On Numerical Approximations of the Koopman Operator
We study numerical approaches to computation of spectral properties of composition operators. We provide a characterization of Koopman Modes in Banach spaces using Generalized Laplace Analysis. We cast the Dynamic Mode Decomposition-type methods in the context of Finite Section theory of infinite di...
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| Format: | Article |
| Language: | English |
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MDPI AG
2022-04-01
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| Series: | Mathematics |
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| Online Access: | https://www.mdpi.com/2227-7390/10/7/1180 |
| _version_ | 1797438430482792448 |
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| author | Igor Mezić |
| author_facet | Igor Mezić |
| author_sort | Igor Mezić |
| collection | DOAJ |
| description | We study numerical approaches to computation of spectral properties of composition operators. We provide a characterization of Koopman Modes in Banach spaces using Generalized Laplace Analysis. We cast the Dynamic Mode Decomposition-type methods in the context of Finite Section theory of infinite dimensional operators, and provide an example of a mixing map for which the finite section method fails. Under assumptions on the underlying dynamics, we provide the first result on the convergence rate under sample size increase in the finite-section approximation. We study the error in the Krylov subspace version of the finite section method and prove convergence in pseudospectral sense for operators with pure point spectrum. Since Krylov sequence-based approximations can mitigate the curse of dimensionality, this result indicates that they may also have low spectral error without an exponential-in-dimension increase in the number of functions needed. |
| first_indexed | 2024-03-09T11:37:47Z |
| format | Article |
| id | doaj.art-14751be089064ec698731a7d90405210 |
| institution | Directory Open Access Journal |
| issn | 2227-7390 |
| language | English |
| last_indexed | 2024-03-09T11:37:47Z |
| publishDate | 2022-04-01 |
| publisher | MDPI AG |
| record_format | Article |
| series | Mathematics |
| spelling | doaj.art-14751be089064ec698731a7d904052102023-11-30T23:38:20ZengMDPI AGMathematics2227-73902022-04-01107118010.3390/math10071180On Numerical Approximations of the Koopman OperatorIgor Mezić0Mechanical Engineering and Mathematics, University of California, Santa Barbara, CA 93106, USAWe study numerical approaches to computation of spectral properties of composition operators. We provide a characterization of Koopman Modes in Banach spaces using Generalized Laplace Analysis. We cast the Dynamic Mode Decomposition-type methods in the context of Finite Section theory of infinite dimensional operators, and provide an example of a mixing map for which the finite section method fails. Under assumptions on the underlying dynamics, we provide the first result on the convergence rate under sample size increase in the finite-section approximation. We study the error in the Krylov subspace version of the finite section method and prove convergence in pseudospectral sense for operators with pure point spectrum. Since Krylov sequence-based approximations can mitigate the curse of dimensionality, this result indicates that they may also have low spectral error without an exponential-in-dimension increase in the number of functions needed.https://www.mdpi.com/2227-7390/10/7/1180koopman operatornumerical analysisdynamical systems |
| spellingShingle | Igor Mezić On Numerical Approximations of the Koopman Operator Mathematics koopman operator numerical analysis dynamical systems |
| title | On Numerical Approximations of the Koopman Operator |
| title_full | On Numerical Approximations of the Koopman Operator |
| title_fullStr | On Numerical Approximations of the Koopman Operator |
| title_full_unstemmed | On Numerical Approximations of the Koopman Operator |
| title_short | On Numerical Approximations of the Koopman Operator |
| title_sort | on numerical approximations of the koopman operator |
| topic | koopman operator numerical analysis dynamical systems |
| url | https://www.mdpi.com/2227-7390/10/7/1180 |
| work_keys_str_mv | AT igormezic onnumericalapproximationsofthekoopmanoperator |