Convergence of the Iterative Rational Krylov Algorithm

TL;DR

This paper proves that for state-space symmetric systems, the IRKA method converges locally to a local minimum of the $ ext{H}_2$ approximation problem, providing theoretical validation for its observed rapid convergence.

Contribution

It establishes the local convergence of IRKA for symmetric systems, filling a gap in the theoretical understanding of this widely used model reduction technique.

Findings

IRKA is a locally convergent fixed point iteration for symmetric systems.

The convergence is towards a local minimum of the $ ext{H}_2$ approximation problem.

Provides theoretical proof for IRKA's rapid convergence observed in practice.

Abstract

The Iterative Rational Krylov Algorithm (IRKA) of [8] is an interpolatory model reduction approach to the optimal $\mathcal{H}_2$ approximation problem. Even though the method has been illustrated to show rapid convergence in various examples, a proof of convergence has not been provided yet. In this note, we show that in the case of state-space symmetric systems, IRKA is a locally convergent fixed point iteration to a local minimum of the underlying $\mathcal{H}_2$ approximation problem.