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

Handle URI:
http://hdl.handle.net/10754/598111
Title:
Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations
Authors:
Carlberg, Kevin; Bou-Mosleh, Charbel; Farhat, Charbel
Abstract:
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. © 2010 John Wiley & Sons, Ltd.
Citation:
Carlberg K, Bou-Mosleh C, Farhat C (2010) Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations. Int J Numer Meth Engng 86: 155–181. Available: http://dx.doi.org/10.1002/nme.3050.
Publisher:
Wiley-Blackwell
Journal:
International Journal for Numerical Methods in Engineering
Issue Date:
28-Oct-2010
DOI:
10.1002/nme.3050
Type:
Article
ISSN:
0029-5981
Sponsors:
The first author acknowledges the partial support by a National Science Foundation Graduate Fellowship and the partial support by a National Defense Science and Engineering Graduate Fellowship. The second and third authors acknowledge the partial support by the Motor Sports Division of the Toyota Motor Corporation under Agreement Number 48737, and the partial support by a research grant from the Academic Excellence Alliance program between King Abdullah University of Science and Technology (KAUST) and Stanford University. All authors also acknowledge the constructive comments received during the review process.
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Full metadata record

DC FieldValue Language
dc.contributor.authorCarlberg, Kevinen
dc.contributor.authorBou-Mosleh, Charbelen
dc.contributor.authorFarhat, Charbelen
dc.date.accessioned2016-02-25T13:12:53Zen
dc.date.available2016-02-25T13:12:53Zen
dc.date.issued2010-10-28en
dc.identifier.citationCarlberg K, Bou-Mosleh C, Farhat C (2010) Efficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximations. Int J Numer Meth Engng 86: 155–181. Available: http://dx.doi.org/10.1002/nme.3050.en
dc.identifier.issn0029-5981en
dc.identifier.doi10.1002/nme.3050en
dc.identifier.urihttp://hdl.handle.net/10754/598111en
dc.description.abstractA 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. © 2010 John Wiley & Sons, Ltd.en
dc.description.sponsorshipThe first author acknowledges the partial support by a National Science Foundation Graduate Fellowship and the partial support by a National Defense Science and Engineering Graduate Fellowship. The second and third authors acknowledge the partial support by the Motor Sports Division of the Toyota Motor Corporation under Agreement Number 48737, and the partial support by a research grant from the Academic Excellence Alliance program between King Abdullah University of Science and Technology (KAUST) and Stanford University. All authors also acknowledge the constructive comments received during the review process.en
dc.publisherWiley-Blackwellen
dc.subjectCompressive approximationen
dc.subjectDiscrete non-linear systemsen
dc.subjectGappy dataen
dc.subjectNon-linear model reductionen
dc.subjectPetrov-Galerkin projectionen
dc.subjectProper orthogonal decompositionen
dc.titleEfficient non-linear model reduction via a least-squares Petrov-Galerkin projection and compressive tensor approximationsen
dc.typeArticleen
dc.identifier.journalInternational Journal for Numerical Methods in Engineeringen
dc.contributor.institutionStanford University, Palo Alto, United Statesen
dc.contributor.institutionNotre Dame University, Lebanon, Zouk Mosbeh, Lebanonen
kaust.grant.programAcademic Excellence Alliance (AEA)en
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