--- _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' ...