Comptes Rendus
Partial differential equations/Numerical analysis
A slack approach to reduced-basis approximation and error estimation for variational inequalities
[Approximation bases réduites et estimateur d'erreur pour les inéquations variationnelles via une approche par variable d'écart]
Comptes Rendus. Mathématique, Volume 354 (2016) no. 3, pp. 283-289.

Nous proposons une nouvelle approche pour le calcul d'approximations bases réduites pour des inégalités variationnelles du premier type. Les trois principales composantes de cette approche sont : (i) une approximation utilisant des variables d'écart pour la solution ; (ii) une approximation primale pour le multiplicateur de Lagrange ; (iii) une borne supérieure a posteriori de l'erreur sur la solution approchée. La stricte faisabilité de l'approximation primale par variable d'écart nous permet deux améliorations majeures par rapport aux méthodes existantes. La première est de pouvoir borner, a posteriori, de façon précise, l'erreur commise. La deuxième est l'utilisation d'une décomposition hors ligne/en ligne grâce à laquelle le coût de calcul de cette borne reste complètement indépendant de la (grande) dimension originale du problème. Les résultats numériques présentent une comparaison des performances entre cette nouvelle approche et les méthodes existantes.

We propose a novel approach for computing certified reduced-basis approximations to solutions to variational inequalities of the first kind. The proposed approach has three components: (i) a slack-based approximation for the solution; (ii) a primal approximation for the Lagrange multiplier; and (iii) a posteriori bounds for the error in the combined primal-slack variable approximation. The strict feasibility of the primal-slack approximations leads to two significant improvements upon existing methods. First, it provides a posteriori error bounds that are significantly sharper than existing bounds. Second, it enables a full offline–online computational decomposition, in which the online cost to compute the error bound is completely independent of the dimension of the original (high-dimensional) problem. Our numerical results allow us to compare the performance of the proposed and existing approaches.

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2015.10.024

Zhenying Zhang 1 ; Eduard Bader 1 ; Karen Veroy 1

1 Aachen Institute for Advanced Study in Computational Engineering Science (AICES), RWTH Aachen University, Germany
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Zhenying Zhang; Eduard Bader; Karen Veroy. A slack approach to reduced-basis approximation and error estimation for variational inequalities. Comptes Rendus. Mathématique, Volume 354 (2016) no. 3, pp. 283-289. doi : 10.1016/j.crma.2015.10.024. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2015.10.024/

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  • Xin Su; Sai-Mang Pun A multiscale method for the heterogeneous Signorini problem, Journal of Computational and Applied Mathematics, Volume 409 (2022), p. 21 (Id/No 114160) | DOI:10.1016/j.cam.2022.114160 | Zbl:1484.65304
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  • Sara Grundel; Volker Mehrmann; Jacquelien M. A. Scherpen; Felix L. Schwenninger Mini-workshop: Mathematics of dissipation – dynamics, data and control. Abstracts from the mini-workshop held May 9–15, 2021 (hybrid meeting), Oberwolfach Rep. 18, No. 2, 1259-1289, 2021 | DOI:10.4171/owr/2021/24 | Zbl:1506.00078
  • Amina Benaceur; Alexandre Ern; Virginie Ehrlacher A reduced basis method for parametrized variational inequalities applied to contact mechanics, International Journal for Numerical Methods in Engineering, Volume 121 (2020) no. 6, pp. 1170-1197 | DOI:10.1002/nme.6261 | Zbl:1548.74460
  • J. Ballani; D. B. P. Huynh; D. J. Knezevic; L. Nguyen; A. T. Patera A component-based hybrid reduced basis/finite element method for solid mechanics with local nonlinearities, Computer Methods in Applied Mechanics and Engineering, Volume 329 (2018), pp. 498-531 | DOI:10.1016/j.cma.2017.09.014 | Zbl:1439.74387
  • Eduard Bader; Martin A. Grepl; Karen Veroy A certified reduced basis approach for parametrized optimal control problems with two-sided control constraints, Model reduction of parametrized systems. Selected contributions based on the presentations at the MoRePaS conference, SISSA, Trieste, Italy, October 13–16, 2015, Cham: Springer, 2017, pp. 37-54 | DOI:10.1007/978-3-319-58786-8_3 | Zbl:6861090
  • E. Bader; Z. Zhang; K. Veroy An empirical interpolation approach to reduced basis approximations for variational inequalities, Mathematical and Computer Modelling of Dynamical Systems, Volume 22 (2016) no. 4, pp. 345-361 | DOI:10.1080/13873954.2016.1198388 | Zbl:1351.49034
  • Eduard Bader; Mark Kärcher; Martin A. Grepl; Karen Veroy Certified reduced basis methods for parametrized distributed elliptic optimal control problems with control constraints, SIAM Journal on Scientific Computing, Volume 38 (2016) no. 6, p. a3921-a3946 | DOI:10.1137/16m1059898 | Zbl:1426.49029

Cité par 9 documents. Sources : zbMATH

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