AUTHOR=Jabarullah Khan Nagoor Kani , Elsheikh Ahmed H. 

TITLE=A Machine Learning Based Hybrid Multi-Fidelity Multi-Level Monte Carlo Method for Uncertainty Quantification

JOURNAL=Frontiers in Environmental Science

VOLUME=Volume 7 - 2019

YEAR=2019

URL=https://www.frontiersin.org/journals/environmental-science/articles/10.3389/fenvs.2019.00105

DOI=10.3389/fenvs.2019.00105

ISSN=2296-665X

ABSTRACT=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 intro-
duced 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) [54] 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 re-
sults 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.