Journal ArticleDOI
Timely Communication: Symmetry and the Karhunen--Loève Analysis
Nejib Smaoui,Dieter Armbruster +1 more
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It is shown that symmetrizing the K--L eigenmodes instead of symmetRIzing the data leads to considerable computational savings if the K-L analysis is done in the snapshot method.Abstract:
The Karhunen--Loeve (K--L) analysis is widely used to generate low-dimensional dynamical systems, which have the same low-dimensional attractors as some large-scale simulations of PDEs. If the PDE is symmetric with respect to a symmetry group G, the dynamical system has to be equivariant under G to capture the full phase space. It is shown that symmetrizing the K--L eigenmodes instead of symmetrizing the data leads to considerable computational savings if the K-L analysis is done in the snapshot method. The feasibility of the approach is demonstrated with an analysis of Kolmogorov flow.read more
Citations
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POD and CVT-based reduced-order modeling of Navier-Stokes flows
TL;DR: Reviews of the POD (proper orthogonal decomposition) and CVT (centroidal Voronoi tessellation) approaches to reduced-order modeling are provided, including descriptions of POD andCVT reduced- order bases, their construction from snapshot sets, and their application to the low-cost simulation of the Navier–Stokes system.
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Low-Dimensional Modelling of Turbulence Using the Proper Orthogonal Decomposition: A Tutorial
TL;DR: This paper describes two different ways to numerically calculate the modes, shows how symmetry considerations can be exploited to simplify and understand them, and describes a generalization of the procedure involving projection onto uncoupled modes that allow streamwise and cross-stream components to evolve independently.
Journal ArticleDOI
Reconstruction equations and the Karhunen—Loéve expansion for systems with symmetry
TL;DR: The main result of this paper is to derive a simple and easily implementable set of reconstruction equations which close the system of ODEs produced by Galerkin projection.
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On Low-Dimensional Galerkin Models for Fluid Flow
TL;DR: It is shown how the restriction to a low-dimensional basis as well as improper treatment of boundary conditions can affect the range of validity of these models.
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Reduced-order modeling of time-dependent PDEs with multiple parameters in the boundary data
TL;DR: In this paper, the proper orthogonal decomposition (POD) approach is applied to the case of multiple parameters in the context of a class of reduced-order models.
References
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Journal ArticleDOI
The Proper Orthogonal Decomposition in the Analysis of Turbulent Flows
TL;DR: The Navier-Stokes equations are well-known to be a good model for turbulence as discussed by the authors, and the results of well over a century of increasingly sophisticated experiments are available at our disposal.
Journal ArticleDOI
Symmetry-increasing bifurcation of chaotic attractors
Pascal Chossat,Martin Golubitsky +1 more
TL;DR: In this article, it was shown that symmetry-increasing bifurcation in the discrete dynamics of symmetric mappings is possible (and is perhaps generic) and that a new attractor should have greater symmetry.
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Preserving symmetries in the proper orthogonal decomposition
TL;DR: It is shown here that a necessary condition for achieving this goal is that the truncated system inherit the symmetry properties of the original infinite-dimensional system, leading to efficient finite truncations.