NDLI: Preserving Symmetries in the Proper Orthogonal Decomposition

Content Provider	Society for Industrial and Applied Mathematics (SIAM)
Author	Aubry, Nadine Titi, Edriss S. Lian, Wen-Yu
Copyright Year	1993
Abstract	The proper orthogonal decomposition (POD) (also called KarhunenLove expansion) has been recently used in turbulence to derive optimally fast converging bases of spatial functions, leading to efficient finite truncations. Whether a finite number of these modes can be used in numerical simulations to derive an accurate finite set of ordinary differential equations, over a certain range of bifurcation parameter values, still remains an open question. 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. In most cases, this leads to a systematic involvement of the symmetry group in deriving a new expansion basis called the symmetric POD basis. The KuramotoSivashinsky equation with periodic boundary conditions is used as a paradigm to illustrate this point of view. However, the conclusion is general and can be applied to other equations, such as the NavierStokes equations, the complex GinzburgLandau equation, and others.
Starting Page	483
Ending Page	505
Page Count	23
File Format	PDF
ISSN	10648275
DOI	10.1137/0914030
e-ISSN	10957197
Issue Number	2
Volume Number	14
Language	English
Publisher	Society for Industrial and Applied Mathematics
Publisher Date	2006-07-13
Access Restriction	Subscribed
Subject Keyword	KuramotoSivashinsky equation KarhunenLove expansion Equations and systems on manifolds empirical eigen-functions Nonlinear initial value problems for linear parabolic equations proper orthogonal decomposition Finite elements, Rayleigh-Ritz and Galerkin methods, finite methods
Content Type	Text
Resource Type	Article
Subject	Applied Mathematics Computational Mathematics

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