Multifidelity Monte Carlo Estimation with Adaptive Low-Fidelity Models

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.