Certified Descent Algorithm for shape optimization driven by fully-computable a posteriori error estimators

Giacomini, Matteo; Pantz, Olivier; Trabelsi, Karim

doi:10.1051/cocv/2016021

Giacomini, Matteo ^1,² ; Pantz, Olivier ³ ; Trabelsi, Karim ²

¹ CMAP Ecole Polytechnique, Route de Saclay, 91128 Palaiseau, France.
² DRI Institut Polytechnique des Sciences Avancées, 15-21 rue M. Grandcoing, 94200 Ivry-sur-Seine, France.
³ Laboratoire J.A. Dieudonné UMR 7351 CNRS-Université de Nice-Sophia Antipolis, Parc Valrose, 06108 Nice cedex 02, France.

ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 3, pp. 977-1001.

Résumé

In this paper we introduce a novel certified shape optimization strategy – named Certified Descent Algorithm (CDA) – to account for the numerical error introduced by the Finite Element approximation of the shape gradient. We present a goal-oriented procedure to derive a certified upper bound of the error in the shape gradient and we construct a fully-computable, constant-free a posteriori error estimator inspired by the complementary energy principle. The resulting CDA is able to identify a genuine descent direction at each iteration and features a reliable stopping criterion. After validating the error estimator, some numerical simulations of the resulting certified shape optimization strategy are presented for the well-known inverse identification problem of Electrical Impedance Tomography.

Reçu le : 2015-09-14
Accepté le : 2016-04-13

MR Zbl

DOI : 10.1051/cocv/2016021

Classification : 49Q10, 65M60, 65N15
Mots-clés : Shape optimization, A posteriori error estimator, Certified Descent Algorithm, Electrical Impedance Tomography

Affiliations des auteurs :

Giacomini, Matteo ^{1, 2} ; Pantz, Olivier ³ ; Trabelsi, Karim ²

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     title = {Certified {Descent} {Algorithm} for shape optimization driven by fully-computable a posteriori error estimators},
     journal = {ESAIM: Control, Optimisation and Calculus of Variations},
     pages = {977--1001},
     publisher = {EDP-Sciences},
     volume = {23},
     number = {3},
     year = {2017},
     doi = {10.1051/cocv/2016021},
     mrnumber = {3660456},
     zbl = {1369.49060},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/cocv/2016021/}
}

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Giacomini, Matteo; Pantz, Olivier; Trabelsi, Karim. Certified Descent Algorithm for shape optimization driven by fully-computable a posteriori error estimators. ESAIM: Control, Optimisation and Calculus of Variations, Tome 23 (2017) no. 3, pp. 977-1001. doi : 10.1051/cocv/2016021. http://www.numdam.org/articles/10.1051/cocv/2016021/

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