The extended adjoint method
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 1, pp. 83-108.

Searching for the optimal partitioning of a domain leads to the use of the adjoint method in topological asymptotic expansions to know the influence of a domain perturbation on a cost function. Our approach works by restricting to local subproblems containing the perturbation and outperforms the adjoint method by providing approximations of higher order. It is a universal tool, easily adapted to different kinds of real problems and does not need the fundamental solution of the problem; furthermore our approach allows to consider finite perturbations and not infinitesimal ones. This paper provides theoretical justifications in the linear case and presents some applications with topological perturbations, continuous perturbations and mesh perturbations. This proposed approach can also be used to update the solution of singularly perturbed problems.

DOI : 10.1051/m2an/2012020
Classification : 49Q10, 49Q12, 74P10, 74P15
Mots-clés : adjoint method, topology optimization, calculus of variations 
@article{M2AN_2013__47_1_83_0,
     author = {Larnier, Stanislas and Masmoudi, Mohamed},
     title = {The extended adjoint method},
     journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
     pages = {83--108},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {1},
     year = {2013},
     doi = {10.1051/m2an/2012020},
     mrnumber = {2968696},
     zbl = {1271.65102},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2012020/}
}
TY  - JOUR
AU  - Larnier, Stanislas
AU  - Masmoudi, Mohamed
TI  - The extended adjoint method
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2013
SP  - 83
EP  - 108
VL  - 47
IS  - 1
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2012020/
DO  - 10.1051/m2an/2012020
LA  - en
ID  - M2AN_2013__47_1_83_0
ER  - 
%0 Journal Article
%A Larnier, Stanislas
%A Masmoudi, Mohamed
%T The extended adjoint method
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2013
%P 83-108
%V 47
%N 1
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2012020/
%R 10.1051/m2an/2012020
%G en
%F M2AN_2013__47_1_83_0
Larnier, Stanislas; Masmoudi, Mohamed. The extended adjoint method. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 47 (2013) no. 1, pp. 83-108. doi : 10.1051/m2an/2012020. http://www.numdam.org/articles/10.1051/m2an/2012020/

[1] G. Allaire, F. De Gournay, F. Jouve and A.-M. Toader, Structural optimization using topological and shape sensitivity via a level set method. Control Cybern. 34 (2005) 59-80. | MR | Zbl

[2] H. Ammari and H. Kang, High-order terms in the asymptotic expansions of the steady-state voltage potentials in the presence of conductivity inhomogeneities of small diameter. SIAM J. Math. Anal. 34 (2003) 1152-1166. | MR | Zbl

[3] H. Ammari and H. Kang, Reconstruction of small inhomogeneities from boundary measurements. Lect. Notes Math. 1846 (2004). | MR | Zbl

[4] H. Ammari and J.K. Seo, An accurate formula for the reconstruction of conductivity inhomogeneities. Adv. Appl. Math. 30 (2003) 679-705. | MR | Zbl

[5] H. Ammari, S. Moskow and M.S. Vogelius, Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume. ESAIM : COCV 9 (2003) 49-66. | Numdam | MR | Zbl

[6] H. Ammari, E. Iakovleva, D. Lesselier and G. Perrusson, MUSIC-type electromagnetic imaging of a collection of small three-dimensional inclusions. SIAM J. Sci. Comput. 29 (2007) 674-709. | MR | Zbl

[7] H. Ammari, E. Bonnetier, Y. Capdeboscq, M. Tanter and M. Fink, Electrical impedance tomography by elastic deformation. SIAM J. Appl. Math. 68 (2008) 1557-1573. | MR | Zbl

[8] H. Ammari, P. Garapon, H. Kang and H. Lee, A method of biological tissues elasticity reconstruction using magnetic resonance elastography measurements. Quart. Appl. Math. 66 (2008) 139-175. | MR | Zbl

[9] H. Ammari, P. Garapon, H. Kang and H. Lee, Separation of scales in elasticity imaging : a numerical study. J. Comput. Math. 28 (2010) 354-370. | Zbl

[10] S. Amstutz, M. Masmoudi and B. Samet, The topological asymptotic for the Helmoltz equation. SIAM J. Control Optim. 42 (2003) 1523-1544. | MR | Zbl

[11] S. Amstutz, I. Horchani and M. Masmoudi, Crack detection by the topological gradient method. Control Cybern. 34 (2005) 81-101. | MR | Zbl

[12] G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing : Partial Differential Equations and the Calculus of Variations. Appl. Math. Sci. 147 (2001). | MR | Zbl

[13] D. Auroux and M. Masmoudi, A one-shot inpainting algorithm based on the topological asymptotic analysis. Comput. Appl. Math. 25 (2006) 251-267. | MR | Zbl

[14] D. Auroux and M. Masmoudi, Image processing by topological asymptotic expansion. J. Math. Imag. Vision 33 (2009) 122-134. | MR

[15] D. Auroux and M. Masmoudi, Image processing by topological asymptotic analysis. ESAIM : Proc. Math. Methods Imag. Inverse Probl. 26 (2009) 24-44. | MR | Zbl

[16] L.J. Belaid, M. Jaoua, M. Masmoudi and L. Siala, Image restoration and edge detection by topological asymptotic expansion. C. R. Acad. Sci. Paris 342 (2006) 313-318. | MR | Zbl

[17] M. Bonnet, Higher-order topological sensitivity for 2-d potential problems. application to fast identification of inclusions. Int. J. Solids Struct. 46 (2009) 2275-2292. | MR | Zbl

[18] M. Bonnet, Fast identification of cracks using higher-order topological sensitivity for 2-d potential problems. Special issue on the advances in mesh reduction methods. In honor of Professor Subrata Mukherjee on the occasion of his 65th birthday. Eng. Anal. Bound. Elem. 35 (2011) 223-235. | MR | Zbl

[19] Y. Capdeboscq and M.S. Vogelius, A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction. ESAIM : M2AN 37 (2003) 159-173. | Numdam | MR | Zbl

[20] Y. Capdeboscq and M.S. Vogelius, Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements. ESAIM : M2AN 37 (2003) 227-240. | Numdam | MR | Zbl

[21] J. Fehrenbach and M. Masmoudi, Coupling topological gradient and Gauss-Newton methods, in IUTAM Symposium on Topological Design Optimization. Edited by M.P. Bendsoe, N. Olhoff and O. Sigmund. Springer (2006).

[22] J. Fehrenbach, M. Masmoudi, R. Souchon and P. Trompette, Detection of small inclusions using elastography. Inverse Probl. 22 (2006) 1055-1069. | MR | Zbl

[23] S. Garreau, P. Guillaume and M. Masmoudi, The topological asymptotic for pde systems : the elasticity case. SIAM J. Control Optim. 39 (2001) 1756-1778. | MR | Zbl

[24] P. Guillaume and M. Hassine, Removing holes in topological shape optimization. ESAIM : COCV 14 (2008) 160-191. | Numdam | MR | Zbl

[25] P. Guillaume and K. Sid Idris, The topological asymptotic expansion for the Dirichlet problem. SIAM J. Control Optim. 41 (2002) 1042-1072. | MR | Zbl

[26] P. Guillaume and K. Sid Idris, The topological sensitivity and shape optimization for the Stokes equations. SIAM J. Control Optim. 43 (2004) 1-31. | MR | Zbl

[27] M. Hassine, S. Jan and M. Masmoudi, From differential calculus to 0-1 topological optimization. SIAM, J. Control Optim. 45 (2007) 1965-1987. | MR | Zbl

[28] S. Larnier and J. Fehrenbach, Edge detection and image restoration with anisotropic topological gradient, in IEEE International Conference on Acoustics Speech and Signal Processing (ICASSP) (2010) 1362-1365.

[29] L. Martin, Conception aérodynamique robuste. Ph.D. thesis, Université Paul Sabatier, Toulouse, France (2011).

[30] M. Masmoudi, The topological asymptotic, in Computational Methods for Control Applications, GAKUTO International Series, edited by R. Glowinski, H. Karawada and J. Periaux. Math. Sci. Appl. 16 (2001) 53-72. | Zbl

[31] B. Mohammadi and O. Pironneau, Shape optimization in fluid mechanics. Annu. Rev. Fluid Mech. 36 (2004) 255-279. | MR | Zbl

[32] J. Ophir, I. Céspedes, H. Ponnekanti, Y. Yazdi and X. Li, Elastography : a quantitative method for imaging the elasticity of biological tissues. Ultrason. Imag. 13 (1991) 111-134.

[33] J. Ophir, S. Alam, B. Garra, F. Kallel, E. Konofagou, T. Krouskop, C. Merritt, R. Righetti, R. Souchon, S. Srinivan and T. Varghese, Elastography : imaging the elastic properties of soft tissues with ultrasound. J. Med. Ultrason. 29 (2002) 155-171.

[34] B. Samet, The topological asymptotic with respect to a singular boundary perturbation. C. R. Math. 336 (2003) 1033-1038. | MR | Zbl

[35] A. Schumacher, Topologieoptimisierung von Bauteilstrukturen unter Verwendung von Lopchpositionierungkrieterien. Ph.D. thesis, Universitat-Gesamthochschule Siegen, Germany (1995).

[36] J. Sokolowski and A. Zochowski, On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999) 1251-1272. | MR | Zbl

[37] Z. Wang, A.C. Bovik, H.R. Sheikh and E.P. Simoncelli, Image quality assessment : from error visibility to structural similarity. IEEE Trans. Image Process. 13 (2004) 600-612.

Cité par Sources :