Local–global model reduction of parameter-dependent, single-phase flow models via balanced truncation

Abstract

In this paper we propose a method for the accurate calculation of output quantities resulting from a parameter-dependent, single-phase flow model. In particular, given a small-dimensional set of inputs (as compared to the fine model), we treat the problem using a combined local–global model reduction technique. The local model reduction is achieved through the use of the Generalized Multiscale Finite Element Method (GMsFEM) where a set of independently calculated basis functions are used in order to construct a suitable coarse approximation space. The multiscale basis function computations are localized to specified coarse subdomains, and follow an offline–online procedure in which a set of eigenvalue problems are used to capture the underlying behavior of the system. Because the offline stage accounts for a one-time preprocessing step, the online coarse space may be cheaply constructed for a given input state. We then apply balanced truncation (BT) to the online coarse system in order to obtain a global reduced-order approximation of the output state. BT recasts the model equation into a systems framework where the input–output mapping may be approximated through the spectral construction of a reduced-order model, and requires the solution of a set of Lyapunov equations. As the Lyapunov equations represent an expensive computation, the efficiency of the proposed method depends on the size of the online coarse space. The combined approach is shown to be flexible with respect to the online space and reduced dimensions, and may be readily modified in order to ensure that the resulting output errors are comparable.