Dynamic mode decomposition for analytic maps

Slipantschuk, Julia; Bandtlow, Oscar F.; Just, Wolfram

doi:10.1016/j.cnsns.2020.105179

Cited by 10 publications

(8 citation statements)

References 37 publications

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“…In the case where Ω is the unit circle in R 2 , Ref. [31] shows that an adequate choice of a function space (namely, Hilbert-Hardy spaces) leads to interesting spectral results. Indeed, by restricting the function space to a class of analytical functions, the adjoint of the Koopman operator (the Perron-Frobenius operator) becomes compact.…”

Section: Koopman Operator and Data-driven Modellingmentioning

confidence: 99%

A quantitative analysis of Koopman operator methods for system identification and predictions

Zhang¹,

Zuazua²

2024

Comptes Rendus. Mécanique

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We give convergence and cost estimates for a data-driven system identification method: given an unknown dynamical system, the aim is to recover its vector field and its flow from trajectory data. It is based on the so-called Koopman operator, which uses the well-known link between differential equations and linear transport equations. Data-driven methods recover specific finite-dimensional approximations of the Koopman operator, which can be understood as a transport operator. We focus on such approximations given by classical finite element spaces, which allow us to give estimates on the approximation of the Koopman operator as well as the solutions of the associated linear transport equation. These approximations are thus relevant objects to solve the system identification problem.We then analyze the convergence of a variant of the generator Extended Dynamic Mode Decomposition (gEDMD) algorithm, one of the main algorithms developed to compute approximations of the Koopman operator from data. We find however that, when combining this algorithm with classical finite element spaces, the results are not satisfactory numerically, as the convergence of the data-driven approximation is too slow for the method to benefit from the accuracy of finite element spaces. In particular, for problems in dimension 1 it is less efficient than direct interpolation methods to recover the vector field. We provide some numerical examples to illustrate this last point.

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Section: Koopman Operator and Data-driven Modellingmentioning

confidence: 99%

A quantitative analysis of Koopman operator methods for system identification and predictions

Zhang¹,

Zuazua²

2024

Comptes Rendus. Mécanique

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“…In the example above, Γ † contains generalized functions (distributions). By enlarging the domain of the Koopman operator, surjectivity of (K − λI) can be mostly restored, resulting in shrinking of the continuous spectrum to a set of discrete values [306].…”

Section: Cavity Flowmentioning

confidence: 99%

“…While the rigged Hilbert space framework has existed since the 1950s and Gelfand, the approach has been used to analyze the Koopman and Perron-Frobenius operators only since the 1990s, and for quantum theory a decade later [209,210]. Only in the past two years have the modern numerical approaches such as DMD, started to connect to this theory [233,306], so we expect the growth of interest in this area in the coming years.…”

Section: Cavity Flowmentioning

confidence: 99%

Modern Koopman Theory for Dynamical Systems

Brunton¹,

Budišić²,

Kaiser³

et al. 2021

Preprint

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The field of dynamical systems is being transformed by the mathematical tools and algorithms emerging from modern computing and data science. First-principles derivations and asymptotic reductions are giving way to data-driven approaches that formulate models in operator theoretic or probabilistic frameworks. Koopman spectral theory has emerged as a dominant perspective over the past decade, in which nonlinear dynamics are represented in terms of an infinitedimensional linear operator acting on the space of all possible measurement functions of the system. This linear representation of nonlinear dynamics has tremendous potential to enable the prediction, estimation, and control of nonlinear systems with standard textbook methods developed for linear systems. However, obtaining finite-dimensional coordinate systems and embeddings in which the dynamics appear approximately linear remains a central open challenge. The success of Koopman analysis is due primarily to three key factors: 1) there exists rigorous theory connecting it to classical geometric approaches for dynamical systems, 2) the approach is formulated in terms of measurements, making it ideal for leveraging big-data and machine learning techniques, and 3) simple, yet powerful numerical algorithms, such as the dynamic mode decomposition (DMD), have been developed and extended to reduce Koopman theory to practice in real-world applications. In this review, we provide an overview of modern Koopman operator theory, describing recent theoretical and algorithmic developments and highlighting these methods with a diverse range of applications. We also discuss key advances and challenges in the rapidly growing field of machine learning that are likely to drive future developments and significantly transform the theoretical landscape of dynamical systems.

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“…In contrast, the Fourier basis is less flexible in stable directions, requiring more modes to capture rapid oscillations, but is extremely efficient at approximating smooth functions and easily captures the smooth variation in unstable directions. A recent alternative non-rigorous collocation-based method of SRB measure approximation has been explored in [32] for certain families (Blaschke products) of analytic Anosov maps. In the case of analytic expanding maps [32], proves that this method produces the true absolutely continuous invariant measure in the limit of increasing numerical resolution.…”

Section: Estimating the Srb Measurementioning

confidence: 99%

Fourier approximation of the statistical properties of Anosov maps on tori

Crimmins

Froyland

2020

Nonlinearity

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We study the stability of statistical properties of Anosov maps on tori by examining the stability of the spectrum of an analytically twisted Perron-Frobenius operator on the anisotropic Banach spaces of Gouëzel and Liverani (2006 Ergod. Theor. Dyn. Syst. 26 189-217). By extending our previous work in Crimmins and Froyland (2019 Ann. Henri Poincaré 20 3113-3161), we obtain the stability of various statistical properties (the variance of a CLT and the rate function of an LDP) of Anosov maps to general perturbations, including new classes of numerical approximations. In particular, we obtain new results on the stability of the rate function under deterministic perturbations. As a key application, we focus on perturbations arising from numerical schemes and develop two new Fourier-analytic methods for efficiently computing approximations of the aforementioned statistical properties. This includes the first example of a rigorous scheme for approximating the peripheral spectral data of the Perron-Frobenius operator of an Anosov map without mollification. We consequently obtain the first rigorous numerical methods for estimating the variance and rate function for Anosov maps.

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Dynamic mode decomposition for analytic maps

Cited by 10 publications

References 37 publications

A quantitative analysis of Koopman operator methods for system identification and predictions

A quantitative analysis of Koopman operator methods for system identification and predictions

Modern Koopman Theory for Dynamical Systems

Fourier approximation of the statistical properties of Anosov maps on tori

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