H2 optimal rational approximation on general domains

TL;DR

This work extends optimal model reduction to systems whose poles occupy general complex domains by introducing the TEXT space, defined through conformal maps $\psi$ and a generalized Hardy space $\mathcal{H}_2(\bar{\mathbb{A}}^{\mathsf{c}})$. It derives first-order optimality conditions in this generalized setting, yielding Hermite interpolation conditions that relate the full and reduced transfer functions via a generalized functional $\mathfrak{F}$. Under practical assumptions on $\psi$ and its derivative, these conditions simplify to explicit interpolation rules, enabling a modified IRKA algorithm that updates interpolation points with $\varphi(\widehat{\lambda}_j)$ rather than mirror images. Numerical experiments on discretized Schrödinger and wave equations demonstrate that the TEXT-IRKA approach can outperform classical IRKA when poles lie on or near the imaginary axis, providing accurate reduced models for unstable or marginally stable dynamics. The framework broadens the applicability of $\mathcal{H}_2$-optimal model reduction to open-domain regions such as left-half planes, unit disks, and ellipses, with potential impact on conservative PDEs and other systems with nonstandard spectra.

Abstract

Optimal model reduction for large-scale linear dynamical systems is studied. In contrast to most existing works, the systems under consideration are not required to be stable, neither in discrete nor in continuous time. As a consequence, the underlying rational transfer functions are allowed to have poles in general domains in the complex plane. In particular, this covers the case of specific conservative partial differential equations such as the linear Schrödinger and the undamped linear wave equation with spectra on the imaginary axis. By an appropriate modification of the classical continuous time Hardy space $\mathcal{H}_2$, a new $\mathcal{H}_2$ like optimal model reduction problem is introduced and first order optimality conditions are derived. As in the classical $\mathcal{H}_2$ case, these conditions exhibit a rational Hermite interpolation structure for which an iterative model reduction algorithm is proposed. Numerical examples demonstrate the effectiveness of the new method.

Figures (5)

• Figure 1: Depiction of the sets introduced in Assumption \ref{['assumption:1']} and the mapping $\psi$ along with its inverse $\psi^{-1}$. Here the dashed closed line indicates the boundary of the set $\{s\in \mathbb{C}\lvert -s^*\in\mathbb{X}\}$.
• Figure 1: Mapping from the left half plane into the interior of a Bernstein ellipse with major and minor axis $(R+R^{-1})/2$ and $(R-R^{-1})/2$, respectively.
• Figure 1: The $\mathcal{H}_2(\bar{\mathbb{A}}^{\mathsf{c}})$ relative error of Algorithm \ref{['algoirka']} and IRKA for different reduced orders $r$ and $n=1000$.
• Figure 2: (Top) the real and imaginary output responses of the FOM ($y$) and ROM ($\widehat{y}_r$) for the chosen input $u$. Here the ROM system matrices are computed with Algorithm 1 for $r=16$. (Middle) the relative error of the reduced model computed with Algorithm 1 and IRKA. (Bottom) trajectory of the chosen control input $u$.
• Figure 3: (Top) the output impulse response of the FOM ($y$) and ROM ($\widehat{y}_r$). Here the system dimension is $n=10000$ and the reduced order is $r=20$. (Bottom) the absolute error.

Theorems & Definitions (15)

• Theorem 2.1
• Theorem 3.1
• Definition 3.2: $\mathcal{H}_2(\bar{\mathbb{A}}^{\mathsf{c}})$ space
• Lemma 3.3
• Proof 1
• Lemma 3.4
• Proof 2
• Corollary 3.5
• Lemma 3.6
• Proof 3