Avoiding spectral pollution for transfer operators using residuals

2025-07-22

AI Summary

Finite-dimensional approximations of the Koopman and Frobenius–Perron operators for nonlinear dynamical systems often suffer from spectral pollution—i.e., spurious eigenvalues—due to subspace truncation. Method: This paper introduces a rigorous, residual-based convergence framework that extends pollution-free numerical spectral theory to general transfer operators, specifically within Hardy–Hilbert spaces. It establishes that even when true eigenfunctions lie outside the approximation subspace, their spectral information is robustly encoded in the residual. Contribution/Results: The method guarantees spectral convergence without spurious modes. Numerical experiments on the Blaschke map family and a molecular dynamics model of protein folding demonstrate substantial improvements in both eigenvalue and eigenfunction estimation accuracy and robustness, validating theoretical claims and broadening applicability to high-dimensional, non-equilibrium systems.

Technology Category

• Machine Learning: Matrix & Tensor Methods
• Intelligent Robots: State Estimation
• Search and Optimization: Non-convex Optimization

Application Category

• Web Mining and Content Analysis: Robustness and generalizability of Web mining methods
• Graph Algorithms and Modeling for the Web: Efficient manipulation of static and dynamic Web-related graphs
• Economics, Online Markets and Human Computation: Sustainability of Web economics

Abstract

Koopman operator theory enables linear analysis of nonlinear dynamical systems by lifting their evolution to infinite-dimensional function spaces. However, finite-dimensional approximations of Koopman and transfer (Frobenius--Perron) operators are prone to spectral pollution, introducing spurious eigenvalues that can compromise spectral computations. While recent advances have yielded provably convergent methods for Koopman operators, analogous tools for general transfer operators remain limited. In this paper, we present algorithms for computing spectral properties of transfer operators without spectral pollution, including extensions to the Hardy-Hilbert space. Case studies--ranging from families of Blaschke maps with known spectrum to a molecular dynamics model of protein folding--demonstrate the accuracy and flexibility of our approach. Notably, we demonstrate that spectral features can arise even when the corresponding eigenfunctions lie outside the chosen space, highlighting the functional-analytic subtleties in defining the "true" Koopman spectrum. Our methods offer robust tools for spectral estimation across a broad range of applications.

Problem

• Avoid spectral pollution in transfer operator computations
• Extend methods to Hardy-Hilbert space for accuracy
• Address functional-analytic subtleties in Koopman spectrum definition

Innovation

• Avoiding spectral pollution in transfer operators
• Using residuals for accurate spectral computations
• Extending methods to Hardy-Hilbert space