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Volume 10, Issue 6
Balanced Truncation Based on Generalized Multiscale Finite Element Method for the Parameter-Dependent Elliptic Problem

Shan Jiang, Anastasiya Protasov & Meiling Sun

Adv. Appl. Math. Mech., 10 (2018), pp. 1527-1548.

Published online: 2018-09

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  • Abstract

In this paper, we combine the generalized multiscale finite element method (GMsFEM) with the balanced truncation (BT) method to address a parameter-dependent elliptic problem. Basically, in progress of a model reduction we try to obtain accurate solutions with less computational resources. It is realized via a spectral decomposition from the dominant eigenvalues, that is used for an enrichment of multiscale basis functions in the GMsFEM. The multiscale bases computations are localized to specified coarse neighborhoods, and follow an offline-online process in which eigenvalue problems are used to capture the underlying system behaviors. In the BT on reduced scales, we present a local-global strategy where it requires the observability and controllability of solutions to a set of Lyapunov equations. As the Lyapunov equations need expensive computations, the efficiency of our combined approach is shown to be readily flexible with respect to the online space and an reduced dimension. Numerical experiments are provided to validate the robustness of our approach for the parameter-dependent elliptic model.

  • Keywords

Generalized multiscale method, balanced truncation, parameter dependent, eigenvalue decomposition, Lyapunov equation.

  • AMS Subject Headings

35J25, 65N12, 65N30

  • Copyright

COPYRIGHT: © Global Science Press

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@Article{AAMM-10-1527, author = {Shan and Jiang and and 19071 and and Shan Jiang and Anastasiya and Protasov and and 19072 and and Anastasiya Protasov and Meiling and Sun and and 19073 and and Meiling Sun}, title = {Balanced Truncation Based on Generalized Multiscale Finite Element Method for the Parameter-Dependent Elliptic Problem}, journal = {Advances in Applied Mathematics and Mechanics}, year = {2018}, volume = {10}, number = {6}, pages = {1527--1548}, abstract = {

In this paper, we combine the generalized multiscale finite element method (GMsFEM) with the balanced truncation (BT) method to address a parameter-dependent elliptic problem. Basically, in progress of a model reduction we try to obtain accurate solutions with less computational resources. It is realized via a spectral decomposition from the dominant eigenvalues, that is used for an enrichment of multiscale basis functions in the GMsFEM. The multiscale bases computations are localized to specified coarse neighborhoods, and follow an offline-online process in which eigenvalue problems are used to capture the underlying system behaviors. In the BT on reduced scales, we present a local-global strategy where it requires the observability and controllability of solutions to a set of Lyapunov equations. As the Lyapunov equations need expensive computations, the efficiency of our combined approach is shown to be readily flexible with respect to the online space and an reduced dimension. Numerical experiments are provided to validate the robustness of our approach for the parameter-dependent elliptic model.

}, issn = {2075-1354}, doi = {https://doi.org/10.4208/aamm.OA-2018-0073}, url = {http://global-sci.org/intro/article_detail/aamm/12721.html} }
TY - JOUR T1 - Balanced Truncation Based on Generalized Multiscale Finite Element Method for the Parameter-Dependent Elliptic Problem AU - Jiang , Shan AU - Protasov , Anastasiya AU - Sun , Meiling JO - Advances in Applied Mathematics and Mechanics VL - 6 SP - 1527 EP - 1548 PY - 2018 DA - 2018/09 SN - 10 DO - http://doi.org/10.4208/aamm.OA-2018-0073 UR - https://global-sci.org/intro/article_detail/aamm/12721.html KW - Generalized multiscale method, balanced truncation, parameter dependent, eigenvalue decomposition, Lyapunov equation. AB -

In this paper, we combine the generalized multiscale finite element method (GMsFEM) with the balanced truncation (BT) method to address a parameter-dependent elliptic problem. Basically, in progress of a model reduction we try to obtain accurate solutions with less computational resources. It is realized via a spectral decomposition from the dominant eigenvalues, that is used for an enrichment of multiscale basis functions in the GMsFEM. The multiscale bases computations are localized to specified coarse neighborhoods, and follow an offline-online process in which eigenvalue problems are used to capture the underlying system behaviors. In the BT on reduced scales, we present a local-global strategy where it requires the observability and controllability of solutions to a set of Lyapunov equations. As the Lyapunov equations need expensive computations, the efficiency of our combined approach is shown to be readily flexible with respect to the online space and an reduced dimension. Numerical experiments are provided to validate the robustness of our approach for the parameter-dependent elliptic model.

Shan Jiang, Anastasiya Protasov & Meiling Sun. (1970). Balanced Truncation Based on Generalized Multiscale Finite Element Method for the Parameter-Dependent Elliptic Problem. Advances in Applied Mathematics and Mechanics. 10 (6). 1527-1548. doi:10.4208/aamm.OA-2018-0073
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