Turbulence, Coherent Structures, Dynamical Systems and Symmetry

By Bernd R. Noack. 2013

1,496 citations recorded by the source corpus.

Subfields: Astronomy and Astrophysics, Statistical and Nonlinear Physics

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Citations in the corpus

Data-driven discovery of partial differential equations (2016, methodology): “This procedure resembles the standard Galerkin projection onto POD modes (18).”
Discovering governing equations from data by sparse identification of nonlinear dynamical systems (2015, methodology): “The ﬁrst two most energetic POD modes capture a signiﬁcant portion of the energy; the steady-state vortex shedding is a limit cycle in these coordinates.”
Data-driven discovery of coordinates and governing equations (2019, methodology): “Principal component analysis (PCA) is one of the most proliﬁc modern coordinate discovery methods, representing high-dimensional data in a low-dimensional linear subspace [41, 42].”
Recursive dynamic mode decomposition of transient and post-transient wake flows (2016, background): “The corresponding time-averaged truncation error χ 2 := 〈‖rm‖2Ω〉M (3.11) can be shown to be minimal; in other words, no other Galerkin expansion with the same number of modes will have a smaller error (Holmes et al. 1998).”
Variants of Dynamic Mode Decomposition: Boundary Condition, Koopman, and Fourier Analyses (2012, methodology): “Modal decompositions such as POD and DMD may then be used to obtain further approximations of surprisingly low dimension, by expanding the solution in terms of a linear combination of modes (Holmes et al. 1996).”
Karhunen-Loève expansion for multi-correlated stochastic processes (2013, neutral): “…constructed, e.g., by using orthogonal polynomials [1] [2], spectral density methods [3] [4] [5] [6], wavelets [7] [8] [9], Karhunen–Loève (KL) decompositions [10] [11] [12] [13] [14] [15] [16], bi-orthogonal techniques [17] [18] [19] or functional principal component analyses [20] [21] [22] [23].”
Stabilization of projection-based reduced order models for linear time-invariant systems via optimization-based eigenvalue reassignment (2014, methodology): “Discussed in detail in Lumley [8] and Holmes et al. [9], POD is a mathematical procedure in which, given an ensemble of data and an inner product, an empirical basis is constructed.”
Exploiting sparsity and equation-free architectures in complex systems (2014, background): “If a low-rank approximation is viable, then reduced-order-models (ROMs) [35,36] can be generated by projecting the governing complex system directly on to the discovered low-dimensional manifold [24,37–41].”
The Significance of Simple Invariant Solutions in Turbulent Flows (2011, methodology): “sets, such as the strange attractors of some highly reduced dynamical systems obtained from the Navier–Stokes equations by proper orthogonal decomposition (see, e.g. Holmes et al. 1996).”
Generalizing Koopman Theory to Allow for Inputs and Control (2016, methodology): “These model reduction techniques typically employ the singular value decomposition to discover low-dimensional subspaces where the dynamics evolve [19].”
Constrained sparse Galerkin regression (2016, background): “Despite this apparent complexity, the dynamics often evolves on a low-dimensional attractor defined by a few dominant coherent structures that contain significant energy or are useful for control (Holmes et al. 2012).”
The engine behind (wall) turbulence: perspectives on scale interactions (2017, background): “The invariant solutions can range from relatively complex strange attractors (Holmes et al. 1996), which – at present, at least – must be extracted from data rather than by direct calculation, to the simpler forms in symmetry-restricted domains, which have been ably reviewed, for example, by…”
Proper Orthogonal Decomposition for Linear-Quadratic Optimal Control (2013, methodology): “For the POD method in complex Hilbert spaces we refer to [75], for instance.”