article published Friedrich Philipp author Manuel Schaller author Karl Worthmann author Sebastian Peitz author 474270000-0002-3389-793X Feliks Nüske author 655 department We consider the data-driven approximation of the Koopman operator for stochastic differential equations on reproducing kernel Hilbert spaces (RKHS). Our focus is on the estimation error if the data are collected from long-term ergodic simulations. We derive both an exact expression for the variance of the kernel cross-covariance operator, measured in the Hilbert-Schmidt norm, and probabilistic bounds for the finite-data estimation error. Moreover, we derive a bound on the prediction error of observables in the RKHS using a finite Mercer series expansion. Further, assuming Koopman-invariance of the RKHS, we provide bounds on the full approximation error. Numerical experiments using the Ornstein-Uhlenbeck process illustrate our results. Springer 2024 eng 2301.0863710.1016/j.acha.2024.101657 71 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> 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>. 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>. @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} } 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>. F. Philipp, M. Schaller, K. Worthmann, S. Peitz, F. Nüske, Applied and Computational Harmonic Analysis 71 (2024). 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> 380312023-01-23T07:03:39Z2024-04-11T12:41:13Z