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

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.