scholarly journals Ergodic Theory, Dynamic Mode Decomposition, and Computation of Spectral Properties of the Koopman Operator

2017 ◽  
Vol 16 (4) ◽  
pp. 2096-2126 ◽  
Author(s):  
Hassan Arbabi ◽  
Igor Mezić
Keyword(s):  
Ergodic Theory ◽  
Spectral Properties ◽  
Dynamic Mode Decomposition ◽  
Dynamic Mode ◽  
Koopman Operator ◽  
Mode Decomposition

scholarly journals A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition

2015 ◽  
Vol 25 (6) ◽  
pp. 1307-1346 ◽  
Author(s):  
Matthew O. Williams ◽  
Ioannis G. Kevrekidis ◽  
Clarence W. Rowley
Keyword(s):  
Dynamic Mode Decomposition ◽  
Data Driven ◽  
Dynamic Mode ◽  
Koopman Operator ◽  
Mode Decomposition

scholarly journals Detecting regime transitions in time series using dynamic mode decomposition

2020 ◽  
Author(s):  
Georg Gottwald ◽  
Federica Gugole
Keyword(s):  
Time Series ◽  
Reconstruction Error ◽  
Reanalysis Data ◽  
Dynamic Mode Decomposition ◽  
Transient Dynamics ◽  
Dynamic Mode ◽  
Koopman Operator ◽  
Regime Changes ◽  
Fast Relaxation ◽  
Mode Decomposition

<p>We employ the framework of the Koopman operator and dynamic mode decomposition to devise a computationally cheap and easily implementable method to detect transient dynamics and regime changes in time series. We argue that typically transient dynamics experiences the full state space dimension with subsequent fast relaxation towards the attractor. In equilibrium, on the other hand, the dynamics evolves on a slower time scale on a lower dimensional attractor. The reconstruction error of a dynamic mode decomposition is used to monitor the inability of the time series to resolve the fast relaxation towards the attractor as well as the effective dimension of the dynamics. We illustrate our method by detecting transient dynamics in the Kuramoto-Sivashinsky equation. We further apply our method to atmospheric reanalysis data; our diagnostics detects the transition from a predominantly negative North Atlantic Oscillation (NAO) to a predominantly positive NAO around 1970, as well as the recently found regime change in the Southern Hemisphere atmospheric circulation around 1970.</p>

scholarly journals Optimal Kernel-Based Dynamic Mode Decomposition

2018 ◽  
Author(s):  
Patrick Héas ◽  
Cédric Herzet
Keyword(s):  
Optimal Solution ◽  
General Setting ◽  
Approximation Problem ◽  
Dynamic Mode Decomposition ◽  
Low Rank ◽  
Dynamic Mode ◽  
Low Rank Approximation ◽  
Koopman Operator ◽  
Mode Decomposition ◽  
Optimal Kernel

The state-of-the-art algorithm known as kernel-based dynamic mode decomposition (K-DMD) provides a sub-optimal solution to the problem of reduced modeling of a dynamical system based on a finite approximation of the Koopman operator. It relies on crude approximations and on restrictive assumptions. The purpose of this work is to propose a kernel-based algorithm solving exactly this low-rank approximation problem in a general setting.

Spectral analysis of nonlinear flows

2009 ◽  
Vol 641 ◽  
pp. 115-127 ◽  
Author(s):  
CLARENCE W. ROWLEY ◽  
IGOR MEZIĆ ◽  
SHERVIN BAGHERI ◽  
PHILIPP SCHLATTER ◽  
DAN S. HENNINGSON
Keyword(s):  
Spectral Analysis ◽  
Temporal Frequency ◽  
Dynamic Mode Decomposition ◽  
Dynamic Mode ◽  
Arnoldi Method ◽  
Koopman Operator ◽  
Infinite Dimensional ◽  
Jet In Crossflow ◽  
Mode Decomposition ◽  
Nonlinear Flows

We present a technique for describing the global behaviour of complex nonlinear flows by decomposing the flow into modes determined from spectral analysis of the Koopman operator, an infinite-dimensional linear operator associated with the full nonlinear system. These modes, referred to as Koopman modes, are associated with a particular observable, and may be determined directly from data (either numerical or experimental) using a variant of a standard Arnoldi method. They have an associated temporal frequency and growth rate and may be viewed as a nonlinear generalization of global eigenmodes of a linearized system. They provide an alternative to proper orthogonal decomposition, and in the case of periodic data the Koopman modes reduce to a discrete temporal Fourier transform. The Arnoldi method used for computations is identical to the dynamic mode decomposition recently proposed by Schmid & Sesterhenn (Sixty-First Annual Meeting of the APS Division of Fluid Dynamics, 2008), so dynamic mode decomposition can be thought of as an algorithm for finding Koopman modes. We illustrate the method on an example of a jet in crossflow, and show that the method captures the dominant frequencies and elucidates the associated spatial structures.

Koopman-mode decomposition of the cylinder wake

2013 ◽  
Vol 726 ◽  
pp. 596-623 ◽  
Author(s):  
Shervin Bagheri
Keyword(s):  
Dynamic Mode Decomposition ◽  
Nonlinear Flow ◽  
Single Frequency ◽  
Dynamic Mode ◽  
Cylinder Wake ◽  
Koopman Operator ◽  
Mode Decomposition ◽  
Analytical Construction ◽  
Vortex Shedding Frequency ◽  
Global Modes

AbstractThe Koopman operator provides a powerful way of analysing nonlinear flow dynamics using linear techniques. The operator defines how observables evolve in time along a nonlinear flow trajectory. In this paper, we perform a Koopman analysis of the first Hopf bifurcation of the flow past a circular cylinder. First, we decompose the flow into a sequence of Koopman modes, where each mode evolves in time with one single frequency/growth rate and amplitude/phase, corresponding to the complex eigenvalues and eigenfunctions of the Koopman operator, respectively. The analytical construction of these modes shows how the amplitudes and phases of nonlinear global modes oscillating with the vortex shedding frequency or its harmonics evolve as the flow develops and later sustains self-excited oscillations. Second, we compute the dynamic modes using the dynamic mode decomposition (DMD) algorithm, which fits a linear combination of exponential terms to a sequence of snapshots spaced equally in time. It is shown that under certain conditions the DMD algorithm approximates Koopman modes, and hence provides a viable method to decompose the flow into saturated and transient oscillatory modes. Finally, the relevance of the analysis to frequency selection, global modes and shift modes is discussed.

Experimental Measurement of In-Plane Rolling Nonpneumatic Tire Vibrations Using High-Speed Imaging

2019 ◽  
Vol 47 (3) ◽  
pp. 196-210
Author(s):  
Meghashyam Panyam ◽  
Beshah Ayalew ◽  
Timothy Rhyne ◽  
Steve Cron ◽  
John Adcox
Keyword(s):  
High Speed ◽  
Image Correlation ◽  
Dynamic Mode Decomposition ◽  
Dynamic Mode ◽  
High Speed Imaging ◽  
Correlation Algorithm ◽  
Mode Decomposition ◽  
Series Of Experiments ◽  
Plane Deformations ◽  
Dominant Frequencies

ABSTRACT This article presents a novel experimental technique for measuring in-plane deformations and vibration modes of a rotating nonpneumatic tire subjected to obstacle impacts. The tire was mounted on a modified quarter-car test rig, which was built around one of the drums of a 500-horse power chassis dynamometer at Clemson University's International Center for Automotive Research. A series of experiments were conducted using a high-speed camera to capture the event of the rotating tire coming into contact with a cleat attached to the surface of the drum. The resulting video was processed using a two-dimensional digital image correlation algorithm to obtain in-plane radial and tangential deformation fields of the tire. The dynamic mode decomposition algorithm was implemented on the deformation fields to extract the dominant frequencies that were excited in the tire upon contact with the cleat. It was observed that the deformations and the modal frequencies estimated using this method were within a reasonable range of expected values. In general, the results indicate that the method used in this study can be a useful tool in measuring in-plane deformations of rolling tires without the need for additional sensors and wiring.

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