Abstract
Dynamical systems are pervasive in almost all engineering and scientific applications. Simulating such systems is computationally very intensive. Hence, Model Order Reduction (MOR) is used to reduce them to a lower dimension. Most of the MOR algorithms require solving large sparse sequences of linear systems. Since using direct methods for solving such systems does not scale well in time with respect to the increase in the input dimension, efficient preconditioned iterative methods are commonly used. In one of our previous works, we have shown substantial improvements by reusing preconditioners for the parametric MOR (Singh et al. 2019). Here, we had proposed techniques for both, the non-parametric and the parametric cases, but had applied them only to the latter. We have three main contributions here. First, we demonstrate that preconditioners can be reused more effectively in the non-parametric case as compared to the parametric one. Second, we show that reusing preconditioners is an art via detailed algorithmic implementations in multiple MOR algorithms. Third and final, we demonstrate that reusing preconditioners for reducing a real-life industrial problem (of size 1.2 million), leads to relative savings of up to 64 % in the total computation time (in absolute terms a saving of 5 days).
Highlights
Dynamical systems arise in many engineering and scientific applications such as weather prediction, machine design, circuit simulation, biomedical engineering, etc
The main goal of this paper is to demonstrate the reuse of preconditioners in the remainder of the algorithms for the first category above (MOR of non-parametric linear second-order dynamical systems) as well as the algorithms for the second category above (MOR of non-parametric bilinear/ quadraticbilinear dynamical systems)
WORK In this work, we have focused on Model Order Reduction (MOR) of non-parametric dynamical systems, on the following three algorithms: Adaptive Iterative Rational Global Arnoldi (AIRGA), Bilinear Iterative Rational Krylov Algorithm (BIRKA), and QB-IHOMM
Summary
Dynamical systems arise in many engineering and scientific applications such as weather prediction, machine design, circuit simulation, biomedical engineering, etc. In this paper we broadly demonstrate the application of our above mentioned framework for MOR of non-parametric dynamical systems. Adaptive Iterative Rational Global Arnoldi (AIRGA) [15] is a Ritz-Galerkin projection based algorithm for MOR of linear second-order MIMO dynamical systems with proportional damping, which for the MIMO case are represented as. Bilinear Iterative Rational Krylov Algorithm (BIRKA) [16] is a Petrov-Galerkin projection based algorithm for MOR of the bilinear first-order dynamical systems, which for the MIMO case are represented as m x(t) = Kx(t) + Njx(t)uj(t) + Fu(t), j=1 y(t) = CT x(t),.
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