assume a hierarchy of models ordered by accuracy per unit costs, e.g. multi-level Monte Carlo (MLMC) [1], with methods that can be applied to un-ordered model ensembles, e.g approximate control variates (ACV) [2] and multi-level best linear unbiased estimators (ML-BLUE) [3]. The talk will then present recent developments in the latter class of non-hierarchical methods. Specifically, we will show that ACV and ML-BLUE are equivalent and present a new method for estimating uncertainty that uses multi-arm bandits to balance the cost of computing the correlation between models (exploration), needed for ACV and ML-BLUE, with the cost of computing the MF estimate of uncertainty (exploitation); the exploration cost is typically ignored by existing methods. The talk will conclude with some vignettes demonstrating the efficacy of MF quadrature on applications in plasma physics and ice-sheet modeling.
References
[1] Michael B. Giles. Multilevel monte carlo methods. Acta Numerica, 24:259–328, 2015.
[2] A.A. Gorodetsky, G. Geraci, M.S. Eldred, and J.D. Jakeman. A generalized approximate control variate framework for multifidelity uncertainty quantification. Journal of Computational Physics, 408:109257, 2020.
[3] Daniel Schaden and Elisabeth Ullmann. On multilevel best linear unbiased estimators. SIAM/ASA Journal on Uncertainty Quantification, 8(2):601–635, 2020.