Balanced Truncation Based on Generalized Multiscale Finite Element Method for the Parameter-Dependent Elliptic Problem
Year: 2018
Author: Shan Jiang, Anastasiya Protasov, Meiling Sun
Advances in Applied Mathematics and Mechanics, Vol. 10 (2018), Iss. 6 : pp. 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.
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Journal Article Details
Publisher Name: Global Science Press
Language: English
DOI: https://doi.org/10.4208/aamm.OA-2018-0073
Advances in Applied Mathematics and Mechanics, Vol. 10 (2018), Iss. 6 : pp. 1527–1548
Published online: 2018-01
AMS Subject Headings: Global Science Press
Copyright: COPYRIGHT: © Global Science Press
Pages: 22
Keywords: Generalized multiscale method balanced truncation parameter dependent eigenvalue decomposition Lyapunov equation.
Author Details
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Generalized Multiscale Finite Element Method and Balanced Truncation for Parameter-Dependent Parabolic Problems
Jiang, Shan
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Mathematics, Vol. 11 (2023), Iss. 24 P.4965
https://doi.org/10.3390/math11244965 [Citations: 1]