Consistent spectral approximation of Koopman operators using resolvent compactification
Year: 2024
Type: article
Abstract: Koopman operators and transfer operators represent dynamical systems through their induced linear action on vector spaces of observables, enabling the use of operator-theoretic techniques to analyze nonlinear dynamics in state space. The extraction of approximate Koopman or transfer operator eigenfunctions (and the associated eigenvalues) from an unknown system is nontrivial, particularly if the system has mixed or continuous spectrum. In this paper, we describe a spectrally accurate approach to approximate the Koopman operator on L 2 for measure-preserving, continuous-time systems via a ‘compactification’ of the resolvent of the generator. This approach employs kernel integral operators to approximate the skew-adjoint Koopman generator by a family of skew-adjoint operators with compact resolvent, whose spectral measures converge in a suitable asymptotic limit, and whose eigenfunctions are approximately periodic. Moreover, we develop a data-driven formulation of our approach, utilizing data sampled on dynamical trajectories and associated dictionaries of kernel eigenfunctions for operator approximation. The data-driven scheme is shown to converge in the limit of large training data under natural assumptions on the dynamical system and observation modality. We explore applications of this technique to dynamical systems on tori with pure point spectra and the Lorenz 63 system as an example with mixing dynamics.
Source: Nonlinearity
Authors: Dimitrios Giannakis, Claire Valva
Language: English
Subfield: Statistical and Nonlinear Physics
Field: Physics and Astronomy
Domain: Physical Sciences
Open Access status: hybrid
Estimated APC paid: $3,165
OpenAlex ID: W4399333359