---
_id: '38031'
abstract:
- lang: eng
  text: "We consider the data-driven approximation of the Koopman operator for\r\nstochastic
    differential equations on reproducing kernel Hilbert spaces (RKHS).\r\nOur focus
    is on the estimation error if the data are collected from long-term\r\nergodic
    simulations. We derive both an exact expression for the variance of the\r\nkernel
    cross-covariance operator, measured in the Hilbert-Schmidt norm, and\r\nprobabilistic
    bounds for the finite-data estimation error. Moreover, we derive\r\na bound on
    the prediction error of observables in the RKHS using a finite\r\nMercer series
    expansion. Further, assuming Koopman-invariance of the RKHS, we\r\nprovide bounds
    on the full approximation error. Numerical experiments using the\r\nOrnstein-Uhlenbeck
    process illustrate our results."
article_number: '101657'
author:
- first_name: Friedrich
  full_name: Philipp, Friedrich
  last_name: Philipp
- first_name: Manuel
  full_name: Schaller, Manuel
  last_name: Schaller
- first_name: Karl
  full_name: Worthmann, Karl
  last_name: Worthmann
- first_name: Sebastian
  full_name: Peitz, Sebastian
  id: '47427'
  last_name: Peitz
  orcid: 0000-0002-3389-793X
- first_name: Feliks
  full_name: Nüske, Feliks
  last_name: Nüske
citation:
  ama: Philipp F, Schaller M, Worthmann K, Peitz S, Nüske F. Error bounds for kernel-based
    approximations of the Koopman operator. <i>Applied and Computational Harmonic
    Analysis </i>. 2024;71. doi:<a href="https://doi.org/10.1016/j.acha.2024.101657">10.1016/j.acha.2024.101657</a>
  apa: Philipp, F., Schaller, M., Worthmann, K., Peitz, S., &#38; Nüske, F. (2024).
    Error bounds for kernel-based approximations of the Koopman operator. <i>Applied
    and Computational Harmonic Analysis </i>, <i>71</i>, Article 101657. <a href="https://doi.org/10.1016/j.acha.2024.101657">https://doi.org/10.1016/j.acha.2024.101657</a>
  bibtex: '@article{Philipp_Schaller_Worthmann_Peitz_Nüske_2024, title={Error bounds
    for kernel-based approximations of the Koopman operator}, volume={71}, DOI={<a
    href="https://doi.org/10.1016/j.acha.2024.101657">10.1016/j.acha.2024.101657</a>},
    number={101657}, journal={Applied and Computational Harmonic Analysis }, publisher={Springer
    }, author={Philipp, Friedrich and Schaller, Manuel and Worthmann, Karl and Peitz,
    Sebastian and Nüske, Feliks}, year={2024} }'
  chicago: Philipp, Friedrich, Manuel Schaller, Karl Worthmann, Sebastian Peitz, and
    Feliks Nüske. “Error Bounds for Kernel-Based Approximations of the Koopman Operator.”
    <i>Applied and Computational Harmonic Analysis </i> 71 (2024). <a href="https://doi.org/10.1016/j.acha.2024.101657">https://doi.org/10.1016/j.acha.2024.101657</a>.
  ieee: 'F. Philipp, M. Schaller, K. Worthmann, S. Peitz, and F. Nüske, “Error bounds
    for kernel-based approximations of the Koopman operator,” <i>Applied and Computational
    Harmonic Analysis </i>, vol. 71, Art. no. 101657, 2024, doi: <a href="https://doi.org/10.1016/j.acha.2024.101657">10.1016/j.acha.2024.101657</a>.'
  mla: Philipp, Friedrich, et al. “Error Bounds for Kernel-Based Approximations of
    the Koopman Operator.” <i>Applied and Computational Harmonic Analysis </i>, vol.
    71, 101657, Springer , 2024, doi:<a href="https://doi.org/10.1016/j.acha.2024.101657">10.1016/j.acha.2024.101657</a>.
  short: F. Philipp, M. Schaller, K. Worthmann, S. Peitz, F. Nüske, Applied and Computational
    Harmonic Analysis  71 (2024).
date_created: 2023-01-23T07:03:39Z
date_updated: 2024-04-11T12:41:13Z
department:
- _id: '655'
doi: 10.1016/j.acha.2024.101657
external_id:
  arxiv:
  - '2301.08637'
intvolume: '        71'
language:
- iso: eng
main_file_link:
- open_access: '1'
  url: https://arxiv.org/pdf/2301.08637
oa: '1'
publication: 'Applied and Computational Harmonic Analysis '
publication_status: published
publisher: 'Springer '
status: public
title: Error bounds for kernel-based approximations of the Koopman operator
type: journal_article
user_id: '47427'
volume: 71
year: '2024'
...