Koopman Operator Spectrum for Random Dynamical Systems
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Koopman Operator Spectrum for Random Dynamical Systems ˇ Nelida Crnjari´ c-Žic1 · Senka Ma´ceši´c1 · Igor Mezi´c2 Received: 23 November 2017 / Accepted: 14 August 2019 © Springer Science+Business Media, LLC, part of Springer Nature 2019
Abstract In this paper, we consider the Koopman operator associated with the discrete and the continuous-time random dynamical system (RDS). We provide results that characterize the spectrum and the eigenfunctions of the stochastic Koopman operator associated with different types of linear RDS. Then we consider the RDS for which the associated Koopman operator family is a semigroup, especially those for which the generator can be determined. We define a stochastic Hankel–DMD algorithm for numerical approximations of the spectral objects (eigenvalues, eigenfunctions) of the stochastic Koopman operator and prove its convergence. We apply the methodology to a variety of examples, revealing objects in spectral expansions of the stochastic Koopman operator and enabling model reduction. Keywords Stochastic Koopman operator · Random dynamical systems · Stochastic differential equations · Dynamic mode decomposition Mathematics Subject Classification 37H10 · 47B33 · 37M99 · 65P99
Communicated by Alain Goriely.
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ˇ Nelida Crnjari´ c-Žic [email protected] Senka Ma´ceši´c [email protected] Igor Mezi´c [email protected]
1
Faculty of Engineering, University of Rijeka, Vukovarska 58, 51000 Rijeka, Croatia
2
Faculty of Mechanical Engineering and Mathematics, University of California, Santa Barbara, Santa Barbara, CA 93106, USA
123
Journal of Nonlinear Science
1 Introduction Prediction and control of the evolution of large complex dynamical systems is a modern-day science and engineering challenge. Some dynamical systems can be modeled well enough by using the standard mathematical tools, such as differential or integral calculus. In these cases the simplifications of the system are typically introduced by neglecting some of the phenomena with a small impact on the behavior of the system. However, there are many dynamical systems for which the mathematical model is too complex or even does not exist, but data can be obtained by monitoring some observables of the system. In this context, data-driven analysis methods need to be developed, including techniques to extract simpler, representative components of the process that can be used for modeling and prediction. One approach for decomposing the complex systems into simpler structures is via the spectral decomposition of the associated Koopman operator. Koopman operator was introduced in Koopman (1931) in the measure-preserving setting, as a composition operator acting on the Hilbert space of square-integrable functions. The increased interest in the spectral operator-theoretic approach to dynamical systems in last decade starts with the works Mezi´c and Banaszuk (2004) and Mezi´c (2005) (see also the earlier Mezi´c and Banaszuk 2000), where the problem of decomposing the evolution of an ergodic dissipative dynamical system from the
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