Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations

A Petrov–Galerkin projection method is proposed for reducing the dimension of a discrete non‐linear static or dynamic computational model in view of enabling its processing in real time. The right reduced‐order basis is chosen to be invariant and is constructed using the Proper Orthogonal Decomposit...

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Vydané v:	International journal for numerical methods in engineering Ročník 86; číslo 2; s. 155 - 181
Hlavní autori:	Carlberg, Kevin, Bou-Mosleh, Charbel, Farhat, Charbel
Médium:	Journal Article
Jazyk:	English
Vydavateľské údaje:	Chichester, UK John Wiley & Sons, Ltd 15.04.2011 Wiley
Predmet:	Approximation compressive approximation Computation Computational efficiency discrete non-linear systems Exact sciences and technology Fluid dynamics Fundamental areas of phenomenology (including applications) gappy data Invariants Mathematical models Mathematics Methods of scientific computing (including symbolic computation, algebraic computation) non-linear model reduction Nonlinear dynamics Nonlinearity Numerical analysis. Scientific computation Petrov-Galerkin projection Physics Projection proper orthogonal decomposition Sciences and techniques of general use Solid mechanics Structural and continuum mechanics Turbulent flows, convection, and heat transfer Vibration, mechanical wave, dynamic stability (aeroelasticity, vibration control...) Invariant Turbulent flow Karhunen Loeve transformation Vibration compressive approximation Gauss Newton method Real time Modeling gappy data Galerkin-Petrov method discrete non-linear systems Static model Petrov―Galerkin projection Least squares method Galerkin method Reduced order systems proper orthogonal decomposition Non linear effect Incomplete information Dynamic model non-linear model reduction
ISSN:	0029-5981, 1097-0207, 1097-0207
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Popis
Shrnutí:	A Petrov–Galerkin projection method is proposed for reducing the dimension of a discrete non‐linear static or dynamic computational model in view of enabling its processing in real time. The right reduced‐order basis is chosen to be invariant and is constructed using the Proper Orthogonal Decomposition method. The left reduced‐order basis is selected to minimize the two‐norm of the residual arising at each Newton iteration. Thus, this basis is iteration‐dependent, enables capturing of non‐linearities, and leads to the globally convergent Gauss–Newton method. To avoid the significant computational cost of assembling the reduced‐order operators, the residual and action of the Jacobian on the right reduced‐order basis are each approximated by the product of an invariant, large‐scale matrix, and an iteration‐dependent, smaller one. The invariant matrix is computed using a data compression procedure that meets proposed consistency requirements. The iteration‐dependent matrix is computed to enable the least‐squares reconstruction of some entries of the approximated quantities. The results obtained for the solution of a turbulent flow problem and several non‐linear structural dynamics problems highlight the merit of the proposed consistency requirements. They also demonstrate the potential of this method to significantly reduce the computational cost associated with high‐dimensional non‐linear models while retaining their accuracy. Copyright © 2010 John Wiley & Sons, Ltd.
Bibliografia:	ArticleID:NME3050 istex:4E8851FB8294A59F96E8405757060C1F0558655E ark:/67375/WNG-KVVT1J7M-C ObjectType-Article-2 SourceType-Scholarly Journals-1 ObjectType-Feature-1 content type line 23
ISSN:	0029-5981 1097-0207 1097-0207
DOI:	10.1002/nme.3050