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A non-intrusive data-based reformulation of a hybrid projection-based model reduction method

Ion Victor Gosea, Serkan Gugercin, Christopher Beattie

TL;DR

The paper addresses non-intrusive model order reduction for large-scale LTI systems by reformulating the intrusive ISRK method into a data-driven algorithm, Quad-ISRK. It replaces explicit Gramian information with a quadrature-based Gramian approximation and constructs reduced-model data solely from transfer-function samples, preserving interpolation properties. The main contribution is a set of data-driven ROM expressions and an iterative algorithm that mirrors ISRK’s convergence behavior while avoiding access to the system matrices. Numerical results on a classical MOR benchmark show Quad-ISRK reproduces the original ISRK performance in both $H_ ext{\infty}$ and $H_2$ senses, with potential for extension to adaptive quadrature and structured/mildly nonlinear systems.

Abstract

We present a novel data-driven reformulation of the iterative SVD-rational Krylov algorithm (ISRK), in its original formulation a Petrov-Galerkin (two-sided) projection-based iterative method for model reduction combining rational Krylov subspaces (on one side) with Gramian/SVD based subspaces (on the other side). We show that in each step of ISRK, we do not necessarily require access to the original system matrices, but only to input/output data in the form of the system's transfer function, evaluated at particular values (frequencies). Numerical examples illustrate the efficiency of the new data-driven formulation.

A non-intrusive data-based reformulation of a hybrid projection-based model reduction method

TL;DR

The paper addresses non-intrusive model order reduction for large-scale LTI systems by reformulating the intrusive ISRK method into a data-driven algorithm, Quad-ISRK. It replaces explicit Gramian information with a quadrature-based Gramian approximation and constructs reduced-model data solely from transfer-function samples, preserving interpolation properties. The main contribution is a set of data-driven ROM expressions and an iterative algorithm that mirrors ISRK’s convergence behavior while avoiding access to the system matrices. Numerical results on a classical MOR benchmark show Quad-ISRK reproduces the original ISRK performance in both and senses, with potential for extension to adaptive quadrature and structured/mildly nonlinear systems.

Abstract

We present a novel data-driven reformulation of the iterative SVD-rational Krylov algorithm (ISRK), in its original formulation a Petrov-Galerkin (two-sided) projection-based iterative method for model reduction combining rational Krylov subspaces (on one side) with Gramian/SVD based subspaces (on the other side). We show that in each step of ISRK, we do not necessarily require access to the original system matrices, but only to input/output data in the form of the system's transfer function, evaluated at particular values (frequencies). Numerical examples illustrate the efficiency of the new data-driven formulation.
Paper Structure (10 sections, 3 theorems, 43 equations, 1 figure, 2 algorithms)

This paper contains 10 sections, 3 theorems, 43 equations, 1 figure, 2 algorithms.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. The system Gramians
  4. The Iterative SVD-Rational Krlyov Method (ISRK)
  5. The proposed data-driven reformulation
  6. Computation of the reduced-order matrices
  7. The proposed data-driven formulation of ISRK
  8. Numerical example: A classical MOR benchmark
  9. Conclusions and future work
  10. Appendix: proof of Lemma \ref{['lem:data_mat']}

Key Result

Corollary 3.1

Let ${\cal K}$ and $\widetilde{{\bf L}}$ as defined in eq:K and quad_L, respectively. Then, where $\boldsymbol{\Phi} = \text{diag}(\phi_1,\phi_2,\ldots,\phi_{{}N_q})$ is a diagonal matrix of quadrature weights.

Figures (1)

  • Figure 1: Relative $\mathcal{H}_2$ norms of the error systems.

Theorems & Definitions (6)

  • Definition 2.1
  • Corollary 3.1
  • Definition 3.2
  • Lemma 3.3
  • Theorem 3.4
  • Remark 3.5