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

TL;DR

This paper introduces a data-driven spectral analysis method for the Koopman operator using pseudo-resolvent construction, effectively reducing spectral pollution and accurately resolving spectral components in dynamical systems.

Contribution

The paper proposes a novel pseudo-resolvent based approach for Koopman spectral analysis, with proven convergence and error bounds, improving upon existing methods like EDMD.

Findings

Effectively suppresses spectral pollution

Accurately resolves closely spaced spectral components

Demonstrates convergence and error bounds in numerical experiments

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.