Preconditioned Iterative Solves in Model Reduction of Second Order Linear Dynamical Systems

TL;DR

This paper enhances the efficiency of the AIRGA algorithm for second order linear dynamical systems by introducing a preconditioned iterative solution approach with SPAI, significantly reducing computation time.

Contribution

It proposes a new technique for updating SPAI preconditioners in AIRGA, improving computational efficiency and demonstrating stability under iterative solution errors.

Findings

SPAI preconditioned CG effectively reduces solution time.

Updating the preconditioner saves about 33% of computation time.

The method is validated on a large beam model.

Abstract

Recently a new algorithm for model reduction of second order linear dynamical systems with proportional damping, the Adaptive Iterative Rational Global Arnoldi (AIRGA) algorithm, has been proposed. The main computational cost of the AIRGA algorithm is in solving a sequence of linear systems. These linear systems do change only slightly from one iteration step to the next. Here we focus on efficiently solving these systems by iterative methods and the choice of an appropriate preconditioner. We propose the use of relevant iterative algorithm and the Sparse Approximate Inverse (SPAI) preconditioner. A technique to cheaply update the SPAI preconditioner in each iteration step of the model order reduction process is given. Moreover, it is shown that under certain conditions the AIRGA algorithm is stable with respect to the error introduced by iterative methods. Our theory is illustrated by experiments. It is demonstrated that SPAI preconditioned Conjugate Gradient (CG) works well for model reduction of a one dimensional beam model with AIRGA algorithm. Moreover, the computation time of preconditioner with update is on an average 2/3 rd of the computation time of preconditioner without update. With average timings running into hours for very large systems, such savings are substantial.