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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Bibliographic Details
Published in:International journal for numerical methods in engineering Vol. 86; no. 2; pp. 155 - 181
Main Authors: Carlberg, Kevin, Bou-Mosleh, Charbel, Farhat, Charbel
Format: Journal Article
Language:English
Published: Chichester, UK John Wiley & Sons, Ltd 15.04.2011
Wiley
Subjects:
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
Online Access:Get full text
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Summary: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.
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content type line 23
ISSN:0029-5981
1097-0207
1097-0207
DOI:10.1002/nme.3050