A Unifying Framework for Interpolatory $\mathcal{L}_2$-optimal Reduced-order Modeling

TL;DR

This paper introduces a comprehensive framework for interpolatory $\\mathcal{L}_2$-optimal reduced-order modeling applicable to various systems, unifying existing conditions and deriving new optimality criteria for different classes of models.

Contribution

It unifies and extends interpolatory optimality conditions across multiple model reduction problems, including $\\mathcal{H}_2$ and $\\mathcal{L}_2$ cases, and introduces novel conditions for parametric and discrete models.

Findings

Framework covers $\\mathcal{H}_2$-optimal model reduction conditions.

Derives new interpolatory conditions for parametric stationary models.

Numerical examples validate the theoretical results.

Abstract

We develop a unifying framework for interpolatory $\mathcal{L}_2$-optimal reduced-order modeling for a wide classes of problems ranging from stationary models to parametric dynamical systems. We first show that the framework naturally covers the well-known interpolatory necessary conditions for $\mathcal{H}_2$-optimal model order reduction and leads to the interpolatory conditions for $\mathcal{H}_2 \otimes \mathcal{L}_2$-optimal model order reduction of multi-input/multi-output parametric dynamical systems. Moreover, we derive novel interpolatory optimality conditions for rational discrete least-squares minimization and for $\mathcal{L}_2$-optimal model order reduction of a class of parametric stationary models. We show that bitangential Hermite interpolation appears as the main tool for optimality across different domains. The theoretical results are illustrated on two numerical examples.