Revisiting IRKA: Connections with pole placement and backward stability

TL;DR

This paper explores the IRKA algorithm for model reduction, revealing its connection to pole placement, analyzing its convergence behavior, and proposing refinements to enhance stability and efficiency.

Contribution

It uncovers the relationship between IRKA and pole assignment, providing insights that lead to improved convergence and backward stability in the algorithm.

Findings

IRKA's convergence is complex but can be improved through pole placement insights.

Refinements to IRKA can enhance backward stability.

New termination criteria can be developed based on the analysis.

Abstract

The iterative rational Krylov algorithm (\textsf{IRKA}) is a popular approach for producing locally optimal reduced-order $\mathcal{H}_2$-approximations to linear time-invariant (LTI) dynamical systems. Overall, \textsf{IRKA} has seen significant practical success in computing high fidelity (locally) optimal reduced models and has been successfully applied in a variety of large-scale settings. Moreover, \textsf{IRKA} has provided a foundation for recent extensions to the systematic model reduction of bilinear and nonlinear dynamical systems. Convergence of the basic \textsf{IRKA} iteration is generally observed to be rapid --- but not always; and despite the simplicity of the iteration, its convergence behavior is remarkably complex and not well understood aside from a few special cases. The overall effectiveness and computational robustness of the basic \textsf{IRKA} iteration is surprising since its algorithmic goals are very similar to a pole assignment problem, which can be notoriously ill-conditioned. We investigate this connection here and discuss a variety of nice properties of the \textsf{IRKA} iteration that are revealed when the iteration is framed with respect to a primitive basis. We find that the connection with pole assignment suggests refinements to the basic algorithm that can improve convergence behavior, leading also to new choices for termination criteria that assure backward stability.