Adaptive choice of near-optimal expansion points for interpolation-based structure-preserving model reduction

TL;DR

This paper addresses the challenge of automatically selecting near-optimal interpolation points and determining the appropriate reduced-order size for structure-preserving model reduction of general transfer-function systems.It introduces StrAIKA, an adaptive, projection-based method that uses a Loewner-derived first-order surrogate to guide iterative updates of interpolation points and model order, with the option to focus accuracy in a chosen frequency region ${\Omega}$.Compared to existing IRKA-like strategies, StrAIKA demonstrates superior accuracy and often lower computational cost across diverse structures (including time delays, viscoelasticity, and rf-gun models) by exploiting region-focused sampling and adaptive pole-dominance selection.The approach preserves internal structure without requiring transfer-function derivatives and provides a practical pathway to high-fidelity, region-specific surrogates, while acknowledging that stability preservation for general structures remains an open challenge.Overall, StrAIKA advances structure-preserving MOR by combining adaptive interpolation-point selection with efficient, data-driven surrogates to tailor reduced models to application-specific frequency ranges.

Abstract

Interpolation-based methods are well-established and effective approaches for the efficient generation of accurate reduced-order surrogate models. Common challenges for such methods are the automatic selection of good or even optimal interpolation points and the appropriate size of the reduced-order model. An approach that addresses the first problem for linear, unstructured systems is the Iterative Rational Krylov Algorithm (IRKA), which computes optimal interpolation points through iterative updates by solving linear eigenvalue problems. However, in the case of preserving internal system structures, optimal interpolation points are unknown, and heuristics based on nonlinear eigenvalue problems result in numbers of potential interpolation points that typically exceed the reasonable size of reduced-order systems. In our work, we propose a projection-based iterative interpolation method inspired by IRKA for generally structured systems to adaptively compute near-optimal interpolation points as well as an appropriate size for the reduced-order system. Additionally, the iterative updates of the interpolation points can be chosen such that the reduced-order model provides an accurate approximation in specified frequency ranges of interest. For such applications, our new approach outperforms the established methods in terms of accuracy and computational effort. We show this in numerical examples with different structures.

Figures (5)

• Figure 1: Approximation of a model in a specified frequency region $\Omega$. Only mirror images with respect to the imaginary axis of poles in the specified frequency region are considered leading to an accurate local approximation of the transfer function.
• Figure 2: First-order system example: All applied methods provide a similar approximation behavior since no special structure needs to be preserved in the example. Insignificant differences are revealed by the pointwise relative errors.
• Figure 3: Time-delay system example: The unstructured approximation computed by TF-IRKA is around two orders of magnitude less accurate than the structure-preserving reduced-order models computed by StrAIKA and SPTF-IRKA for smaller to medium frequencies. Overall StrAIKA provides the most accurate approximation.
• Figure 4: Viscoelastic beam example: All methods succeed in computing reasonably accurate reduced-order models. The relative approximation error of the reduced-order model obtained by StrAIKA is around three orders of magnitude smaller compared to the other methods.
• Figure 5: Radio frequency gun example: The reduced-order model computed by StrAIKA shows a relatively uniform accuracy in $\Omega$, while the surrogate obtained form SPTF-IRKA is only accurate for less than 60rad. The first-order system computed by TF-IRKA is not able to approximate the dynamics of the original system at all.

Theorems & Definitions (1)

• Proposition 1: Structured interpolation BeaG09