Stability Analysis of Bilinear Iterative Rational Krylov Algorithm

TL;DR

This paper proves the stability of the BIRKA algorithm for bilinear model reduction when using inexact iterative linear solvers, and analyzes the impact of solver errors on the reduced model's accuracy.

Contribution

It provides a theoretical stability analysis of BIRKA with inexact linear solves and supports findings with numerical experiments.

Findings

BIRKA remains stable under certain conditions despite inexact solves.

The accuracy of the reduced model is maintained with controlled solver errors.

Numerical experiments confirm theoretical stability and accuracy results.

Abstract

Models coming from different physical applications are very large in size. Simulation with such systems is expensive so one usually obtains a reduced model (by model reduction) that replicates the input-output behaviour of the original full model. A recently proposed algorithm for model reduction of bilinear dynamical systems, Bilinear Iterative Rational Krylov Algorithm (BIRKA), does so in a locally optimal way. This algorithm requires solving very large linear systems of equations. Usually these systems are solved by direct methods (e.g., LU), which are very expensive. A better choice is iterative methods (e.g., Krylov). However, iterative methods introduce errors in linear solves because they are not exact. They solve the given linear system up to a certain tolerance. We prove that under some mild assumptions BIRKA is stable with respect to the error introduced by the inexact linear solves. We also analyze the accuracy of the reduced system obtained from using these inexact solves and support all our results by numerical experiments.