---
_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."
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. arXiv:230108637. Published online
2023.
apa: Philipp, F., Schaller, M., Worthmann, K., Peitz, S., & Nüske, F. (2023).
Error bounds for kernel-based approximations of the Koopman operator. In arXiv:2301.08637.
bibtex: '@article{Philipp_Schaller_Worthmann_Peitz_Nüske_2023, title={Error bounds
for kernel-based approximations of the Koopman operator}, journal={arXiv:2301.08637},
author={Philipp, Friedrich and Schaller, Manuel and Worthmann, Karl and Peitz,
Sebastian and Nüske, Feliks}, year={2023} }'
chicago: Philipp, Friedrich, Manuel Schaller, Karl Worthmann, Sebastian Peitz, and
Feliks Nüske. “Error Bounds for Kernel-Based Approximations of the Koopman Operator.”
ArXiv:2301.08637, 2023.
ieee: F. Philipp, M. Schaller, K. Worthmann, S. Peitz, and F. Nüske, “Error bounds
for kernel-based approximations of the Koopman operator,” arXiv:2301.08637.
2023.
mla: Philipp, Friedrich, et al. “Error Bounds for Kernel-Based Approximations of
the Koopman Operator.” ArXiv:2301.08637, 2023.
short: F. Philipp, M. Schaller, K. Worthmann, S. Peitz, F. Nüske, ArXiv:2301.08637
(2023).
date_created: 2023-01-23T07:03:39Z
date_updated: 2023-01-23T07:04:12Z
department:
- _id: '655'
external_id:
arxiv:
- '2301.08637'
language:
- iso: eng
main_file_link:
- open_access: '1'
url: https://arxiv.org/pdf/2301.08637
oa: '1'
publication: arXiv:2301.08637
status: public
title: Error bounds for kernel-based approximations of the Koopman operator
type: preprint
user_id: '47427'
year: '2023'
...