Article2019

Multifidelity Monte Carlo Estimation with Adaptive Low-Fidelity Models

47 citations0/0 files storedSIAM/ASA Journal on Uncertainty Quantification

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Probabilistic and Robust Engineering Design Advanced Multi-Objective Optimization Algorithms Gaussian Processes and Bayesian Inference

Abstract

Multifidelity Monte Carlo (MFMC) estimation combines low- and high-fidelity models to speed up the estimation of statistics of the high-fidelity model outputs. MFMC optimally samples the low- and high-fidelity models such that the MFMC estimator has minimal mean-squared error (MSE) for a given computational budget. In the setup of MFMC, the low-fidelity models are static; i.e., they are given and fixed and cannot be changed and adapted. We introduce the adaptive MFMC (AMFMC) method that splits the computational budget between adapting the low-fidelity models to improve their approximation quality and sampling the low- and high-fidelity models to reduce the MSE of the estimator. Our AMFMC approach derives the quasi-optimal balance between adaptation and sampling in the sense that our approach minimizes an upper bound of the MSE, instead of the error directly. We show that the quasi-optimal number of adaptations of the low-fidelity models is bounded even in the limit of an infinite budget. This shows that adapting low-fidelity models in MFMC beyond a certain approximation accuracy is unnecessary and can even be wasteful. Our AMFMC approach trades off adaptation and sampling and so avoids overadaptation of the low-fidelity models. Besides the costs of adapting low-fidelity models, our AMFMC approach can also take into account the costs of the initial construction of the low-fidelity models (``offline costs''), which is critical if low-fidelity models are computationally expensive to build such as reduced models and data-fit surrogate models. Numerical results demonstrate that our adaptive approach can achieve orders of magnitude speedups compared to MFMC estimators with static low-fidelity models and compared to Monte Carlo estimators that use the high-fidelity model alone.

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References

2008Multilevel Monte Carlo Path Simulation 1964A New Method of Locating the Maximum Point of an Arbitrary Multipeak Curve in the Presence of Noise 2018Survey of Multifidelity Methods in Uncertainty Propagation, Inference, and Optimization 2004Proceedings of the 2004 Winter Simulation Conference, 2004.2011Multilevel Monte Carlo methods and applications to elliptic PDEs with random coefficients 2008A Knowledge-Gradient Policy for Sequential Information Collection 20082008 Winter Simulation Conference 2016Optimal Model Management for Multifidelity Monte Carlo Estimation

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