Spectral analysis of nonlinear flows

By Clarence W. Rowley, Igor Mezi'c, Shervin Bagheri, Philipp Schlatter, Dan S. Henningson. 2009

2,379 citations recorded by the source corpus.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics

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In Computational Mechanics

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Citations in the corpus

Model Reduction for Flow Analysis and Control (2017, methodology): “The connection between DMD and the Koopman operator was first presented in Rowley et al. (2009).”
Data-driven discovery of Koopman eigenfunctions for control (2017, methodology): “Current data-driven methods to approximate the Koopman operator include extended DMD (EDMD) [70,71,29] and the variational approach of conformation dynamics (VAC) [49,50].”
Characterizing and correcting for the effect of sensor noise in the dynamic mode decomposition (2014, background): “DMD has a strong connection to Koopman operator theory (Koopman, 1931; Mezi´c, 2005), as exposed in Rowley et al. (2009), and further reviewed in Mezi´c (2013), which can justify its use in analyzing nonlinear dynamical systems.”
Data-Driven Sparse Sensor Placement for Reconstruction: Demonstrating the Benefits of Exploiting Known Patterns (2017, neutral): “[21] C.”
Compressive Sensing and Low-Rank Libraries for Classification of Bifurcation Regimes in Nonlinear Dynamical Systems (2013, methodology): “(ii) The algorithm works equally well with experimental data in an equation-free context, for instance by using dynamic mode decomposition [37, 39] or equation-free modeling [26] in place of POD, (iii) Given the low-rank space in which the algorithm works, it is ideal for use with control…”
Constrained sparse Galerkin regression (2016, methodology): “In contrast, data-driven approaches are becoming increasingly popular and encompass a large variety of different techniques such as the eigensystem realisation algorithm (ERA) (Juang and Pappa, 1985), dynamic mode decomposition (DMD) (Kutz et al., 2016; Rowley et al., 2009; Schmid, 2010), Koopman th”
Dynamic Mode Decomposition with Control (2014, methodology): “Second, DMD has acquired popularity as a method for systems with nonlinear dynamics, due to a strong connection between DMD and Koopman operator theory [24], [26], [30], [6], [27].”
Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis (2017, methodology): “This tendency is explained more rigorously by the connection between DMD and Koopman operator theory (Mezić, 2005; Rowley et al., 2009).”
Isostables, isochrons, and Koopman spectrum for the action-angle representation of stable fixed point dynamics (2013, background): “and the vectors vk1···kn are the so-called Koopman modes [23], i.”
De-biasing the dynamic mode decomposition for applied Koopman spectral analysis of noisy datasets (2015, methodology): “It was shown in [42] that DMD approximates the Koopman operator [31], an infinite-dimensional linear operator that describes the evolution of a nonlinear dynamical system by its action on observables (defined precisely in the next section).”
Modal Analysis of Fluid Flows: An Overview (2017, methodology): “Inmanyways,DMDmaybeviewed as combining favorable aspects of both thePODand the discreteFourier transform [105,108], resulting in spatiotemporal coherent structures identified purely from data.”
Sparse Identification of Nonlinear Dynamics with Control (SINDYc) (2016, methodology): “Many techniques in system identification (Ljung, 1999), including dynamic mode decomposition (DMD) (Schmid and Sesterhenn, 2008; Rowley et al., 2009; Schmid, 2010; Tu et al., 2014) and DMD with control (DMDc) (Proctor et al., 2016a), are designed to handle high-dimensional data with the assumption…”
A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition (2014, methodology): “(2014), this generalization is related to the Krylov method presented in Rowley et al. (2009). Now,wewill prove that theKoopmanmodes computed using EDMDare equivalent to DMDmodes, if the dictionary used for EDMD consists of scalar observables of the formψi (x) = e∗ i x for i = 1, .”
Applications of the dynamic mode decomposition (2011, methodology): “The analysis of a data sequence, generated by a general nonlinear dynamical system, by a linear snapshot-to-snapshot mapping is linked to Koopman analysis [12,15] which has recently been extended and applied to fluid mechanical problems, such as direct numerical simulations of a jet in crossflow [19”