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/

Bibliographie
Cité par

L. Afraites, M. Dambrine and D. Kateb, Shape methods for the transmission problem with a single measurement. Numer. Func. Anal. Opt. 28 (2007) 519–551. | DOI | MR | Zbl

L. Afraites, M. Dambrine and D. Kateb, On second order shape optimization methods for electrical impedance tomography. SIAM J. Control Optim. 47 (2008) 1556–1590. | DOI | MR | Zbl

F. Alauzet, B. Mohammadi and O. Pironneau, Mesh adaptivity and optimal shape design for aerospace. In Variational Analysis and Aerospace Engineering: Mathematical Challenges for Aerospace Design, edited by G. Buttazzo and A. Frediani. Optimization and Its Applications. Springer US (2012) 323–337.

G. Allaire and O. Pantz, Structural optimization with FreeFem++. Struct. Multidisc. Optim. 32 (2006) 173–181. | DOI | MR | Zbl

H. Ammari, E. Bossy, J. Garnier and L. Seppecher, Acousto-electromagnetic tomography. SIAM J. Appl. Math. 72 (2012) 1592–1617. | DOI | MR | Zbl

H. Azegami, S. Kaizu, M. Shimoda and E. Katamine, Irregularity of shape optimization problems and an improvement technique. In Computer Aided Optimization Design of Structures V, edited by S. Hernandez and C. Brebbia. Computational Mechanics Publications (1997) 309–326.

N. Banichuk, F.-J. Barthold, A. Falk and E. Stein, Mesh refinement for shape optimization. Struct. Optim. 9 (1995) 46–51. | DOI

L. Borcea, Electrical impedance tomography. Inverse Probl. 18 (2002) R99. | DOI | MR | Zbl

A. Calderón, On an inverse boundary value problem. In Seminar on Numerical Analysis and its Applications to Continuum Physics (Rio de Janeiro 1980). Soc. Brasil. Mat. (1980) 65–73. | MR | Zbl

A. Carpio and M.-L. Rapún, Hybrid topological derivative and gradient-based methods for electrical impedance tomography. Inverse Probl. 28 (2012) 095010. | DOI | MR | Zbl

J. Céa, Conception optimale ou identification de formes, calcul rapide de la dérivée directionnelle de la fonction coût. ESAIM: M2AN 20 (1986) 371–402. | DOI | Numdam | MR | Zbl

M. Cheney, D. Isaacson and J. Newell, Electrical impedance tomography. SIAM Rev. 41 (1999) 85–101. | DOI | MR | Zbl

E. Chung, T. Chan and X.-C. Tai, Electrical impedance tomography using level set representation and total variational regularization. J. Comput. Phys. 205 (2005) 357–372. | DOI | MR | Zbl

M. Delfour and J.-P. Zolésio, Shapes and geometries: analysis, differential calculus, and optimization. SIAM, Philadelphia, USA (2001). | MR | Zbl

G. Dogǎn, P. Morin, R. Nochetto and M. Verani, Discrete gradient flows for shape optimization and applications. Special Issue Honoring the 80th Birthday of Professor Ivo Babuka. Comput. Methods Appl. Mech. Engrg. 196 (2007) 3898–3914. | DOI | MR | Zbl

K. Eppler and H. Harbrecht, A regularized Newton method in electrical impedance tomography using shape Hessian information. Control Cybernet. 34 (2005) 203–225. | MR | Zbl

L. Formaggia, S. Micheletti and S. Perotto, Anisotropic mesh adaption in computational fluid dynamics: application to the advection-diffusion-reaction and the Stokes problems. Appl. Numer. Math. 51 (2004) 511–533. | DOI | MR | Zbl

T. Grätsch and K.-J. Bathe, Goal-oriented error estimation in the analysis of fluid flows with structural interactions. Comput. Methods Appl. Mech. Engrg. 195 (2006) 5673–5684. | DOI | MR | Zbl

F. Hecht, New development in FreeFem++. J. Numer. Math. 20 (2012) 251–265. | DOI | MR | Zbl

M. Hintermüller and A. Laurain, Electrical impedance tomography: from topology to shape. Control Cybern. 37 (2008) 913–933. | MR | Zbl

M. Hintermüller, A. Laurain and A.A. Novotny, Second-order topological expansion for electrical impedance tomography. Adv. Comput. Math. 36 (2012) 235–265. | DOI | MR | Zbl

R. Hiptmair, A. Paganini and S. Sargheini, Comparison of approximate shape gradients. BIT Numer. Math. 54 (2014) 1–27. | MR | Zbl

D. Holder, Electrical Impedance Tomography: Methods, History and Applications. Series in Medical Physics and Biomedical Engineering. CRC Press (2004).

B. Jin and P. Maass, An analysis of electrical impedance tomography with applications to Tikhonov regularization. ESAIM: COCV 18 (2012) 1027–1048. | Numdam | MR | Zbl

N. Kikuchi, K. Chung, T. Torigaki and J. Taylor, Adaptive Finite Element Methods for shape optimization of linearly elastic structures. Comput. Methods Appl. Mech. Eng. 57 (1986) 67–89. | DOI | MR | Zbl

R. Kohn and M. Vogelius, Relaxation of a variational method for impedance computed tomography. Comm. Pure Appl. Math. 40 (1987) 745–777. | DOI | MR | Zbl

A. Laurain and K. Sturm, Distributed shape derivative via averaged adjoint method and applications. ESAIM: M2AN 50 (2016) 1241–1267. | DOI | Numdam | MR | Zbl

P. Morin, R. Nochetto, M. Pauletti and M. Verani, Adaptive finite element method for shape optimization. ESAIM: COCV 18 (2012) 1122–1149. | Numdam | MR | Zbl

J. Oden and S. Prudhomme, Goal-oriented error estimation and adaptivity for the Finite Element Method. Comput. Math. Appl. 41 (2001) 735–756. | DOI | MR | Zbl

O. Pantz, Sensibilité de l’équation de la chaleur aux sauts de conductivité. C. R. Acad. Sci. Paris, Ser. I (2005) 333–337. | MR | Zbl

G. Porta, S. Perotto and F. Ballio, Anisotropic mesh adaptation driven by a recovery-based error estimator for shallow water flow modeling. Int. J. Numer. Methods Fluids 70 (2012) 269–299. | DOI | MR | Zbl

S. Prudhomme, J. Oden, T. Westermann, J. Bass and M. Botkin, Practical methods for a posteriori error estimation in engineering applications. Int. J. Numer. Methods Engrg. 56 (2003) 1193–1224. | DOI | MR | Zbl

S. Repin, A posteriori error estimation for variational problems with uniformly convex functionals. Math. Comput. 69 (2000) 481–500. | DOI | MR | Zbl

J. Roche, Adaptive Newton-like method for shape optimization. Control Cybern. 34 (2005) 363–377. | MR | Zbl

M. Rüter, T. Gerasimov and E. Stein, Goal-oriented explicit residual-type error estimates in XFEM. Comput. Mech. 52 (2013) 361–376. | DOI | MR | Zbl

A. Schleupen, K. Maute and E. Ramm, Adaptive FE-procedures in shape optimization. Struct. Multidisc. Optim. 19 (2000) 282–302. | DOI

J. Sokołowski and J. Zolésio, Introduction to shape optimization: shape sensitivity analysis. Springer-Verlag (1992). | MR | Zbl

J. Sylvester and G. Uhlmann, A global uniqueness theorem for an inverse boundary value problem. Ann. Math. 125 (1987) 153–169. | DOI | MR | Zbl

T. Vejchodský, Complementary error bounds for elliptic systems and applications. Appl. Math. Comput. 219 (2013) 7194–7205. | DOI | MR | Zbl

A. Wexler, B. Fry and M.R. Neuman, Impedance-computed tomography algorithm and system. Appl. Opt. 24 (1985) 3985–3992. | DOI

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