Multilevel Monte Carlo methods for computing failure probability of porous media flow systems

Fagerlund, Fritjof; Hellman, Fredrik; Målqvist, Axel; Niemi, Auli

doi:10.1016/j.advwatres.2016.06.007

Cited by 8 publications

(7 citation statements)

References 29 publications

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“…23 However, MLMC has been shown to converge and perform well in numerous applications. 23,36,39,40 Another limitation is that the MLMC Theorem requires many constants to be known or approximated, 23 which adds the need for preliminary sampling when these constants are not available. However, the computational time for MLMC has been shown to be orders of magnitude less than for standard MC sampling, 22,23 which compensates for the time spent on preliminary sampling.…”

Section: Extension To Multiple Observablesmentioning

confidence: 99%

“…The main limitation of the MLMC method is that the algorithm is heuristic and is not guaranteed to converge . However, MLMC has been shown to converge and perform well in numerous applications . Another limitation is that the MLMC Theorem requires many constants to be known or approximated, which adds the need for preliminary sampling when these constants are not available.…”

Section: Multilevel Monte Carlomentioning

confidence: 99%

“…Since the original publication, 22 the MLMC method has been expanded to estimating the expected values of various types of ordinary differential equations (ODEs), 23,24 continuous time Markov chains, 25 computing mean exit times, 26 central statistical moments of arbitrary order, 27 and even probability density functions. [28][29][30][31] Furthermore, MLMC has been applied to uncertainty analysis in flows through porous media, [31][32][33][34][35][36][37][38] statistical solutions of the Navier-Stokes equations, 39 reliability analysis of engineered systems, 40 and surfactant/polymer enhanced-oil-recovery. 41 In all cases, improvements in computational costs associated with obtaining accurate results have been reported.…”

Section: Introductionmentioning

confidence: 99%

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Multilevel Monte Carlo applied to chemical engineering systems subject to uncertainty

Kimaev

Ricardez‐Sandoval

2017

AIChE Journal

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The aim of this study is to evaluate the performance of Multilevel Monte Carlo (MLMC) sampling technique for uncertainty quantification in chemical engineering systems. Three systems (a mixing tank, a wastewater treatment plant, and a ternary distillation column, all subject to uncertainty) were considered. The expected values of the systems' observables were estimated using MLMC, Power Series and Polynomial Chaos expansions, and standard Monte Carlo (MC) sampling. The MLMC technique achieved results of significantly greater accuracy than other methods at a lower computational cost than standard MC. This study highlights the nuances of adapting the MLMC technique to chemical engineering systems and the advantages of using MLMC for uncertainty quantification.

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Section: Extension To Multiple Observablesmentioning

confidence: 99%

Section: Multilevel Monte Carlomentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Multilevel Monte Carlo applied to chemical engineering systems subject to uncertainty

Kimaev

Ricardez‐Sandoval

2017

AIChE Journal

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“…Elsakout et al (2015) demonstrated the performance of MLMCMC for uncertainty quantification tasks involving reservoir simulation with less computational cost in comparison to the standard Markov Chain Monte Carlo method. Fagerlund et al (2016) combined selective refinement technique with the MLMC for estimating the sweep efficiency in a two-phase flow scenario where an absolute accuracy of failure probability in a magnitude 5 to 10 percent is required. Lu et al (2016) applied MLMC method for estimating cumulative distribution functions of QoI obtained from the numerical approximation of largescale stochastic subsurface simulations.…”

Section: Introductionmentioning

confidence: 99%

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

Khan

Elsheikh

2019

Front. Environ. Sci.

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This paper focuses on reducing the computational cost of the Monte Carlo method for uncertainty propagation. Recently, Multi-Fidelity Monte Carlo (MFMC) method (Ng, 2013; Peherstorfer et al., 2016) and Multi-Level Monte Carlo (MLMC) method (Müller et al., 2013; Giles, 2015) 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) (Xiao et al., 2017) 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.

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“…The model hierarchy then often corresponds to different discretizations of the underlying PDE. There are extensions [11,12,13] of the multilevel Monte Carlo method [17,16] for rare event probability estimation, which are based on variance reduction with control variates, instead of importance sampling. The subset method [1,37] is another approach that has been extended to exploit a hierarchy of coarse-grid approximations in [35].…”

Section: Introductionmentioning

confidence: 99%

Multifidelity Preconditioning of the Cross-Entropy Method for Rare Event Simulation and Failure Probability Estimation

Peherstorfer¹,

Krämer

Willcox

2018

SIAM/ASA J. Uncertainty Quantification

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Accurately estimating rare event probabilities with Monte Carlo can become costly if for each sample a computationally expensive high-fidelity model evaluation is necessary to approximate the system response. Variance reduction with importance sampling significantly reduces the number of required samples if a suitable biasing density is used. This work introduces a multifidelity approach that leverages a hierarchy of low-cost surrogate models to efficiently construct biasing densities for importance sampling. Our multifidelity approach is based on the cross-entropy method that derives a biasing density via an optimization problem. We approximate the solution of the optimization problem at each level of the surrogate-model hierarchy, reusing the densities found on the previous levels to precondition the optimization problem on the subsequent levels. With the preconditioning, an accurate approximation of the solution of the optimization problem at each level can be obtained from a few model evaluations only. In particular, at the highest level, only few evaluations of the computationally expensive high-fidelity model are necessary. Our numerical results demonstrate that our multifidelity approach achieves speedups of several orders of magnitude in a thermal and a reacting-flow example compared to the single-fidelity cross-entropy method that uses a single model alone.

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Multilevel Monte Carlo methods for computing failure probability of porous media flow systems

Cited by 8 publications

References 29 publications

Multilevel Monte Carlo applied to chemical engineering systems subject to uncertainty

Multilevel Monte Carlo applied to chemical engineering systems subject to uncertainty

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

Multifidelity Preconditioning of the Cross-Entropy Method for Rare Event Simulation and Failure Probability Estimation

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