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The extended adjoint method

Published online by Cambridge University Press:  31 July 2012

Stanislas Larnier  and
Mohamed Masmoudi
Stanislas Larnier
Affiliation:
UniversitéPaul Sabatier, Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse, France. [email protected]; [email protected]
Mohamed Masmoudi
Affiliation:
UniversitéPaul Sabatier, Institut de Mathématiques de Toulouse, 118 route de Narbonne, 31062 Toulouse, France. [email protected]; [email protected]
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Abstract

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

Keywords

Adjoint methodtopology optimizationcalculus of variations
Type
Research Article
Information
ESAIM: Mathematical Modelling and Numerical Analysis , Volume 47 , Issue 1 , January 2013 , pp. 83 - 108
Copyright
© EDP Sciences, SMAI, 2012

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References

Allaire, G., de Gournay, F., Jouve, F. and Toader, A.-M., Structural optimization using topological and shape sensitivity via a level set method. Control Cybern. 34 (2005) 59–80. Google Scholar
Ammari, H. and Kang, H., 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. Google Scholar
H. Ammari and H. Kang, Reconstruction of small inhomogeneities from boundary measurements. Lect. Notes Math. 1846 (2004).
Ammari, H. and Seo, J.K., An accurate formula for the reconstruction of conductivity inhomogeneities. Adv. Appl. Math. 30 (2003) 679–705. Google Scholar
Ammari, H., Moskow, S. and Vogelius, M.S., Boundary integral formulae for the reconstruction of electric and electromagnetic inhomogeneities of small volume. ESAIM : COCV 9 (2003) 49–66. Google Scholar
Ammari, H., Iakovleva, E., Lesselier, D. and Perrusson, G., MUSIC-type electromagnetic imaging of a collection of small three-dimensional inclusions. SIAM J. Sci. Comput. 29 (2007) 674–709. Google Scholar
Ammari, H., Bonnetier, E., Capdeboscq, Y., Tanter, M. and Fink, M., Electrical impedance tomography by elastic deformation. SIAM J. Appl. Math. 68 (2008) 1557–1573. Google Scholar
Ammari, H., Garapon, P., Kang, H. and Lee, H., A method of biological tissues elasticity reconstruction using magnetic resonance elastography measurements. Quart. Appl. Math. 66 (2008) 139–175. Google Scholar
Ammari, H., Garapon, P., Kang, H. and Lee, H., Separation of scales in elasticity imaging : a numerical study. J. Comput. Math. 28 (2010) 354–370. Google Scholar
Amstutz, S., Masmoudi, M. and Samet, B., The topological asymptotic for the Helmoltz equation. SIAM J. Control Optim. 42 (2003) 1523–1544. Google Scholar
Amstutz, S., Horchani, I. and Masmoudi, M., Crack detection by the topological gradient method. Control Cybern. 34 (2005) 81–101. Google Scholar
G. Aubert and P. Kornprobst, Mathematical Problems in Image Processing : Partial Differential Equations and the Calculus of Variations. Appl. Math. Sci. 147 (2001).
Auroux, D. and Masmoudi, M., A one-shot inpainting algorithm based on the topological asymptotic analysis. Comput. Appl. Math. 25 (2006) 251–267. Google Scholar
Auroux, D. and Masmoudi, M., Image processing by topological asymptotic expansion. J. Math. Imag. Vision 33 (2009) 122–134. Google Scholar
Auroux, D. and Masmoudi, M., Image processing by topological asymptotic analysis. ESAIM : Proc. Math. Methods Imag. Inverse Probl. 26 (2009) 24–44. Google Scholar
Belaid, L.J., Jaoua, M., Masmoudi, M. and Siala, L., Image restoration and edge detection by topological asymptotic expansion. C. R. Acad. Sci. Paris 342 (2006) 313–318. Google Scholar
Bonnet, M., Higher-order topological sensitivity for 2-d potential problems. application to fast identification of inclusions. Int. J. Solids Struct. 46 (2009) 2275–2292. Google Scholar
Bonnet, M., 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. Google Scholar
Capdeboscq, Y. and Vogelius, M.S., A general representation formula for boundary voltage perturbations caused by internal conductivity inhomogeneities of low volume fraction. ESAIM : M2AN 37 (2003) 159–173. Google Scholar
Capdeboscq, Y. and Vogelius, M.S., Optimal asymptotic estimates for the volume of internal inhomogeneities in terms of multiple boundary measurements. ESAIM : M2AN 37 (2003) 227–240. Google Scholar
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).
Fehrenbach, J., Masmoudi, M., Souchon, R. and Trompette, P., Detection of small inclusions using elastography. Inverse Probl. 22 (2006) 1055–1069. Google Scholar
Garreau, S., Guillaume, P. and Masmoudi, M., The topological asymptotic for pde systems : the elasticity case. SIAM J. Control Optim. 39 (2001) 1756–1778. Google Scholar
Guillaume, P. and Hassine, M., Removing holes in topological shape optimization. ESAIM : COCV 14 (2008) 160–191. Google Scholar
Guillaume, P. and Sid Idris, K., The topological asymptotic expansion for the Dirichlet problem. SIAM J. Control Optim. 41 (2002) 1042–1072. Google Scholar
Guillaume, P. and Sid Idris, K., The topological sensitivity and shape optimization for the Stokes equations. SIAM J. Control Optim. 43 (2004) 1–31. Google Scholar
Hassine, M., Jan, S. and Masmoudi, M., From differential calculus to 0-1 topological optimization. SIAM, J. Control Optim. 45 (2007) 1965–1987. Google Scholar
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.
L. Martin, Conception aérodynamique robuste. Ph.D. thesis, Université Paul Sabatier, Toulouse, France (2011).
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.
Mohammadi, B. and Pironneau, O., Shape optimization in fluid mechanics. Annu. Rev. Fluid Mech. 36 (2004) 255–279. Google Scholar
Ophir, J., Céspedes, I., Ponnekanti, H., Yazdi, Y. and Li, X., Elastography : a quantitative method for imaging the elasticity of biological tissues. Ultrason. Imag. 13 (1991) 111–134. Google Scholar
Ophir, J., Alam, S., Garra, B., Kallel, F., Konofagou, E., Krouskop, T., Merritt, C., Righetti, R., Souchon, R., Srinivan, S. and Varghese, T., Elastography : imaging the elastic properties of soft tissues with ultrasound. J. Med. Ultrason. 29 (2002) 155–171. Google ScholarPubMed
Samet, B., The topological asymptotic with respect to a singular boundary perturbation. C. R. Math. 336 (2003) 1033–1038. Google Scholar
A. Schumacher, Topologieoptimisierung von Bauteilstrukturen unter Verwendung von Lopchpositionierungkrieterien. Ph.D. thesis, Universitat-Gesamthochschule Siegen, Germany (1995).
Sokolowski, J. and Zochowski, A., On the topological derivative in shape optimization. SIAM J. Control Optim. 37 (1999) 1251–1272. Google Scholar
Wang, Z., Bovik, A.C., Sheikh, H.R. and Simoncelli, E.P., Image quality assessment : from error visibility to structural similarity. IEEE Trans. Image Process. 13 (2004) 600–612. Google Scholar