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Data-Driven Spectral Analysis Through Pseudo-Resolvent Koopman Operator in Dynamical Systems

Yuanchao Xu, Itsushi Sakata, Isao Ishikawa

TL;DR

This work develops a data-driven framework for spectral analysis of the Koopman operator by directly constructing a pseudo-resolvent from time-series data using the Sherman–Morrison–Woodbury identity. The method yields a resolvent-based spectral indicator that suppresses spectral pollution and provides pseudoeigenfunctions as directions of maximal amplification. The authors establish Hausdorff convergence of the approximate spectrum for isolated eigenvalues, preserve algebraic multiplicities under strong stability, and derive computable error bounds for eigenvalues. Numerical experiments on pendulum, Lorenz, and coupled oscillators demonstrate improved spectral localization and robust separation of closely spaced spectral components, highlighting practical benefits for data-driven dynamical analysis.

Abstract

We present a data-driven method for spectral analysis of the Koopman operator based on direct construction of the pseudo-resolvent from time-series data. Finite-dimensional approximation of the Koopman operator, such as those obtained from Extended Dynamic Mode Decomposition, are known to suffer from spectral pollution. To address this issue, we construct the pseudo-resolvent operator using the Sherman-Morrison-Woodbury identity whose norm serves as a spectral indicator, and pseudoeigenfunctions are extracted as directions of maximal amplification. We establish convergence of the approximate spectrum to the true spectrum in the Hausdorff metric for isolated eigenvalues, with preservation of algebraic multiplicities, and derive error bounds for eigenvalue approximation. Numerical experiments on pendulum, Lorenz, and coupled oscillator systems demonstrate that the method effectively suppresses spectral pollution and resolves closely spaced spectral components.

Data-Driven Spectral Analysis Through Pseudo-Resolvent Koopman Operator in Dynamical Systems

TL;DR

This work develops a data-driven framework for spectral analysis of the Koopman operator by directly constructing a pseudo-resolvent from time-series data using the Sherman–Morrison–Woodbury identity. The method yields a resolvent-based spectral indicator that suppresses spectral pollution and provides pseudoeigenfunctions as directions of maximal amplification. The authors establish Hausdorff convergence of the approximate spectrum for isolated eigenvalues, preserve algebraic multiplicities under strong stability, and derive computable error bounds for eigenvalues. Numerical experiments on pendulum, Lorenz, and coupled oscillators demonstrate improved spectral localization and robust separation of closely spaced spectral components, highlighting practical benefits for data-driven dynamical analysis.

Abstract

We present a data-driven method for spectral analysis of the Koopman operator based on direct construction of the pseudo-resolvent from time-series data. Finite-dimensional approximation of the Koopman operator, such as those obtained from Extended Dynamic Mode Decomposition, are known to suffer from spectral pollution. To address this issue, we construct the pseudo-resolvent operator using the Sherman-Morrison-Woodbury identity whose norm serves as a spectral indicator, and pseudoeigenfunctions are extracted as directions of maximal amplification. We establish convergence of the approximate spectrum to the true spectrum in the Hausdorff metric for isolated eigenvalues, with preservation of algebraic multiplicities, and derive error bounds for eigenvalue approximation. Numerical experiments on pendulum, Lorenz, and coupled oscillator systems demonstrate that the method effectively suppresses spectral pollution and resolves closely spaced spectral components.
Paper Structure (12 sections, 7 theorems, 41 equations, 7 figures)

This paper contains 12 sections, 7 theorems, 41 equations, 7 figures.

Table of Contents

  1. Introduction
  2. Preliminary
  3. Koopman Operator
  4. Resolvent and Pseudo-Resolvent Operator
  5. A Data-Driven Computational Approach for Pseudo-Resolvent Estimation
  6. Convergence Analysis of Spectrum
  7. Error Bounds Analysis
  8. Experiments
  9. Pendulum
  10. Comparing pseudospectra contour
  11. Clustering two oscillators with thermal noise
  12. Conclusion

Key Result

Lemma 4.6

Let $U$ be a compact subset of the resolvent set $\rho(\mathcal{K})$. If Eq.eq:stable_cvg from Assumption ass:stable_cvg holds for all $z \in U$, then there exists $N_U \in \mathbb{N}$ and a constant $M_U < \infty$ such that

Figures (7)

  • Figure 1: As $z$ approaches the unit circle, the value of $1/||\mathbf{R}(z, \mathbf{K}_N)||$ decreases.
  • Figure 2: Effect of increasing the dictionary size $N$ (number of basis functions) on the detection of spectral points.
  • Figure 3: Evaluation of $1/\|\mathbf{R}(x, \mathbf{A}_N)\|$ at $x = \pm 0.01$.
  • Figure 4: Pseudospectral contours computed by ResDMD (left) and Resolvent DMD (right) for the Lorenz system. The contours represent the level sets of the inverse resolvent norm, $1/\|\mathbf{R}_N(z)\|$. The pink dots indicate the eigenvalues computed by standard EDMD. Resolvent DMD demonstrates sharper localization along the real axis with significantly reduced spectral pollution compared to the distortions observed in ResDMD.
  • Figure 5: Clustering in the embedded feature space for ResDMD (left) and Resolvent DMD (right). Resolvent DMD produces three clearly separated clusters, whereas for ResDMD the regions corresponding to Cluster 2 and Cluster 3 lie close together with a blurred boundary.
  • ...and 2 more figures

Theorems & Definitions (26)

  • Remark 3.1
  • Remark 3.2
  • Definition 4.1
  • Remark 4.3
  • Definition 4.4
  • Lemma 4.6
  • proof
  • Lemma 4.7: chaitin1983spectral
  • Remark 4.8
  • Corollary 4.9
  • ...and 16 more