Inexact Solves in Interpolatory Model Reduction

TL;DR

This paper explores the use of inexact solves in interpolatory model reduction, providing bounds on perturbations, demonstrating backward stability, and showing robustness and effectiveness in large-scale ${ m H}_2$ approximation.

Contribution

It introduces a framework for using inexact solves in interpolatory model reduction, establishing stability, robustness, and practical strategies for large-scale problems.

Findings

Inexact solves induce bounded perturbations in the reduced model.

Petrov-Galerkin framework ensures backward stability of the reduced model.

Inexact solves are effective in large-scale ${ m H}_2$ optimal model reduction.

Abstract

We investigate the use of inexact solves for interpolatory model reduction and consider associated perturbation effects on the underlying model reduction problem. We give bounds on system perturbations induced by inexact solves and relate this to termination criteria for iterative solution methods. We show that when a Petrov-Galerkin framework is employed for the inexact solves, the associated reduced order model is an exact interpolatory model for a nearby full-order system; thus demonstrating backward stability. We also give evidence that for $\h2$-optimal interpolation points, interpolatory model reduction is robust with respect to perturbations due to inexact solves. Finally, we demonstrate the effecitveness of direct use of inexact solves in optimal ${\mathcal H}_2$ approximation. The result is an effective model reduction strategy that is applicable in realistically large-scale settings.