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