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

Jiang, Lijian; Li, Qiuqi

doi:10.1016/j.jcp.2017.02.055

Cited by 24 publications

(20 citation statements)

References 34 publications

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“…Thus, we can obtain a sequence of RGMsB approximation spaces satisfying

X_{H}^{1} \subset X_{H}^{2} \subset ⋯ \subset X_{H}^{N} \subset X_{h}

and a set of RB functions

false{φ_{i, n}^{r g} : i = 1, ⋯, N_{c}, n = 1, ⋯, N false}

. Then, following the main ideas in Jiang and Li (, ), a set of orthonormal basis functions can be obtained as reduced generalized multiscale basis by applying POD to the set

false{φ_{i, n}^{r g} : i = 1, ⋯, N_{c}, n = 1, ⋯, N false}

in the

{false(\cdot, \cdot false)}_{X_{h}}

inner product. Finally, we denote the set of constructed reduced generalized multiscale basis by

{false{ϕ}_{i}^{r g} : i = 1, ⋯, N_{c} \times N false}

with a single index, which spans the same space as the

span false{φ_{i, n}^{r g} : i = 1, ⋯, N_{c}, n = 1, ⋯, N false}

.…”

Section: Reduced Generalized Multiscale Basis Methodsmentioning

confidence: 99%

“…(2) Compute

N false(N \leq M_{g}^{i} false)

eigenvectors v j of matrix Y i corresponding the first N largest eigenvalues λ j ( j = 1,…, N ). (3) For any i ( i = 1,…, N c ), the POD GMsFE basis functions

ϕ_{i, j}^{r g}

can be represented by (Jiang & Li, )

ϕ_{i, j}^{r g} = \frac{1}{\sqrt{λ_{j}}} true \sum_{n = 1}^{M_{g}^{i}} v_{j, n} φ_{i, n}^{g m s}, j = 1, ⋯, N,

where v j , n is the n th element of vector v j . Thus, given N and Ξ opt , we can construct the reduced GMsFE space

X_{H}^{N} = span {false{ϕ}_{i, j}^{r g} : i = 1, ⋯, N_{c}, j = 1, ⋯, N false}

by POD method.…”

Section: Strategies For Rgmsbmmentioning

confidence: 99%

“…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. To get parameter‐independent multiscale basis functions, one can construct generalized multiscale basis functions from a set of samples in the parameter space.…”

Section: Introductionmentioning

confidence: 99%

“…However, this usually results in a high‐dimensional GMsFE space, which brings great challenge for approximation. To alleviate the high dimensionality of the GMsFE space, Jiang and Li () proposed a reduced mixed GMsFE basis method for effectively solving the parametrized elliptic partial differential equations with multiscale coefficients by identifying a set of RB functions from the high‐dimensional GMsFE space.…”

Section: Introductionmentioning

confidence: 99%

“…It may be computationally very expensive and possibly infeasible to simulate such stochastic multiscale models in traditional numerical methods. Therefore, the interest in developing efficient multiscale methods and model reduction methods for the stochastic multiscale models has steadily grown in recent years (e.g., Chung et al, ; Dostert et al, ; Efendiev et al, ; Ganapathysubramanian & Zabaras, ; He et al, ; Jiang & Li, ; Nguyen, ; Shi et al, ).…”

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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“…Thus, we can obtain a sequence of RGMsB approximation spaces satisfying

X_{H}^{1} \subset X_{H}^{2} \subset ⋯ \subset X_{H}^{N} \subset X_{h}

and a set of RB functions

false{φ_{i, n}^{r g} : i = 1, ⋯, N_{c}, n = 1, ⋯, N false}

. Then, following the main ideas in Jiang and Li (, ), a set of orthonormal basis functions can be obtained as reduced generalized multiscale basis by applying POD to the set

false{φ_{i, n}^{r g} : i = 1, ⋯, N_{c}, n = 1, ⋯, N false}

in the

{false(\cdot, \cdot false)}_{X_{h}}

inner product. Finally, we denote the set of constructed reduced generalized multiscale basis by

{false{ϕ}_{i}^{r g} : i = 1, ⋯, N_{c} \times N false}

with a single index, which spans the same space as the

span false{φ_{i, n}^{r g} : i = 1, ⋯, N_{c}, n = 1, ⋯, N false}

.…”

Section: Reduced Generalized Multiscale Basis Methodsmentioning

confidence: 99%

“…(2) Compute

N false(N \leq M_{g}^{i} false)

eigenvectors v j of matrix Y i corresponding the first N largest eigenvalues λ j ( j = 1,…, N ). (3) For any i ( i = 1,…, N c ), the POD GMsFE basis functions

ϕ_{i, j}^{r g}

can be represented by (Jiang & Li, )

ϕ_{i, j}^{r g} = \frac{1}{\sqrt{λ_{j}}} true \sum_{n = 1}^{M_{g}^{i}} v_{j, n} φ_{i, n}^{g m s}, j = 1, ⋯, N,

where v j , n is the n th element of vector v j . Thus, given N and Ξ opt , we can construct the reduced GMsFE space

X_{H}^{N} = span {false{ϕ}_{i, j}^{r g} : i = 1, ⋯, N_{c}, j = 1, ⋯, N false}

by POD method.…”

Section: Strategies For Rgmsbmmentioning

confidence: 99%

Section: Introductionmentioning

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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Model reduction for nonlinear multiscale parabolic problems using dynamic mode decomposition

Jiang

2020

Numerical Meth Engineering

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SummaryIn this article, a model reduction technique is presented to solve nonlinear multiscale parabolic problems using dynamic mode decomposition. The multiple scales and nonlinearity bring great challenges for simulating the problems. To overcome this difficulty, we develop a model reduction method for the nonlinear multiscale dynamic problems by integrating constraint energy minimizing generalized multiscale finite element method (CEM‐GMsFEM) with dynamic mode decomposition (DMD). CEM‐GMsFEM has shown great efficiency to solve linear multiscale problems in a coarse space. However, using CEM‐GMsFEM to directly solve multiscale nonlinear parabolic models involves dynamically computing the residual and the Jacobian on a fine grid. This may be very computationally expensive because the evaluation of the nonlinear term is implemented in a high‐dimensional fine scale space. As a data‐driven method, DMD can use observation data and give an explicit expression to accurately describe the underlying nonlinear dynamic system. To efficiently compute the multiscale nonlinear parabolic problems, we propose a CEM‐DMD model reduction by combing CEM‐GMsFEM and DMD. The CEM‐DMD reduced model is a coarsen linear model, which avoids the nonlinear solver in the fine space. It is crucial to judiciously choose observation in DMD. Only proper observation can render an accurate DMD model. In the context of CEM‐DMD, we introduce two different observations: fine scale observation and coarse scale observation. In the construction of DMD model, the coarse scale observation requires much less computation than the fine scale observation. The CEM‐DMD model using the coarse scale observation gives a complete coarse model for the nonlinear multiscale dynamic systems and significantly improves the computation efficiency. To show the performance of the CEM‐DMD using the different observations, we present a few numerical results for the nonlinear multiscale parabolic problems in heterogeneous porous media.

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Inverse modeling of tracer flow via a mass conservative generalized multiscale finite volume/element method and stochastic collocation

Presho

2018

Comp. Appl. Math.

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

Cited by 24 publications

References 34 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 reduction for nonlinear multiscale parabolic problems using dynamic mode decomposition

Inverse modeling of tracer flow via a mass conservative generalized multiscale finite volume/element method and stochastic collocation

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