{"intvolume":"        71","volume":71,"_id":"38031","status":"public","type":"journal_article","publication":"Applied and Computational Harmonic Analysis ","publisher":"Springer ","publication_status":"published","user_id":"47427","article_number":"101657","date_updated":"2024-04-11T12:41:13Z","language":[{"iso":"eng"}],"title":"Error bounds for kernel-based approximations of the Koopman operator","year":"2024","author":[{"last_name":"Philipp","first_name":"Friedrich","full_name":"Philipp, Friedrich"},{"first_name":"Manuel","full_name":"Schaller, Manuel","last_name":"Schaller"},{"full_name":"Worthmann, Karl","first_name":"Karl","last_name":"Worthmann"},{"id":"47427","last_name":"Peitz","orcid":"0000-0002-3389-793X","full_name":"Peitz, Sebastian","first_name":"Sebastian"},{"last_name":"Nüske","first_name":"Feliks","full_name":"Nüske, Feliks"}],"department":[{"_id":"655"}],"date_created":"2023-01-23T07:03:39Z","doi":"10.1016/j.acha.2024.101657","abstract":[{"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.","lang":"eng"}],"main_file_link":[{"open_access":"1","url":"https://arxiv.org/pdf/2301.08637"}],"external_id":{"arxiv":["2301.08637"]},"citation":{"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>.","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>.","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>","short":"F. Philipp, M. Schaller, K. Worthmann, S. Peitz, F. Nüske, Applied and Computational Harmonic Analysis  71 (2024).","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>","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>."},"oa":"1"}