The extended adjoint method
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 1, p. 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 : https://doi.org/10.1051/m2an/2012020
Classification:  49Q10,  49Q12,  74P10,  74P15
Keywords: 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 - Mod\'elisation Math\'ematique et Analyse Num\'erique},
     publisher = {EDP-Sciences},
     volume = {47},
     number = {1},
     year = {2013},
     pages = {83-108},
     doi = {10.1051/m2an/2012020},
     zbl = {1271.65102},
     mrnumber = {2968696},
     language = {en},
     url = {http://www.numdam.org/item/M2AN_2013__47_1_83_0}
}
Larnier, Stanislas; Masmoudi, Mohamed. The extended adjoint method. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 47 (2013) no. 1, pp. 83-108. doi : 10.1051/m2an/2012020. http://www.numdam.org/item/M2AN_2013__47_1_83_0/

[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 2211063 | Zbl 1167.49324

[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 2001663 | Zbl 1036.35050

[3] H. Ammari and H. Kang, Reconstruction of small inhomogeneities from boundary measurements. Lect. Notes Math. 1846 (2004). | MR 2168949 | Zbl 1113.35148

[4] H. Ammari and J.K. Seo, An accurate formula for the reconstruction of conductivity inhomogeneities. Adv. Appl. Math. 30 (2003) 679-705. | MR 1977849 | Zbl 1040.78008

[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 1957090 | Zbl 1075.78010

[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 2306264 | Zbl 1132.78308

[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 2424952 | Zbl 1156.35101

[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 2396655 | Zbl 1143.35384

[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 1222.92051

[10] S. Amstutz, M. Masmoudi and B. Samet, The topological asymptotic for the Helmoltz equation. SIAM J. Control Optim. 42 (2003) 1523-1544. | MR 2046373 | Zbl 1051.49029

[11] S. Amstutz, I. Horchani and M. Masmoudi, Crack detection by the topological gradient method. Control Cybern. 34 (2005) 81-101. | MR 2211064 | Zbl 1167.74437

[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 2244145 | Zbl 1110.35001

[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 2321652 | Zbl 1182.94006

[14] D. Auroux and M. Masmoudi, Image processing by topological asymptotic expansion. J. Math. Imag. Vision 33 (2009) 122-134. | MR 2480980

[15] D. Auroux and M. Masmoudi, Image processing by topological asymptotic analysis. ESAIM : Proc. Math. Methods Imag. Inverse Probl. 26 (2009) 24-44. | MR 2498137 | Zbl 1183.68679

[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 2201955 | Zbl 1086.68141

[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 2517928 | Zbl 1217.74095

[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 2740347 | Zbl 1259.74025

[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 1972656 | Zbl 1137.35346

[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 1991198 | Zbl 1137.35347

[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 2235654 | Zbl 1099.74028

[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 1825864 | Zbl 0990.49028

[24] P. Guillaume and M. Hassine, Removing holes in topological shape optimization. ESAIM : COCV 14 (2008) 160-191. | Numdam | MR 2375755 | Zbl 1140.49029

[25] P. Guillaume and K. Sid Idris, The topological asymptotic expansion for the Dirichlet problem. SIAM J. Control Optim. 41 (2002) 1042-1072. | MR 1972502 | Zbl 1053.49031

[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 2081970 | Zbl 1093.49029

[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 2285710 | Zbl 1139.49039

[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 1082.93584

[31] B. Mohammadi and O. Pironneau, Shape optimization in fluid mechanics. Annu. Rev. Fluid Mech. 36 (2004) 255-279. | MR 2062314 | Zbl 1076.76020

[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 1993977 | Zbl 1028.65123

[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 1691940 | Zbl 0940.49026

[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.