This paper focuses on reducing the computational cost of the Monte Carlo method for uncertainty propagation. Recently, Multi-Fidelity Monte Carlo (MFMC) method [46, 48] and Multi-Level Monte Carlo (MLMC) method [44, 29] were introduced to reduce the computational cost of Monte Carlo method by making use of low- fidelity models that are cheap to an evaluation in addition to the high-fidelity models. In this paper, we use machine learning techniques to combine the features of both the MFMC method and the MLMC method into a single framework called Multi-Fidelity- Multi-Level Monte Carlo (MFML-MC) method. In MFML-MC method, we use a hierarchy of proper orthogonal decomposition (POD) based approximations of high- fidelity outputs to formulate a MLMC framework. Next, we utilize Gradient Boosted Tree Regressor (GBTR) to evolve the dynamics of POD based reduced order model (ROM)  on every level of the MLMC framework. Finally, we incorporate MFMC method in order to exploit the POD ROM as a level specific low-fidelity model in the MFML-MC method. We compare the performance of MFML-MC method with the Monte Carlo method that uses either a high-fidelity model or a single low-fidelity model on two subsurface flow problems with random permeability field. Numerical results suggest that MFML-MC method provides an unbiased estimator with speedups by orders of magnitude in comparison to Monte Carlo method that uses high-fidelity model only.
Khan, N. K. J., & Elsheikh, A. H. (2019). A machine learning based hybrid Multi-Fidelity Multi-Level Monte Carlo method for uncertainty quantification. Frontiers in Environmental Science, 7(JUN). https://doi.org/10.3389/fenvs.2019.00105