Abstract
The spectral theory of linear operators enables the analysis of their properties on stable subspaces. The Koopman operator allows to extend these approaches to a large class of nonlinear operators in a surprising way. This is even applicable for numerical analysis of time dependent data of simulations and measurements. We give here some remarks on the numerical approach, link it to spectral analysis by the Herglotz-Bochner theorem and are doing some steps for significance for partial differential equations.
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References
Bellow, A., Losert, V.: The weighted pointwise ergodic theorem and the individual ergodic theorem along subsequences. Trans. Am. Math. Soc. 288(1), 307–345 (1985). https://doi.org/10.1090/S0002-9947-1985-0773063-8
Besicovitch, A.S.: On generalized almost periodic functions. Proc. Lond. Math. Soc. s2-25, 495–512 (1926). doi:10.1112/plms/s2-25.1.495
Budišić, M., Mohr, R., Mezić, I.: Applied Koopmanism. Chaos 22, 047510 (2012). doi:10.1063/1.4772195. http://dx.doi.org/10.1063/1.4772195
Chen, K.K., Tu, J.H., Rowley, C.W.: Variants of dynamic mode decomposition: boundary condition, Koopman, and Fourier analyse. J. Nonlinear Sci. 22(6), 887–915 (2012)
Eisner, T., Farkas, B., Haase, M., Nagel, R.: Operator Theoretic Aspects of Ergodic Theory. Graduate Texts in Mathematics. Springer, Cham (2015)
Koopman, B.O.: Hamiltonian systems and transformations in Hilbert space. Proc. Natl. Acad. Sci. U. S. A. 17(5), 315–318 (1931)
Küster, U.: The spectral structure of a nonlinear operator and its approximation. In: Sustained Simulation Performance 2015: Proceedings of the Joint Workshop on Sustained Simulation Performance, University of Stuttgart (HLRS) and Tohoku University, pp. 109–123. Springer, Cham (2015) ISBN:978-3-319-20340-9. doi:10.1007/978-3-319-20340-9_9
Küster, U.: The spectral structure of a nonlinear operator and its approximation II. In: Sustained Simulation Performance 2016: Proceedings of the Joint Workshop on Sustained Simulation Performance, University of Stuttgart (HLRS) and Tohoku University (2016). ISBN:978-3-319-46735-1
Mirzaee, H., Henn, T., Krause, M. J., Goubergrits, L., Schumann, C., Neugebauer, M., Kuehne, T., Preusser, T., Hennemuth, A.: MRI-based computational hemodynamics in patients with aortic coarctation using the lattice Boltzmann methods: clinical validation study. J. Magn. Reson. Imaging 45(1), 139–146 (2016). doi:10.1002/jmri.25366
Rellich, F.: Störungstheorie der Spektralzerlegung I., Analytische Störung der isolierten Punkteigenwerte eines beschränkten Operators. Math. Ann. 113, 600–619 (1937)
Ruopp, A., Schneider, R., MRI-based computational hemodynamics in patients. In: Resch, M.M., Bez, W., Focht, E. (eds.) Sustained Simulation Performance 2017 (abbrev. WSSP 2017). Springer, Cham (2017). doi:10.1007/978-3-319-66896-3
Schmid, P.J.: Dynamic mode decomposition of numerical and experimental data. J. Fluid Mech. 656, 24 (2010)
Schröder, E.: Ueber iterirte Functionen. Math. Ann. 3(2), 296–322 (1870). doi:10.1007/BF01443992
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Küster, U., Schneider, R., Ruopp, A. (2017). The Numerical Approximation of Koopman Modes of a Nonlinear Operator Along a Trajectory. In: Resch, M., Bez, W., Focht, E., Gienger, M., Kobayashi, H. (eds) Sustained Simulation Performance 2017 . Springer, Cham. https://doi.org/10.1007/978-3-319-66896-3_3
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