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Compression and Distillation of Data Quadruplets in Non-intrusive Reduced-order Modeling

Umair Zulfiqar

TL;DR

The paper develops a cohesive, data-driven, non-intrusive framework for reduced-order modeling that unifies BT and IRKA through the compression and distillation of data quadruplets. By leveraging transfer function samples and impulse responses within the Loewner framework, it introduces DD-ADI-BT, three data-driven IRKA variants (CT/DT) using frequency-, impulse-, or transfer-function data, and PORK-based DT methods, all without requiring new samples at every iteration. The methods show performance comparable to intrusive BT and IRKA across continuous-time and discrete-time benchmarks, while providing a unifying perspective on data-driven interpolation as a compression/distillation process. This approach enables practical MOR from available data and suggests a versatile route for data-driven model reduction in large-scale systems.

Abstract

This paper introduces a quadrature-free, data-driven approach to balanced truncation for both continuous-time and discrete-time systems. The method non-intrusively constructs reduced-order models using available transfer function samples from the right half of the $s$-plane. It is highlighted that the proposed data-driven balanced truncation and existing quadrature-based balanced truncation algorithms share a common feature: both compress their respective data quadruplets to derive reduced-order models. Additionally, it is demonstrated that by using different compression strategies, these quadruplets can be utilized to develop three data-driven formulations of the IRKA for both continuous-time and discrete-time systems. These formulations non-intrusively generate reduced models using transfer function samples from the $jω$-axis or the right half of the $s$-plane, or impulse response samples. Notably, these IRKA formulations eliminate the necessity of computing new transfer function samples as IRKA iteratively updates the interpolation points. The efficacy of the proposed algorithms is validated through numerical examples, which show that the proposed data-driven approaches perform comparably to their intrusive counterparts.

Compression and Distillation of Data Quadruplets in Non-intrusive Reduced-order Modeling

TL;DR

The paper develops a cohesive, data-driven, non-intrusive framework for reduced-order modeling that unifies BT and IRKA through the compression and distillation of data quadruplets. By leveraging transfer function samples and impulse responses within the Loewner framework, it introduces DD-ADI-BT, three data-driven IRKA variants (CT/DT) using frequency-, impulse-, or transfer-function data, and PORK-based DT methods, all without requiring new samples at every iteration. The methods show performance comparable to intrusive BT and IRKA across continuous-time and discrete-time benchmarks, while providing a unifying perspective on data-driven interpolation as a compression/distillation process. This approach enables practical MOR from available data and suggests a versatile route for data-driven model reduction in large-scale systems.

Abstract

This paper introduces a quadrature-free, data-driven approach to balanced truncation for both continuous-time and discrete-time systems. The method non-intrusively constructs reduced-order models using available transfer function samples from the right half of the -plane. It is highlighted that the proposed data-driven balanced truncation and existing quadrature-based balanced truncation algorithms share a common feature: both compress their respective data quadruplets to derive reduced-order models. Additionally, it is demonstrated that by using different compression strategies, these quadruplets can be utilized to develop three data-driven formulations of the IRKA for both continuous-time and discrete-time systems. These formulations non-intrusively generate reduced models using transfer function samples from the -axis or the right half of the -plane, or impulse response samples. Notably, these IRKA formulations eliminate the necessity of computing new transfer function samples as IRKA iteratively updates the interpolation points. The efficacy of the proposed algorithms is validated through numerical examples, which show that the proposed data-driven approaches perform comparably to their intrusive counterparts.

Paper Structure

This paper contains 35 sections, 2 theorems, 117 equations, 13 figures, 8 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Review of Interpolation Theory beattie2017model
  4. Iterative Rational Krylov Algorithm (IRKA) gugercin2008h_2
  5. Pseudo-optimal Rational Krylov (PORK) Algorithm wolf2014h
  6. Interpolatory Loewner framework mayo2007framework
  7. Balanced Truncation (BT) moore1981principal
  8. Data-driven Quadrature-based Balanced Truncation (QuadBT)goseaQuad
  9. Low-rank ADI-based Non-intrusive Balanced Truncation for Continuous-time Systems
  10. Avoiding the Computation of $Q_s$ and $P_s$
  11. Extension to Tangential-ADI Method
  12. DD-ADI-BT via Input-Output Time-domain data
  13. Data-driven Implementations of IRKA for Continuous-time Systems
  14. Using Available Frequency Response Data
  15. Using Available Impulse Response Data
  16. ...and 20 more sections

Key Result

Theorem 6.1

By setting $E_r = I$ and $B_r = \bar{Q}_s^{-1}\bar{L}_b^*$, the following properties hold:

Figures (13)

  • Figure 1: Working Principle
  • Figure 2: Comparison of Hankel singular values
  • Figure 3: Comparison of relative error $\frac{||G(s)-G_r(s)||_{\mathcal{H}_\infty}}{||G(s)||_{\mathcal{H}_\infty}}$
  • Figure 4: Frequency responses of $G(s)$ and $G_r(s)$
  • Figure 5: Comparison of Hankel singular values
  • ...and 8 more figures

Theorems & Definitions (4)

  • Theorem 6.1
  • proof
  • Theorem 6.2
  • proof