Balanced Truncation Based on Generalized Multiscale Finite Element Method for the Parameter-Dependent Elliptic Problem

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.