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Front. Environ. Sci. | doi: 10.3389/fenvs.2019.00105

A machine learning based hybrid Multi-Fidelity Multi-Level Monte Carlo method for uncertainty quantification

  • 1Heriot-Watt University, United Kingdom

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.

Keywords: uncertainty quantification, POD, Multi-Fidelity Monte Carlo method, multi-level Monte Carlo method, machine learning

Received: 27 Nov 2018; Accepted: 20 Jun 2019.

Copyright: © 2019 Jabarullah Khan and H. Elsheikh. This is an open-access article distributed under the terms of the Creative Commons Attribution License (CC BY). The use, distribution or reproduction in other forums is permitted, provided the original author(s) and the copyright owner(s) are credited and that the original publication in this journal is cited, in accordance with accepted academic practice. No use, distribution or reproduction is permitted which does not comply with these terms.

* Correspondence: Mr. Nagoor Kani Jabarullah Khan, Heriot-Watt University, Edinburgh, United Kingdom, nj7@hw.ac.uk