Geodesic
Certified Descent Algorithm for shape optimization driven by fully-computable a posteriori error estimators
Giacomini, Matteo1, 2  ; Pantz, Olivier3  ; Trabelsi, Karim2 
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

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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 :
Accepté le :
DOI : 10.1051/cocv/2016021
Classification : 49Q10, 65M60, 65N15
Keywords: Shape optimization, A posteriori error estimator, Certified Descent Algorithm, Electrical Impedance Tomography

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. 
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     author = {Giacomini, Matteo and Pantz, Olivier and Trabelsi, Karim},
     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},
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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

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