Time-Domain Iterative Rational Krylov Method

TL;DR

This work develops TD-IRKA, a time-domain data-driven approach to construct $ abla_{H_2}$-optimal DDROMs from a single trajectory, circumventing the need for repeated frequency data evaluations required by TF-IRKA. It builds a conditioning-aware data-informativity framework, deriving explicit formulas for the extreme eigenvalues of rank-1 perturbations to orthogonal projections to minimize linear-system ill-conditioning during frequency data recovery. The method introduces an optimal z-scaling and an overflow-avoidance strategy, enabling reliable recovery of $H(\sigma)$ and $H'(\sigma)$ from time-domain data for $|\sigma|>1$, and then uses those values to perform a TD-IRKA iteration that yields high-fidelity DDROMs. Numerical experiments on linear advection, ISS, and a heat PDE demonstrate that TD-IRKA matches or closely tracks TF-IRKA performance while requiring only a single time-domain simulation, including a black-box PDE case, supporting practical relevance for large-scale, data-driven model reduction.

Abstract

The Realization Independent Iterative Rational Krylov Algorithm (TF-IRKA) is a frequency-based data-driven reduced order modeling (DDROM) method that constructs $\mathcal H_2$ optimal DDROMs. However, as the $\mathcal H_2$ optimal approximation theory dictates, TF-IRKA requires repeated sampling of frequency data, that is, values of the system transfer function and its derivative, outside the unit circle. This repeated evaluation of frequency data requires repeated full model computations and may not be feasible. The data-informativity framework for moment matching provides a method for obtaining such frequency data from a single time-domain simulation. However, this framework usually requires solving linear systems with prohibitively ill-conditioned matrices, especially when recovering frequency data from off the unit circle as required for optimality. In this paper, building upon our previous work with the data informativity framework for moment matching, we provide a formula for the nonzero extreme eigenvalues of a symmetric rank-$1$ perturbation to an orthogonal projection, which then leads to an optimal scaling of the aforementioned linear systems. We also establish connections between the underlying dynamical system and conditioning of these linear systems. This analysis then leads to our algorithmic development, time-domain IRKA, which allows us to implement a time-domain variant of TF-IRKA, constructing $\mathcal H_2$ optimal DDROMs from a single time-domain simulation without requiring repeated frequency data evaluations. The numerical examples illustrate the effectiveness of the proposed algorithm.

Figures (7)

• Figure 1: Condition number of $\mathbf{U}\mathbf{z}(\sigma)$ grows rapidly with $|\sigma|$
• Figure 1: Errors $\epsilon_1(\hat{n})$ and $\epsilon_d(\hat{n})$ dependence on $\hat{n}$
• Figure 1: Relative $\mathcal{H}_2$ errors of TF-IRKA and TD-IRKA DDROMs approximating $H_2$.
• Figure 2: Condition number of $\mathbf{U}\hat{\mathbf{z}}(\sigma)$ as $|\sigma|$ increases.
• Figure 2: Relative $\mathcal{H}_2$ errors of TF-IRKA and TD-IRKA DDROMs approximating $H_1$.

Theorems & Definitions (15)

• Theorem 2.1
• Remark 2.2
• Lemma 2.3
• Remark 2.4
• Theorem 3.1: Weyl's theorem SymetricEigenProb_BoundsweylAsymptotischeVerteilungsgesetzEigenwerte1912
• Lemma 3.2
• Proof 1
• Theorem 3.3
• Proof 2
• Remark 3.4