Reduced multiscale finite element basis methods for elliptic PDEs with parameterized inputs

Jiang, Lijian; Li, Qiuqi

doi:10.1016/j.cam.2016.01.033

Cited by 13 publications

(14 citation statements)

References 19 publications

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“…If the snapshots have multiscale features, then we have to use a computationally fine mesh to resolve the features in all scales, which may be very expensive. To overcome this difficulty, Efendiev et al () and Jiang and Li () suggested using the MsFEM to compute the local snapshots on a coarse grid. Next, following the works in Hou and Wu (), Efendiev et al (), and Jiang and Li (, ), we introduce how to construct the GMsFE basis functions and reduced GMsFE basis functions.…”

Section: Reduced Generalized Multiscale Basis Methodsmentioning

confidence: 99%

“…To reduce computational cost and maintain simulation efficiency, many multiscale methods, including the multiscale finite element method (MsFEM; Hou & Wu, ), variational multiscale method (Hughes et al, ), and heterogeneous multiscale method (E & Engquist, ), have been proposed for solving the problems with highly heterogeneous coefficients on a coarse grid. The MsFEM is one of the pioneer multiscale methods for multiscale systems, and many other multiscale methods share its similarity (Efendiev & Hou, ; Jiang & Li, ). Its main idea is to incorporate the small‐scale information into multiscale basis functions in local regions and capture the impact of small‐scale features on the large scale through finite element formulation.…”

Section: Introductionmentioning

confidence: 99%

“…For parametrized multiscale models, if we utilize a traditional finite element method to compute snapshot functions in the global region for the RB model, the computation in the off‐line phase may be quite expensive due to the cost required for resolving all scales of the model inputs on a fine‐scale grid. To mitigate the computational cost, several recent studies have suggested using the MsFEM to compute local snapshots on a coarse‐scale grid for the construction of ROM (Efendiev et al, ; Jiang & Li, ). Although the GMsFEM is one of the accurate and efficient methods to solve the multiscale problems on a coarse grid, the generalized multiscale finite element (GMsFE) basis functions usually depend on the random parameters in parametrized partial differential equations when the parameterized inputs vary with the realizations of random variables (Jiang & Li, ), which substantially impacts on the computation efficiency.…”

Section: Introductionmentioning

confidence: 99%

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A Reduced Generalized Multiscale Basis Method for Parametrized Groundwater Flow Problems in Heterogeneous Porous Media

Jiang

2019

Water Resources Research

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In this paper, we develop a reduced generalized multiscale basis method for efficiently solving the parametrized groundwater flow problems in heterogeneous porous media. Recently proposed generalized multiscale finite element (GMsFE) method is one of the accurate and efficient methods to solve the multiscale problems on a coarse grid. However, the GMsFE basis functions usually depend on the random parameters in the parametrized model, which substantially impacts on the computation efficiency when the parametrized equation needs to be solved many times for many instances of parameters. To enhance computation efficiency, we construct reduced generalized multiscale basis functions independent of parameters from the high‐dimensional GMsFE space and generate a reduced‐order multiscale model by projecting the parameterized flow governing equation onto the low‐dimensional reduced GMsFE space. Then, in order to improve the online computation of reduced‐order model, we apply a matrix discrete empirical interpolation method to affinely decompose the nonaffine parametrized systems arising from the discretization of governing equation. Finally, a few numerical experiments are carried out for the parametrized transient flow problems in heterogeneous porous media to illustrate the efficiency and accuracy of the proposed model reduction method. The results show that the proposed reduced generalized multiscale basis method can significantly improve computation efficiency while maintaining comparative accuracy for the groundwater flow problems by selecting a suitable coarse mesh size and an optimal number of reduced GMsFE basis functions at each of coarse nodes.

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Section: Reduced Generalized Multiscale Basis Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A Reduced Generalized Multiscale Basis Method for Parametrized Groundwater Flow Problems in Heterogeneous Porous Media

Jiang

2019

Water Resources Research

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“…The cross-validation method is devoted to selecting the optimal parameters for multiscale basis in [25]. In the paper, we will extend the idea to identify reduced mixed GMsFE basis functions from the snapshots Σ.…”

Section: Basis-oriented Cross-validation Methods For Reduced Mixed Gmsmentioning

confidence: 99%

“…Remark 5.1. We can use the cross-validation method in [25] to choose the first optimal sample µ 1 from the training set Ξ train . This choice can improve the accuracy.…”

Section: This Implies Thatêmentioning

confidence: 99%

Model's sparse representation based on reduced mixed GMsFE basis methods

Jiang

2017

Journal of Computational Physics

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In this paper, we propose a model's sparse representation based on reduced mixed generalized multiscale finite element (GMsFE) basis methods for elliptic PDEs with random inputs. Mixed generalized multiscale finite element method (GMsFEM) is one of the accurate and efficient approaches to solve multiscale problem in a coarse grid with local mass conservation. When the inputs of the PDEs are parameterized by the random variables, the GMsFE basis functions usually depend on the random parameters. This leads to a large number degree of freedoms for the mixed GMsFEM and substantially impacts on the computation efficiency. In order to overcome the difficulty, we develop reduced mixed GMsFE basis methods such that the multiscale basis functions are independent of the random parameters and span a low-dimensional space. To this end, a greedy algorithm is used to find a set of optimal samples from a training set scattered in the parameter space. Reduced mixed GMsFE basis functions are constructed based on the optimal samples using two optimal sampling strategies: basis-oriented cross-validation and proper orthogonal decomposition. Although the dimension of the space spanned by the reduced mixed GMsFE basis functions is much smaller than the dimension of the original full order model, the online computation still depends on the number of coarse degree of freedoms. To significantly improve the online computation, we integrate the reduced mixed GMsFE basis methods with sparse tensor approximation and obtain a sparse representation for the model's outputs. The sparse representation is very efficient for evaluating the model's outputs for many instances of parameters. To illustrate the efficacy of the proposed methods, we present a few numerical examples for multsicale problems with random inputs.Comment: 37 page

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Multiscale model reduction for fluid infiltration simulation through dual-continuum porous media with localized uncertainties

Wang

Vasilyeva

2018

Journal of Computational and Applied Mathematics

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Reduced multiscale finite element basis methods for elliptic PDEs with parameterized inputs

Cited by 13 publications

References 19 publications

A Reduced Generalized Multiscale Basis Method for Parametrized Groundwater Flow Problems in Heterogeneous Porous Media

A Reduced Generalized Multiscale Basis Method for Parametrized Groundwater Flow Problems in Heterogeneous Porous Media

Model's sparse representation based on reduced mixed GMsFE basis methods

Multiscale model reduction for fluid infiltration simulation through dual-continuum porous media with localized uncertainties

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