Spectral Properties of Dynamical Systems, Model Reduction and Decompositions

By Igor Mezi'c. 2005

1,510 citations recorded by the source corpus.

Subfields: Computational Mechanics, Statistical and Nonlinear Physics, Statistics and Probability

How researchers see this paper

The researchers whose trusted networks rank it highest, strongest advocate first. Every researcher gets a different order.

In Computational Mechanics

In Statistical and Nonlinear Physics

In Statistics and Probability

Citations in the corpus

Deep Generative Markov State Models (2018, methodology): “The analogous approach for deterministic dynamical systems such as complex fluid flows is called Koopman analysis, where time propagation is approximated by a linear model in a suitable function space transformation of the flow variables [16, 26, 29, 4].”
On Convergence of Extended Dynamic Mode Decomposition to the Koopman Operator (2017, methodology): “6], 4) the Extended DMD [27]. The relationship between these methods and the spectral operator properties of the Koopman operator was rst noticed in [20], based on the spectral expansion developed in [15]. However, rigorous results in this direction are sparse. Williams et al. [27] provided a result”
Nonlinear normal modes and spectral submanifolds: existence, uniqueness and use in model reduction (2016, background): “Mezić [27]) in their arguments, but all spectral subspaces of a linear mapping are well defined without the need to view them as zero sets of Koopman eigenfunctions.”
Data-Driven Sparse Sensor Placement for Reconstruction: Demonstrating the Benefits of Exploiting Known Patterns (2017, background): “short [74] I.”
Data-Driven Model Reduction and Transfer Operator Approximation (2017, methodology): “ eigenvalues and eigenfunctions of transfer operators contain importantinformationabouttheglobalpropertiesofthesystemsuchasmetastablesetsor fastandslowprocessesandcanalsobeusedasreducedcoordinates,see[31,28,10,32,29,2] andreferencestherein. Example2.6.Theeigenvalues ‘andeigenfunctions’ ‘ofK ˝associa”
Chaos as an intermittently forced linear system (2016, methodology): “For a detailed discussion on the Koopman operator, there are many excellent research articles [70, 82, 16, 17, 61] and reviews [18, 71].”
Spectral proper orthogonal decomposition and its relationship to dynamic mode decomposition and resolvent analysis (2017, background): “This tendency is explained more rigorously by the connection between DMD and Koopman operator theory (Mezić, 2005; Rowley et al., 2009).”
Multi-scale proper orthogonal decomposition of complex fluid flows (2018, methodology): “This theory was then used by Mezić (2005) to show how finite dimensional approximations of such Koopman (linear) dynamical system could be used for MOR applications.”
Sparse identification of nonlinear dynamics for model predictive control in the low-data limit (2017, background): “Delay coordinates also deﬁne intrinsic coordinates for the Koopman operator [41], which provides a simple linear embedding of nonlinear systems [78,79].”
A Data–Driven Approximation of the Koopman Operator: Extending Dynamic Mode Decomposition (2014, methodology): “5, we experiment by applying EDMD to stochastic data and show it approximates the eigenfunctions of the SKO for Markov processes.”