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IRKA is a Riemannian Gradient Descent Method

Petar Mlinarić, Christopher A. Beattie, Zlatko Drmač, Serkan Gugercin

TL;DR

This work reframes the IRKA fixed-point method for $\mathcal{H}_{2}$-optimal model order reduction as a Riemannian gradient descent on the embedded manifold of stable rational functions with fixed McMillan degree, within the Hardy space $\mathcal{H}_{2}$. It establishes the manifold structure, derives the Riemannian gradient of the $\mathcal{H}_{2}$ objective, and provides a geometric interpretation that links IRKA’s interpolation conditions to orthogonality on the tangent space. The authors then introduce IRKA2, a variable-step-size variant with backtracking line search that preserves stability and guarantees a decrease in $\mathcal{H}_{2}$ error, while remaining interpretable as a sequence of bitangential Hermite interpolants. Numerical experiments demonstrate that IRKA2 yields faster convergence and greater robustness, especially in challenging SISO cases and in MIMO benchmarks. The work thus provides a solid geometric foundation for MOR and offers a practical, line-search-enabled alternative to IRKA with improved stability guarantees.

Abstract

The iterative rational Krylov algorithm (IRKA) is a commonly used fixed-point iteration developed to minimize the $\mathcal{H}_2$ model order reduction error. In this work, IRKA is recast as a Riemannian gradient descent method with a fixed step size over the manifold of rational functions having fixed degree. This interpretation motivates the development of a Riemannian gradient descent method utilizing as a natural extension variable step size and line search. Comparisons made between IRKA and this extension on a few examples demonstrate significant benefits.

IRKA is a Riemannian Gradient Descent Method

TL;DR

This work reframes the IRKA fixed-point method for -optimal model order reduction as a Riemannian gradient descent on the embedded manifold of stable rational functions with fixed McMillan degree, within the Hardy space . It establishes the manifold structure, derives the Riemannian gradient of the objective, and provides a geometric interpretation that links IRKA’s interpolation conditions to orthogonality on the tangent space. The authors then introduce IRKA2, a variable-step-size variant with backtracking line search that preserves stability and guarantees a decrease in error, while remaining interpretable as a sequence of bitangential Hermite interpolants. Numerical experiments demonstrate that IRKA2 yields faster convergence and greater robustness, especially in challenging SISO cases and in MIMO benchmarks. The work thus provides a solid geometric foundation for MOR and offers a practical, line-search-enabled alternative to IRKA with improved stability guarantees.

Abstract

The iterative rational Krylov algorithm (IRKA) is a commonly used fixed-point iteration developed to minimize the model order reduction error. In this work, IRKA is recast as a Riemannian gradient descent method with a fixed step size over the manifold of rational functions having fixed degree. This interpretation motivates the development of a Riemannian gradient descent method utilizing as a natural extension variable step size and line search. Comparisons made between IRKA and this extension on a few examples demonstrate significant benefits.
Paper Structure (32 sections, 10 theorems, 74 equations, 6 figures, 1 algorithm)

This paper contains 32 sections, 10 theorems, 74 equations, 6 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. H2-optimal Model Order Reduction
  3. Model Order Reduction Problem
  4. Hardy Space
  5. Interpolatory Necessary Conditions for Optimality
  6. Bitangential Hermite Interpolation
  7. The IRKA Iteration
  8. Riemannian Optimization Background
  9. General Manifolds
  10. General Tangent Space and Smooth Functions
  11. Embedded Submanifolds
  12. Tangent Vectors, Tangent Spaces, Tangent Bundle, and Normal Spaces of Embedded Submanifolds
  13. Riemannian Manifolds and Submanifolds
  14. Riemannian Optimization
  15. Rational Functions Form a Manifold
  16. ...and 17 more sections

Key Result

Theorem 2.1

Let $H, \widehat{H} \in \Re\mathcal{H}_{2}$ be such that $\widehat{H}$ has the pole-residue form where $\lambda_i$ are pairwise distinct. Let $\widehat{H}$ be an $\mathcal{H}_{2}$-optimal approximation for $H$ as in eq:h2-opt-prob. Then for $i = 1, 2, \ldots, r$

Figures (6)

  • Figure 1: Necessary $\mathcal{H}_{2}$-optimality conditions in terms of orthogonality. $H$ is the full-order transfer function, $\Sigma_{r, m, p}^-(\mathbb{C})$ is the manifold of stable rational functions of McMillan degree $r$, $\widehat{H}$ is the $\mathcal{H}_{2}$-optimal reduced-order transfer function, $\mathrm{T}_{\widehat{H}} \Sigma_{r, m, p}^-(\mathbb{C})$ is the tangent space of $\Sigma_{r, m, p}^-(\mathbb{C})$ at $\widehat{H}$.
  • Figure 2: Two methods for $\mathcal{H}_{2}$-optimal model order reduction. $H$ is the full-order transfer function, $\Sigma_{r, m, p}^-(\mathbb{C})$ is the manifold of stable rational functions of degree $r$, $\widehat{H}_k$ is the current reduced-order transfer function, $\mathrm{T}_{\widehat{H}_k} \Sigma_{r, m, p}^-(\mathbb{C})$ is the tangent space of $\Sigma_{r, m, p}^-(\mathbb{C})$ at $\widehat{H}_k$, and $\widehat{H}_{k + 1}$ is the next iterate
  • Figure 3: IRKA and irka2 results for the GugAB08 example and $r = 1$. Red crosses represent unstable models
  • Figure 4: IRKA and irka2 results for the GugAB08 example, $r = 2$, and initial Cauchy index of $0$
  • Figure 5: IRKA and irka2 results for the GugAB08 example, $r = 2$, and initial Cauchy index of $2$
  • ...and 1 more figures

Theorems & Definitions (19)

  • Theorem 2.1
  • Theorem 4.1
  • proof
  • Lemma 6.1
  • proof
  • Theorem 6.2
  • proof
  • Corollary 6.3
  • Corollary 6.4
  • proof
  • ...and 9 more