Corpus ID: 235265875

Anti-Koopmanism

@article{Gonzlez2021AntiKoopmanism,
  title={Anti-Koopmanism},
  author={Efra{\'i}n Gonz{\'a}lez and Moad Abudia and Michael T. Jury and Rushikesh Kamalapurkar and Joel A. Rosenfeld},
  journal={ArXiv},
  year={2021},
  volume={abs/2106.00106}
}
This article addresses several longstanding misconceptions concerning Koopman operators, including the existence of lattices of eigenfunctions, common eigenfunctions between Koopman operators, and boundedness and compactness of Koopman operators, among others. Counterexamples are provided for each misconception. This manuscript also proves that the Gaussian RBF’s native space only supports bounded Koopman operator corresponding to affine dynamics, which shows that the assumption of boundedness… Expand
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References

SHOWING 1-10 OF 33 REFERENCES
On the numerical approximation of the Perron-Frobenius and Koopman operator
TLDR
Different methods that have been developed over the last decades to compute finite-dimensional approximations of infinite-dimensional operators - e.g. Ulam's method and Extended Dynamic Mode Decomposition (EDMD) - are reviewed. Expand
Global Stability Analysis Using the Eigenfunctions of the Koopman Operator
  • A. Mauroy, I. Mezić
  • Mathematics, Computer Science
  • IEEE Transactions on Automatic Control
  • 2016
TLDR
The main results establish the (necessary and sufficient) relationship between the existence of specific eigenfunctions of the Koopman operator and the global stability property of hyperbolic fixed points and limit cycles. Expand
Applied Koopmanism.
TLDR
The Koopman framework is showing potential for crossing over from academic and theoretical use to industrial practice, and the paper highlights its strengths in applied and numerical contexts. Expand
Learning Koopman Invariant Subspaces for Dynamic Mode Decomposition
TLDR
This paper proposes minimization of the residual sum of squares of linear least-squares regression to estimate a set of functions that transforms data into a form in which the linear regression fits well. Expand
A kernel-based method for data-driven koopman spectral analysis
TLDR
A data-driven, kernel-based method for approximating the leading Koopman eigenvalues, eigenfunctions, and modes in problems with high-dimensional state spaces is presented, using a set of scalar observables that are defined implicitly by the feature map associated with a user-defined kernel function. Expand
Pseudogenerators of Spatial Transfer Operators
TLDR
It is shown that even though the family of spatial transfer operators is not a semigroup, it possesses a well-defined generating structure and makes collocation methods particularly easy to implement and computationally efficient, which in turn may open the door for furt... Expand
A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition
TLDR
This approach is an extension of dynamic mode decomposition (DMD), which has been used to approximate the Koopman eigenvalues and modes, and if the data provided to the method are generated by a Markov process instead of a deterministic dynamical system, the algorithm approximates the eigenfunctions of the Kolmogorov backward equation. Expand
Koopman Invariant Subspaces and Finite Linear Representations of Nonlinear Dynamical Systems for Control
TLDR
This work presents a data-driven strategy to identify relevant observable functions for Koopman analysis by leveraging a new algorithm to determine relevant terms in a dynamical system by ℓ1-regularized regression of the data in a nonlinear function space and demonstrates the usefulness of nonlinear observable subspaces in the design of Koop man operator optimal control laws for fully nonlinear systems using techniques from linear optimal control. Expand
Composition Operators: And Classical Function Theory
The study of composition operators forges links between fundamental properties of linear operators and results from the classical theory of analytic functions. This book provides a self-containedExpand
Spectral Properties of Dynamical Systems, Model Reduction and Decompositions
In this paper we discuss two issues related to model reduction of deterministic or stochastic processes. The first is the relationship of the spectral properties of the dynamics on the attractor ofExpand
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