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Open AccessFeature PaperArticle

Toward Optimality of Proper Generalised Decomposition Bases

1
IBNM, Leibniz Universität Hannover, Appelstraße 9a, 30167 Hannover, Germany
2
LMT, ENS Paris-Saclay, CNRS, Université Paris Saclay, 61 Avenue du Président Wilson, 94235 Cachan, France
*
Author to whom correspondence should be addressed.
Math. Comput. Appl. 2019, 24(1), 30; https://doi.org/10.3390/mca24010030
Received: 31 January 2019 / Revised: 28 February 2019 / Accepted: 1 March 2019 / Published: 5 March 2019
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Abstract

The solution of structural problems with nonlinear material behaviour in a model order reduction framework is investigated in this paper. In such a framework, greedy algorithms or adaptive strategies are interesting as they adjust the reduced order basis (ROB) to the problem of interest. However, these greedy strategies may lead to an excessive increase in the size of the ROB, i.e., the solution is no more represented in its optimal low-dimensional expansion. Here, an optimised strategy is proposed to maintain, at each step of the greedy algorithm, the lowest dimension of a Proper Generalized Decomposition (PGD) basis using a randomised Singular Value Decomposition (SVD) algorithm. Comparing to conventional approaches such as Gram–Schmidt orthonormalisation or deterministic SVD, it is shown to be very efficient both in terms of numerical cost and optimality of the ROB. Examples with different mesh densities are investigated to demonstrate the numerical efficiency of the presented method. View Full-Text
Keywords: model order reduction (MOR); low-rank approximation; proper generalised decomposition (PGD); PGD compression; randomised SVD; nonlinear material behaviour model order reduction (MOR); low-rank approximation; proper generalised decomposition (PGD); PGD compression; randomised SVD; nonlinear material behaviour
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Alameddin, S.; Fau, A.; Néron, D.; Ladevèze, P.; Nackenhorst, U. Toward Optimality of Proper Generalised Decomposition Bases. Math. Comput. Appl. 2019, 24, 30.

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