Balanced Truncation Based on Generalized Multiscale Finite Element Method for the Parameter-Dependent Elliptic Problem (Advances in Applied Mathematics and Mechanics, 2021)
In this paper, we combine the generalized multiscale finite element method(GMsFEM) with the balanced truncation (BT) method to address a parameter-dependentelliptic problem. Basically, in progress of a model reduction we try to obtainaccurate solutions with less computational resources. It is realized via a spectraldecomposition from the dominant eigenvalues, that is used for an enrichment of multiscalebasis functions in the GMsFEM. The multiscale bases computations are localizedto specified coarse neighborhoods, and follow an offline-online process in whicheigenvalue problems are used to capture the underlying system behaviors. In the BTon reduced scales, we present a local-global strategy where it requires the observabilityand controllability of solutions to a set of Lyapunov equations. As the Lyapunovequations need expensive computations, the efficiency of our combined approach isshown to be readily flexible with respect to the online space and an reduced dimension.Numerical experiments are provided to validate the robustness of our approachfor the parameter-dependent elliptic model.