Bayesian Multiscale Finite Element Methods. Modeling Missing Subgrid Information Probabilistically

Efendiev, Yalchin; Leung, Wing Tat; Cheung, Siu Wun; Guha, Nilabja; Hoáng, Viêt Há; Mallick, Bani K.

doi:10.1615/intjmultcompeng.2017019851

Cited by 6 publications

(9 citation statements)

References 34 publications

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“…A more important measure of the usefulness of the trained neural network is the predicted multiscale solution u pred ms (κ) given by (14)- (15). We compare the predicted solution to u ms defined by (10)- (11), and compute the relative error in L 2 and energy norm, i.e.…”

Section: Numerical Resultsmentioning

confidence: 99%

Prediction of Discretization of GMsFEM Using Deep Learning

et al. 2019

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In this paper, we propose a deep-learning-based approach to a class of multiscale problems. THe Generalized Multiscale Finite Element Method (GMsFEM) has been proven successful as a model reduction technique of flow problems in heterogeneous and high-contrast porous media. The key ingredients of GMsFEM include mutlsicale basis functions and coarse-scale parameters, which are obtained from solving local problems in each coarse neighborhood. Given a fixed medium, these quantities are precomputed by solving local problems in an offline stage, and result in a reduced-order model. However, these quantities have to be re-computed in case of varying media. The objective of our work is to make use of deep learning techniques to mimic the nonlinear relation between the permeability field and the GMsFEM discretizations, and use neural networks to perform fast computation of GMsFEM ingredients repeatedly for a class of media. We provide numerical experiments to investigate the predictive power of neural networks and the usefulness of the resultant multiscale model in solving channelized porous media flow problems.

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Section: Numerical Resultsmentioning

confidence: 99%

Prediction of Discretization of GMsFEM Using Deep Learning

et al. 2019

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“…The source function is taken as f = 1. An experiment with a similar set-up was performed in [24]. We will compare the solutions at the time instant T = 0.02.…”

Section: Numerical Resultsmentioning

confidence: 99%

“…The thresholds are set as σ L = 9 × 10 −6 and σ d = 1 × 10 −7 . We also compare our proposed method with the Bayesian method in [24], in which a residual-minimizing likelihood is used.…”

Section: Numerical Resultsmentioning

confidence: 99%

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Dynamic data-driven Bayesian GMsFEM

Cheung

Guha

2019

Journal of Computational and Applied Mathematics

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In this paper, we propose a Bayesian approach for multiscale problems with the availability of dynamic observational data. Our method selects important degrees of freedom probabilistically in a Generalized multiscale finite element method framework. Due to scale disparity in many multiscale applications, computational models can not resolve all scales. Dominant modes in the Generalized Multiscale Finite Element Method are used as "permanent" basis functions, which we use to compute an inexpensive multiscale solution and the associated uncertainties. Through our Bayesian framework, we can model approximate solutions by selecting the unresolved scales probabilistically. We consider parabolic equations in heterogeneous media. The temporal domain is partitioned into subintervals. Using residual information and given dynamic data, we design appropriate prior distribution for modeling missing subgrid information. The likelihood is designed to minimize the residual in the underlying PDE problem and the mismatch of observational data. Using the resultant posterior distribution, the sampling process identifies important degrees of freedom beyond permanent basis functions. The method adds important degrees of freedom in resolving subgrid information and ensuring the accuracy of the observations.

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“…The main idea of GMsFEM is to extract local dominant modes by carefully designed local spectral problems in coarse regions, and the convergence of the GMsFEM is related to eigenvalue decay of local spectral problems. For a more detailed discussion on GMsFEM, we refer the readers to [27,24,26,21,16,11,31,7,9,46,50,48,8] and the references therein. Through the design of local spectral problems, our method results in the minimal degree of freedom in representing high-contrast features.…”

Section: Introductionmentioning

confidence: 99%

Explicit and Energy-Conserving Constraint Energy Minimizing Generalized Multiscale Discontinuous Galerkin Method for Wave Propagation in Heterogeneous Media

Cheung¹,

Chung²,

Efendiev³

et al. 2020

Preprint

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In this work, we propose a local multiscale model reduction approach for the timedomain scalar wave equation in a heterogenous media. A fine mesh is used to capture the heterogeneities of the coefficient field, and the equation is solved globally on a coarse mesh in the discontinuous Galerkin discretization setting. The main idea of the model reduction approach is to extract dominant modes in local spectral problems for representation of important features, construct multiscale basis functions in coarse oversampled regions by constraint energy minimization problems, and perform a Petrov-Galerkin projection and a symmetrization onto the coarse grid. The method is expicit and energy conserving, and exhibits both coarse-mesh and spectral convergence, provided that the oversampling size is appropriately chosen. We study the stability and convergence of our method. We also present numerical results on the Marmousi model in order to test the performance of the method and verify the theoretical results.

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Bayesian Multiscale Finite Element Methods. Modeling Missing Subgrid Information Probabilistically

Cited by 6 publications

References 34 publications

Prediction of Discretization of GMsFEM Using Deep Learning

Prediction of Discretization of GMsFEM Using Deep Learning

Dynamic data-driven Bayesian GMsFEM

Explicit and Energy-Conserving Constraint Energy Minimizing Generalized Multiscale Discontinuous Galerkin Method for Wave Propagation in Heterogeneous Media

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