About this Journal Submit a Manuscript Table of Contents
Abstract and Applied Analysis
Volume 2013 (2013), Article ID 816134, 20 pages
http://dx.doi.org/10.1155/2013/816134
Research Article

Hybrid Topological Derivative-Gradient Based Methods for Nondestructive Testing

1Departamento de Matemática Aplicada, Universidad Complutense de Madrid, 28040 Madrid, Spain
2Departamento de Fundamentos Matematicos, Universidad Politécnica de Madrid, 28040 Madrid, Spain

Received 26 December 2012; Accepted 9 June 2013

Academic Editor: Ağacik Zafer

Copyright © 2013 A. Carpio and M.-L. Rapún. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Linked References

  1. A. Liseno and R. Pierre, “Imaging of voids by means of a physical-optics-based shape-reconstrution algorithm,” Journal of the Optical Society of America A, vol. 21, no. 6, pp. 968–974, 2004. View at Publisher · View at Google Scholar · View at Scopus
  2. A. J. Devaney, “Geophysical diffraction tomography,” IEEE Transactions on Geoscience and Remote Sensing, vol. 22, no. 1, pp. 3–13, 1984. View at Publisher · View at Google Scholar · View at Scopus
  3. D. Colton, K. Giebermann, and P. Monk, “A regularized sampling method for solving three-dimensional inverse scattering problems,” SIAM Journal on Scientific Computing, vol. 21, no. 6, pp. 2316–2330, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  4. D. Colton and A. Kirsch, “A simple method for solving inverse scattering problems in the resonance region,” Inverse Problems, vol. 12, no. 4, pp. 383–393, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  5. F. Natterer and F. Wübbeling, “A propagation-backpropagation method for ultrasound tomography,” Inverse Problems, vol. 11, no. 6, pp. 1225–1232, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  6. R. E. Kleinman and P. M. van den Berg, “A modified gradient method for two-dimensional problems in tomography,” Journal of Computational and Applied Mathematics, vol. 42, no. 1, pp. 17–35, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  7. F. Hettlich, “Fréchet derivatives in inverse obstacle scattering,” Inverse Problems, vol. 11, no. 2, pp. 371–382, 1995. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  8. A. Kirsch, “The domain derivative and two applications in inverse scattering theory,” Inverse Problems, vol. 9, no. 1, pp. 81–96, 1993. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  9. M. Masmoudi, Outils pour la conception optimale des formes [Thése d'Etat en Sciences Mathématiques], Université de Nice, 1987.
  10. R. Potthast, “Fréchet differentiability of the solution to the acoustic Neumann scattering problem with respect to the domain,” Journal of Inverse and Ill-Posed Problems, vol. 4, no. 1, pp. 67–84, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  11. A. Litman, D. Lesselier, and F. Santosa, “Reconstruction of a two-dimensional binary obstacle by controlled evolution of a level-set,” Inverse Problems, vol. 14, no. 3, pp. 685–706, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  12. F. Santosa, “A level-set approach for inverse problems involving obstacles,” Optimisation et Calcul des Variations, vol. 1, pp. 17–33, 1996. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  13. G. R. Feijoo, “A new method in inverse scattering based on the topological derivative,” Inverse Problems, vol. 20, no. 6, pp. 1819–1840, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  14. M. Bonnet and B. B. Guzina, “Sounding of finite solid bodies by way of topological derivative,” International Journal for Numerical Methods in Engineering, vol. 61, no. 13, pp. 2344–2373, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  15. B. B. Guzina and M. Bonnet, “Small-inclusion asymptotic of misfit functionals for inverse problems in acoustics,” Inverse Problems, vol. 22, no. 5, pp. 1761–1785, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  16. B. Samet, S. Amstutz, and M. Masmoudi, “The topological asymptotic for the Helmholtz equation,” SIAM Journal on Control and Optimization, vol. 42, no. 5, pp. 1523–1544, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  17. M. Bonnet and A. Constantinescu, “Inverse problems in elasticity,” Inverse Problems, vol. 21, no. 2, pp. R1–R50, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  18. O. Dorn and D. Lesselier, “Level set methods for inverse scattering,” Inverse Problems, vol. 22, no. 4, pp. R67–R131, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  19. M. Burger, B. Hackl, and W. Ring, “Incorporating topological derivatives into level set methods,” Journal of Computational Physics, vol. 194, no. 1, pp. 344–362, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  20. S. Garreau, P. Guillaume, and M. Masmoudi, “The topological asymptotic for PDE systems: the elasticity case,” SIAM Journal on Control and Optimization, vol. 39, no. 6, pp. 1756–1778, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  21. A. Carpio and M.-L. Rapún, “Solving inhomogeneous inverse problems by topological derivative methods,” Inverse Problems, vol. 24, no. 4, Article ID 045014, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  22. A. Carpio and M.-L. Rapún, “An iterative method for parameter identification and shape reconstruction,” Inverse Problems in Science and Engineering, vol. 18, no. 1, pp. 35–50, 2010. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  23. A. Carpio and M.-L. Rapún, “Hybrid topological derivative and gradient-based methods for electrical impedance tomography,” Inverse Problems, vol. 28, no. 9, Article ID 095010, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  24. H. Ammari, J. Garnier, V. Jugnon, and H. Kang, “Direct reconstruction methods in ultrasound imaging of small anomalies,” in Mathematical Modelling in Biomedical Imaging II, vol. 2035 of Lecture Notes in Mathematics, Springer, Berlin, Germany, 2012. View at Publisher · View at Google Scholar
  25. H. Ammari, J. Garnier, V. Jugnon, and H. Kang, “Stability and resolution analysis for a topological derivative based imaging functional,” SIAM Journal on Control and Optimization, vol. 50, no. 1, pp. 48–76, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  26. L. Borcea, G. Papanicolaou, C. Tsogka, and J. Berryman, “Imaging and time reversal in random media,” Inverse Problems, vol. 18, no. 5, pp. 1247–1279, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  27. A. Carpio and M.-L. Rapún, “Domain reconstruction using photothermal techniques,” Journal of Computational Physics, vol. 227, no. 17, pp. 8083–8106, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  28. L. Borcea, “Electrical impedance tomography,” Inverse Problems, vol. 18, no. 6, pp. R99–R136, 2002. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  29. M. Cheney, D. Isaacson, and J. C. Newell, “Electrical impedance tomography,” SIAM Review, vol. 41, no. 1, pp. 85–101, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  30. M. Brühl and M. Hanke, “Numerical implementation of two noniterative methods for locating inclusions by impedance tomography,” Inverse Problems, vol. 16, no. 4, pp. 1029–1042, 2000. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  31. E. T. Chung, T. F. Chan, and X.-C. Tai, “Electrical impedance tomography using level set representation and total variational regularization,” Journal of Computational Physics, vol. 205, no. 1, pp. 357–372, 2005. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  32. M. Hintermüller and A. Laurain, “Electrical impedance tomography: from topology to shape,” Control and Cybernetics, vol. 37, no. 4, pp. 913–933, 2008. View at MathSciNet
  33. M. Hintermüller, A. Laurain, and A. A. Novotny, “Second-order topological expansion for electrical impedance tomography,” Advances in Computational Mathematics, vol. 36, no. 2, pp. 235–265, 2012. View at Publisher · View at Google Scholar · View at Zentralblatt MATH
  34. B. Hofmann, “Approximation of the inverse electrical impedance tomography problem by an inverse transmission problem,” Inverse Problems, vol. 14, no. 5, pp. 1171–1187, 1998. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  35. K. Ito, K. Kunisch, and Z. Li, “Level-set function approach to an inverse interface problem,” Inverse Problems, vol. 17, no. 5, pp. 1225–1242, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  36. M. Hanke and M. Brühl, “Recent progress in electrical impedance tomography,” Inverse Problems, vol. 19, no. 6, pp. S65–S90, 2003. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  37. L. Borcea, “A nonlinear multigrid for imaging electrical conductivity and permittivity at low frequency,” Inverse Problems, vol. 17, no. 2, pp. 329–359, 2001. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  38. K. A. Dines and R. J. Lytle, “Analysis of electrical conductivity imaging,” Geophysics, vol. 46, no. 7, pp. 1025–1036, 1981. View at Publisher · View at Google Scholar · View at Scopus
  39. T. J. Yorkey, J. G. Webster, and W. J. Tompkins, “Comparing reconstruction algorithms for electrical impedance tomography,” IEEE Transactions on Biomedical Engineering, vol. 34, no. 11, pp. 843–852, 1987. View at Scopus
  40. B. B. Guzina and I. Chikichev, “From imaging to material identification: a generalized concept of topological sensitivity,” Journal of the Mechanics and Physics of Solids, vol. 55, no. 2, pp. 245–279, 2007. View at Publisher · View at Google Scholar · View at MathSciNet
  41. M. Abramovitz and I. A. Stegun, Handbook of Mathematical Functions, Dover, New York, NY, USA, 1972.
  42. J. Sokołowski and A. Żochowski, “On the topological derivative in shape optimization,” SIAM Journal on Control and Optimization, vol. 37, no. 4, pp. 1251–1272, 1999. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  43. A. Carpio and M. L. Rapún, “Topological derivatives for shape reconstruction,” in Inverse Problems and Imaging, vol. 1943 of Lecture Notes in Mathematics, pp. 85–133, Springer, Berlin, Germany, 2008. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  44. G. R. Feijóo, A. A. Oberai, and P. M. Pinsky, “An application of shape optimization in the solution of inverse acoustic scattering problems,” Inverse Problems, vol. 20, no. 1, pp. 199–228, 2004. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  45. M.-L. Rapún and F.-J. Sayas, “Indirect methods with Brakhage-Werner potentials for Helmholtz transmission problems,” in Numerical Mathematics and Advanced Applications: Proceedings of ENUMATH 2005, the 6th European Conference on Numerical Mathematics and Advanced Applications Santiago de Compostela, Spain, July 2005, pp. 1146–1154, Springer, Berlin, Germany, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  46. M.-L. Rapún and F.-J. Sayas, “A mixed-FEM and BEM coupling for the approximation of the scattering of thermal waves in locally non-homogeneous media,” ESAIM: Mathematical Modelling and Numerical Analysis, vol. 40, no. 5, pp. 871–896, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  47. A. Carpio, B. T. Johansson, and M.-L. Rapún, “Determining planar multiple sound-soft obstacles from scattered acoustic fields,” Journal of Mathematical Imaging and Vision, vol. 36, no. 2, pp. 185–199, 2010. View at Publisher · View at Google Scholar · View at MathSciNet
  48. D. Colton and B. D. Sleeman, “Uniqueness theorems for the inverse problem of acoustic scattering,” IMA Journal of Applied Mathematics, vol. 31, no. 3, pp. 253–259, 1983. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  49. T. Johansson and B. D. Sleeman, “Reconstruction of an acoustically sound-soft obstacle from one incident field and the far-field pattern,” IMA Journal of Applied Mathematics, vol. 72, no. 1, pp. 96–112, 2007. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  50. M. Cheney, D. Isaacson, and E. Somersalo, “Existence and uniqueness for electrode models for electric current computed tomography,” SIAM Journal on Applied Mathematics, vol. 52, no. 4, pp. 1023–1040, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  51. A. Lechleiter and A. Rieder, “Newton regularizations for impedance tomography: a numerical study,” Inverse Problems, vol. 22, no. 6, pp. 1967–1987, 2006. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet
  52. D. C. Dobson, “Convergence of a reconstruction method for the inverse conductivity problem,” SIAM Journal on Applied Mathematics, vol. 52, no. 2, pp. 442–458, 1992. View at Publisher · View at Google Scholar · View at Zentralblatt MATH · View at MathSciNet