Journal of Elliptic and Parabolic Equations

, Volume 4, Issue 1, pp 177–205 | Cite as

One-iteration reconstruction algorithm for geometric inverse source problem

Article
  • 22 Downloads

Abstract

In this paper, we address the reconstruction of characteristic source functions \(\delta g^*\) in the elliptic partial differential equations \(-\Delta u+u=\delta g^*\) from the knowledge of the boundary measurements. We will detect the shape and location of the unknown source term from additional boundary conditions. We propose a new reconstruction method based on the Kohn–Vogelius formulation and the topological gradient method. The inverse problem is formulated as a topological optimization one. An asymptotic expansion for an energy function is derived with respect to a small topological perturbation of the source term. The unknown source is reconstructed using a level-set curve of the topological gradient. A non-iterative reconstruction procedure based on the topological sensitivity is implemented. The efficiency and accuracy of the reconstruction algorithm are illustrated by some numerical results.

Keywords

Geometric inverse source problem Topological sensitivity analysis Topological optimization Kohn–Vogelius formulation 

Mathematics Subject Classification

35A15 35B20 49K40 65R32 

References

  1. 1.
    Abda, A.B., Hassine, M., Jaoua, M., Masmoudi, M.: Topological sensitivity analysis for the location of small cavities in stokes flow. SIAM J. Control Optim. 48, 2871–2900 (2009)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    El Badia, A., El Hajj, A.: Identification of dislocations in materials from boundary measurements. SIAM J. Appl. Math. 73, 84–103 (2013)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Abdelaziz, B., El Badia, A., El Hajj, A.: Direct algorithms for solving some inverse source problems in \(2D\) elliptic equations. Inverse Probl. 31, 105002 (2015)MathSciNetCrossRefMATHGoogle Scholar
  4. 4.
    Atkinson, K., Han,W.: Theoretical Numerical Analysis: A Functional Analysis Framework, In: Texts in Applied Mathematics. vol 39, 3rd edn. Springer (2009)Google Scholar
  5. 5.
    Amstutz, S.: Sensitivity analysis with respect to a local perturbation of the material property. Asymptot. Anal. 49, 87–108 (2006)MathSciNetMATHGoogle Scholar
  6. 6.
    Amstutz, S., Giusti, S.M., Novotny, V.V., de Souza Neto, E.A.: Topological derivative for multi-scale linear elasticity models applied to the synthesis of microstructures. Int. J. Numer. Methods Eng. 84, 733–756 (2010)MathSciNetCrossRefMATHGoogle Scholar
  7. 7.
    Amstutz, S., Horchani, I., Masmoudi, M.: Crack detection by the topological gradient method. Control Cybern. 34, 81–101 (2005)MathSciNetMATHGoogle Scholar
  8. 8.
    Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer (2010)Google Scholar
  9. 9.
    Cabayan, H.S., Belford, G.G.: On computing a stable least squares solution to the inverse problem for a planar newtonian potential. SIAM J. Appl. Math. 20, 51–61 (1971)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Caubet, F., Conca, C., Godoy, M.: On the detection of several obstacles in 2D Stokes flow: topological sensitivity and combination with shape derivatives. Inverse Probl. Imaging. 10, 327–67 (2016)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Canelas, A., Laurain, A., Novotny, A.A.: A new reconstruction method for the inverse potential problem. J. Comput. Phys. 268, 417–431 (2012)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Canelas, A., Laurain, A., Novotny, A.A.: A new reconstruction method for the inverse source problem from partial boundary measurements. Inverse Probl. 31, 075009 (2015)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Chaabane, S., Masmoudi, M., Meftahi, H.: Topological and shape gradient strategy for solving geometrical inverse problems. J. Math. Anal. Appl. 400, 724–742 (2013)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Céa, J., Garreau, S., Guillaume, P., Masmoudi, M.: The shape and topological optimizations connection. Compt. Methods Appl. Mech. Eng. 188, 713–726 (2000)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Cheng, X., Gong, R., Han, W.: A new general mathematical framework for bioluminescence tomography. Comput. Methods Appl. Mech. Eng. 197, 524–535 (2008)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    Cheng, X., Gong, R., Han, W., Zheng, X.: A novel coupled complex boundary method for solving inverse source problems. Inverse Probl. 30, 055002 (2014)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    Dominguez, N., Gibiat, V., Esquerre, Y.: Time domain topological gradient and time reversal analogy: an inverse method for ultrasonic target detection. Wave Motion. 42, 31–52 (2005)MathSciNetCrossRefMATHGoogle Scholar
  18. 18.
    Ferchichi, J., Hassine, M., Khenous, H.: Detection of point-forces location using topological algorithm in stokes flows. Appl. Math. Comput. 219, 7056–7074 (2013)MathSciNetMATHGoogle Scholar
  19. 19.
    Garreau, S., Guillaume, P., Masmoudi, M.: The topological asymptotic for pde systems: the elasticity case. SIAM J. Control Optim. 39, 1756–1778 (2001)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Guillaume, P., Idris, K.S.: Topological sensitivity and shape optimization for the stokes equations. SIAM J. Control Optim. 43, 1–31 (2004)MathSciNetCrossRefMATHGoogle Scholar
  21. 21.
    Han, W., Cong, W., Wang, G.: Mathematical theory and numerical analysis of bioluminescence tomography. Inverse Probl. 22, 1659–1675 (2006)MathSciNetCrossRefMATHGoogle Scholar
  22. 22.
    Han, W., Kazmi, K., Cong, W., Wang, G.: Bioluminescence tomography with optimized optical parameters. Inverse Probl. 23, 1215–1228 (2007)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    Hassine, M., Hrizi, M.: Topological sensitivity analysis for reconstruction of the inverse source problem from boundary measurement. Int. J. Math. Comput. Phys. Electr. Comput. Eng. 11, 26–32 (2017)Google Scholar
  24. 24.
    Hassine, M., Jan, S., Masmoudi, M.: From differential calculus to \(0-1\) topological optimization. SIAM J Control Optim. 45, 1965–1987 (2007)MathSciNetCrossRefMATHGoogle Scholar
  25. 25.
    Hassine, M., Masmoudi, M.: The topological asymptotic expansion for the quasi-stokes problem. ESAIM Control Optim. Calcu. Var. 10, 478–504 (2004)MathSciNetCrossRefMATHGoogle Scholar
  26. 26.
    Isakov, V.: Inverse Source Problems, vol. 34. American Mathematical Soc, providence (1990)MATHGoogle Scholar
  27. 27.
    Isakov, V.: Inverse Problems for Partial Differential Equations, vol 127. Springer (2006)Google Scholar
  28. 28.
    Isakov, V., Leung, S., Qian, J.: A fast local level set method for inverse gravimetry. Commun. Comput. Phys. 10(4), 1044–1070 (2011)MathSciNetCrossRefMATHGoogle Scholar
  29. 29.
    Jleli, M., Samet, B., Vial, V.: Topological sensitivity analysis for the modified Helmholtz equation under an impedance condition on the boundary of a hole. J. Math. Pures Appl. 103, 557–574 (2015)MathSciNetCrossRefMATHGoogle Scholar
  30. 30.
    Kohn, R., Vogelius, M.: Determining conductivity by boundary measurements. Commun. Pure Appl. Math. 37, 289–298 (1984)MathSciNetCrossRefMATHGoogle Scholar
  31. 31.
    Larnier, S., Masmoudi, M.: The extended adjoint method. ESAIM Math. Model. Numer. Anal. 47, 83–108 (2013)MathSciNetCrossRefMATHGoogle Scholar
  32. 32.
    Lv, Y., Tian, J., Cong, W., Wang, G., Luo, J., Yang, W., Li, H.: A multilevel adaptive finite element algorithm for bioluminescence tomography. Opt. Express. 14, 8211–8223 (2006)CrossRefGoogle Scholar
  33. 33.
    Ring, W.: Identification of a core from boundary data. SIAM J. Appl. Math. 55, 677–706 (1995)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    Sabelli, A., Aquino, W.: A source sensitivity approach for source localization in steady-state linear systems. Inverse Probl. 29, 095005 (2013)MathSciNetCrossRefMATHGoogle Scholar
  35. 35.
    Sokolowski, J., Zolésio, J.-P.: Introduction to Shape Optimization. Springer, Berlin, Heiderlberg, New York (1992)CrossRefMATHGoogle Scholar
  36. 36.
    Wang, T., Gao, S., Zhang, L., Wu, Y., He, X., Hou, Y., Huang, H., Tian, J.: Overlap domain decomposition method for bioluminescence tomography (blt). Int. J. Numer. Methods Biomed. Eng. 26, 511–523 (2010)MathSciNetMATHGoogle Scholar

Copyright information

© Orthogonal Publishing and Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Faculty of SciencesMonastir UniversityMonastirTunisia

Personalised recommendations