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Scalable Pseudospectral Analysis via Low-Rank Approximations of Dynamical Systems

Vladimir R. Kostic, Dragana Lj. Cvetkovic, Ljiljana Cvetkovic

TL;DR

The paper addresses the high computational cost of pseudospectral analysis for large, often dense, nonnormal operators by introducing a unified low-rank framework. The core theoretical result (Theorem 1) gives an exact characterization of the $\varepsilon$-pseudospectrum of a low-rank matrix $A=UV^*$ in terms of a small $2r\times2r$ matrix, reducing resolvent-norm computations to eigenvalue problems of size proportional to the rank. It further provides inclusion guarantees for low-rank approximations (via truncated and randomized SVD) and perturbation bounds for Kreiss constants, with computational costs that scale with the effective rank rather than the ambient dimension. The framework extends naturally to data-driven transfer operators (Koopman/Perron--Frobenius) learned from trajectory data, enabling scalable pseudospectral analysis in nonlinear and stochastic dynamics, supported by numerical experiments showing orders-of-magnitude speedups.

Abstract

Pseudospectral analysis is fundamental for quantifying the sensitivity and transient behavior of nonnormal matrices, yet its computational cost scales cubically with dimension, rendering it prohibitive for large-scale systems. While existing research on scalable pseudospectral computation has focused on exploiting sparsity structures, common in discretizations of differential operators, these approaches are ill-suited for machine learning and data-driven dynamical systems, where operators are typically dense but approximately low-rank. In this paper, we develop a comprehensive low-rank framework that dramatically reduces this computational burden. Our core theoretical contribution is an exact characterization of the pseudospectrum of arbitrary low-rank matrices, reducing the evaluation of resolvent norms to eigenvalue problems of dimension proportional to the rank. Building on this foundation, we derive rigorous inclusion sets for the pseudospectra of general matrices via truncated and randomized low-rank approximations, with explicit perturbation bounds. These results enable efficient estimators for key stability quantities, including distance to instability and Kreiss constants, at a cost that scales with the effective rank rather than the ambient dimension. We further demonstrate how our framework naturally extends to data-driven settings, providing pseudospectral analysis of transfer operators learned from nonlinear and stochastic dynamical systems. Numerical experiments confirm orders-of-magnitude speedups while preserving accuracy, opening pseudospectral analysis to previously intractable high-dimensional problems in computational PDEs, control theory, and data-driven dynamics.

Scalable Pseudospectral Analysis via Low-Rank Approximations of Dynamical Systems

TL;DR

The paper addresses the high computational cost of pseudospectral analysis for large, often dense, nonnormal operators by introducing a unified low-rank framework. The core theoretical result (Theorem 1) gives an exact characterization of the -pseudospectrum of a low-rank matrix in terms of a small matrix, reducing resolvent-norm computations to eigenvalue problems of size proportional to the rank. It further provides inclusion guarantees for low-rank approximations (via truncated and randomized SVD) and perturbation bounds for Kreiss constants, with computational costs that scale with the effective rank rather than the ambient dimension. The framework extends naturally to data-driven transfer operators (Koopman/Perron--Frobenius) learned from trajectory data, enabling scalable pseudospectral analysis in nonlinear and stochastic dynamics, supported by numerical experiments showing orders-of-magnitude speedups.

Abstract

Pseudospectral analysis is fundamental for quantifying the sensitivity and transient behavior of nonnormal matrices, yet its computational cost scales cubically with dimension, rendering it prohibitive for large-scale systems. While existing research on scalable pseudospectral computation has focused on exploiting sparsity structures, common in discretizations of differential operators, these approaches are ill-suited for machine learning and data-driven dynamical systems, where operators are typically dense but approximately low-rank. In this paper, we develop a comprehensive low-rank framework that dramatically reduces this computational burden. Our core theoretical contribution is an exact characterization of the pseudospectrum of arbitrary low-rank matrices, reducing the evaluation of resolvent norms to eigenvalue problems of dimension proportional to the rank. Building on this foundation, we derive rigorous inclusion sets for the pseudospectra of general matrices via truncated and randomized low-rank approximations, with explicit perturbation bounds. These results enable efficient estimators for key stability quantities, including distance to instability and Kreiss constants, at a cost that scales with the effective rank rather than the ambient dimension. We further demonstrate how our framework naturally extends to data-driven settings, providing pseudospectral analysis of transfer operators learned from nonlinear and stochastic dynamical systems. Numerical experiments confirm orders-of-magnitude speedups while preserving accuracy, opening pseudospectral analysis to previously intractable high-dimensional problems in computational PDEs, control theory, and data-driven dynamics.
Paper Structure (9 sections, 7 theorems, 54 equations, 4 figures)

This paper contains 9 sections, 7 theorems, 54 equations, 4 figures.

Table of Contents

  1. Introduction
  2. The problem, background and preliminaries
  3. Transient dynamics and robustness of stability.
  4. Transfer operators of ergodic processes.
  5. Machine learning methods for transfer operators.
  6. Pseudospectrum of low-rank matrices
  7. Low rank approximation of pseudospectral properties
  8. Nonlinear stochastic dynamics
  9. Concluding remarks

Key Result

Theorem 3.1

Given $d>r\geq1$, and any $U,V\in\mathbb{C}^{d,r}$, let Then, for all $z\in\mathbb C$, $\Lambda_{}(\mathcal{M}_{U,V}(z))\subset\mathbb R_{+}$ and $\mu_{U,V}(z):=\sqrt{\lambda_{min} (\mathcal{M}_{U,V}(z))} = \sigma_{\min}(z I- UV^*)$, and, consequentially, the following characterization of the ${\varepsilon}$-pseudospectrum of $A=UV^*$ holds true: $\Lambd

Figures (4)

  • Figure 1: Results for the Example \ref{['ex:toy_1']}. Left panel: pseudospectrum of a $100\times 100$ rank 10 matrix with its localization set based on rank $\ell=5$ truncated SVD. The test for the intersection (magenta markers) is performed by applying Proposition \ref{['prop:d2i']}. Right pannel: Comparison of the computational time speedup (logarithm of the CPU time ratio between standard computation and the low-rank version) for various dimensions and ranks over 10 random trials.
  • Figure 1: Pseudospectrum of a rank 21 integral operator of Example \ref{['ex:logistic']} and its localization set based on rank $r=7$ truncated SVD. On the left is the ground truth computed numerically, while on the right is the result obtained by learning from data. The test for the intersection (magenta markers) is performed by applying Proposition \ref{['prop:d2i']}.
  • Figure 2: Estimated pseudospectrum of the transfer operator for the normal Ornstein–Uhlenbeck process of Example \ref{['ex:ou']} via Theorem \ref{['thm:ps_rrr']}. The drift is a symmetric matrix with entries $a_{11}=a_{22}=-0.7$ and $a_{12}=a_{21}=0.3$, the RRR estimator \ref{['eq:rrr']} is learned from $n=10^4$ samples, with rank $r= 20$, Tikhnov regularization $\gamma=10^{-6}$ and the Gaussian kernel. In the top row we show the pseudospectrum and Kreiss constant of $\widehat{G}_{\gamma,r}\colon\mathcal{H}\to\mathcal{H}$ computed via \ref{['eq:rkhs_lrps']}, while in the bottom row we show the $\mathcal{L}^2_\pi$ estimation obtained by \ref{['eq:l2_lrps']}. On the left we see pseudospectrum and on the right the values over which Kreiss constant is obtained as the maximum.
  • Figure 3: Estimated pseudospectrum of the transfer operator for the nonnormal Ornstein–Uhlenbeck process of Example \ref{['ex:ou']} via Theorem \ref{['thm:ps_rrr']}. The drift is a non-normal matrix with entries $a_{11}=a_{22}=-0.7$, $a_{12}=100$ and $a_{21}=-0.1$. In the top row we show the pseudospectrum and Kreiss constant of $\widehat{G}_{\gamma,r}\colon\mathcal{H}\to\mathcal{H}$ computed via \ref{['eq:rkhs_lrps']}, while in the bottom row we show the $\mathcal{L}^2_\pi$ estimation obtained by \ref{['eq:l2_lrps']}. On the left we see pseudospectrum and on the right the values over which Kreiss constant is obtained as the maximum.

Theorems & Definitions (15)

  • Theorem 3.1
  • Proof 1
  • Proof 2
  • Proposition 3.3
  • Proof 3
  • Proposition 3.4
  • Proof 4
  • Lemma 4.1: TE2020
  • Theorem 4.2
  • Lemma 4.3: Kostic-ICML2024
  • ...and 5 more