@Article{AAMM-10-6,
author = {Shan, Jiang and Protasov, Anastasiya and Sun, Meiling},
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 = {<p style="text-align: justify;">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.</p>},
issn = {2075-1354},
doi = {https://doi.org/10.4208/aamm.OA-2018-0073},
url = {https://global-sci.com/article/73233/balanced-truncation-based-on-generalized-multiscale-finite-element-method-for-the-parameter-dependent-elliptic-problem}
}