Reduced Order Modeling For Large-Scale Linear Systems

Abstract

A large variety of physical phenomena can be described by large-scale systems of linear ordinary differential equations (ODEs) obtained by one of the discretization methods, in particular one of the methods of Computational Fluid Dynamics (CFD). The solution of such ODE systems is relatively straightforward with well-developed methods, which makes the large-scale linear systems one of the powerful ways of analyzing physical phenomena. Their practical applicability is, however, severely limited by the computational expense. Days or even weeks may be needed to simulate an unsteady behavior of a system with typical 106 or more degrees of freedom. This limits applications in many important areas, from the demand for extensive solution results for fastpaced optimization design to the need for industrial online predictive control. Therefore, efficient yet accurate models that approximate large-scale linear systems are critically needed. We focus on two major application scenarios: thermal management system in battery packs of electrical/hybrid electric vehicles and the prediction of airborne transmission of respiratory infections, e.g., SARSCOV-2, in indoor environments. The reduced-order modeling (ROM) Krylov-subspace method is developed to reduce the computational effort of CFD. It is based on the projection of the original model onto a Krylov subspace by the Arnoldi-type algorithms. Versions of the method for both single-input and multiple-input systems are presented. The algorithms do not require access the original system matrix, which is usually inaccessible from commercial CFD software. The comparison between the results using the ROM and the original CFD models shows a reduction by a factor of 10^3 in computational time without significant loss in the accuracy of the results.