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.
Highlights
Effective propagation of uncertainties through nonlinear dynamical systems has become an essential task for model based engineering applications (Elsheikh et al, 2013; Petvipusit et al, 2014; Kani and Elsheikh, 2018)
We propose a Multi-Fidelity-Multi-Level Monte Carlo (MFML-MC) method to address some of the limitations of standard MLMC method with Galerkin projection based reduced order model (ROM) (Antoulas et al, 2001; Lassila et al, 2014; Codina et al, 2015) as low-fidelity models, in particular for large scale nonlinear uncertainty quantification (UQ) problems
To the best of our knowledge, this paper presents the first attempt to combine the features of Multi-Fidelity Monte Carlo (MFMC) method and MLMC method using machine learning techniques for UQ analysis of nonlinear dynamical systems representing multi-phase porous media flow with uncertainty in the permeability field
Summary
Effective propagation of uncertainties through nonlinear dynamical systems has become an essential task for model based engineering applications (e.g., water resources management, petroleum reservoir management) (Elsheikh et al, 2013; Petvipusit et al, 2014; Kani and Elsheikh, 2018). Multi-Fidelity Multi-Level Monte Carlo Method of uncertainties through multi-phase porous media flow models remains challenging because of high dimensionality of input parameter space (e.g., heterogeneous permeability) and the nonpolynomial model nonlinearities (Elsheikh et al, 2012, 2013) For this class of problems, probabilistic techniques, including stochastic Galerkin (Ghanem and Spanos, 1991; Stefanou, 2009), and stochastic collocation methods (Babuška et al, 2007; Doostan and Owhadi, 2011) have limited applicability despite they are computationally very effective for quasi-linear flow models with the small number of random variables (Li and Zhang, 2007; Lin and Tartakovsky, 2009)
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