Skip to main content
SpringerLink
Log in

One-iteration reconstruction algorithm for geometric inverse source problem

  • Published:
Journal of Elliptic and Parabolic Equations Aims and scope Submit manuscript

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.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

References

  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)

    Article  MathSciNet  MATH  Google Scholar 

  2. El Badia, A., El Hajj, A.: Identification of dislocations in materials from boundary measurements. SIAM J. Appl. Math. 73, 84–103 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  4. Atkinson, K., Han,W.: Theoretical Numerical Analysis: A Functional Analysis Framework, In: Texts in Applied Mathematics. vol 39, 3rd edn. Springer (2009)

  5. Amstutz, S.: Sensitivity analysis with respect to a local perturbation of the material property. Asymptot. Anal. 49, 87–108 (2006)

    MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  7. Amstutz, S., Horchani, I., Masmoudi, M.: Crack detection by the topological gradient method. Control Cybern. 34, 81–101 (2005)

    MathSciNet  MATH  Google Scholar 

  8. Brezis, H.: Functional Analysis, Sobolev Spaces and Partial Differential Equations. Springer (2010)

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  11. Canelas, A., Laurain, A., Novotny, A.A.: A new reconstruction method for the inverse potential problem. J. Comput. Phys. 268, 417–431 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  14. Céa, J., Garreau, S., Guillaume, P., Masmoudi, M.: The shape and topological optimizations connection. Compt. Methods Appl. Mech. Eng. 188, 713–726 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  15. Cheng, X., Gong, R., Han, W.: A new general mathematical framework for bioluminescence tomography. Comput. Methods Appl. Mech. Eng. 197, 524–535 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

  19. Garreau, S., Guillaume, P., Masmoudi, M.: The topological asymptotic for pde systems: the elasticity case. SIAM J. Control Optim. 39, 1756–1778 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  20. Guillaume, P., Idris, K.S.: Topological sensitivity and shape optimization for the stokes equations. SIAM J. Control Optim. 43, 1–31 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  21. Han, W., Cong, W., Wang, G.: Mathematical theory and numerical analysis of bioluminescence tomography. Inverse Probl. 22, 1659–1675 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  22. Han, W., Kazmi, K., Cong, W., Wang, G.: Bioluminescence tomography with optimized optical parameters. Inverse Probl. 23, 1215–1228 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  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. Hassine, M., Jan, S., Masmoudi, M.: From differential calculus to \(0-1\) topological optimization. SIAM J Control Optim. 45, 1965–1987 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  25. Hassine, M., Masmoudi, M.: The topological asymptotic expansion for the quasi-stokes problem. ESAIM Control Optim. Calcu. Var. 10, 478–504 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  26. Isakov, V.: Inverse Source Problems, vol. 34. American Mathematical Soc, providence (1990)

    MATH  Google Scholar 

  27. Isakov, V.: Inverse Problems for Partial Differential Equations, vol 127. Springer (2006)

  28. Isakov, V., Leung, S., Qian, J.: A fast local level set method for inverse gravimetry. Commun. Comput. Phys. 10(4), 1044–1070 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  MathSciNet  MATH  Google Scholar 

  30. Kohn, R., Vogelius, M.: Determining conductivity by boundary measurements. Commun. Pure Appl. Math. 37, 289–298 (1984)

    Article  MathSciNet  MATH  Google Scholar 

  31. Larnier, S., Masmoudi, M.: The extended adjoint method. ESAIM Math. Model. Numer. Anal. 47, 83–108 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  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)

    Article  Google Scholar 

  33. Ring, W.: Identification of a core from boundary data. SIAM J. Appl. Math. 55, 677–706 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  34. Sabelli, A., Aquino, W.: A source sensitivity approach for source localization in steady-state linear systems. Inverse Probl. 29, 095005 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  35. Sokolowski, J., Zolésio, J.-P.: Introduction to Shape Optimization. Springer, Berlin, Heiderlberg, New York (1992)

    Book  MATH  Google Scholar 

  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)

    MathSciNet  MATH  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, Faculty of Sciences, Monastir University, Avenue de l’Environnement, 5000, Monastir, Tunisia

    Mourad Hrizi & Maatoug Hassine

Authors
  1. Mourad Hrizi
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Maatoug Hassine
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mourad Hrizi.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Hrizi, M., Hassine, M. One-iteration reconstruction algorithm for geometric inverse source problem. J Elliptic Parabol Equ 4, 177–205 (2018). https://doi.org/10.1007/s41808-018-0015-4

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s41808-018-0015-4

Keywords

Mathematics Subject Classification

Search

Navigation