Approximate Global Convergence in Imaging of Land Mines from Backscattered Data

  • Larisa Beilina
  • Michael V. Klibanov
Conference paper
Part of the Springer Proceedings in Mathematics & Statistics book series (PROMS, volume 48)


We present new model of an approximate globally convergent method in the most challenging case of the backscattered data. In this case data for the coefficient inverse problem are given only at the backscattered side of the medium which should be reconstructed. We demonstrate efficiency and robustness of the proposed technique on the numerical solution of the coefficient inverse problem in two dimensions with the time-dependent backscattered data. Goal of our tests is to reconstruct dielectrics in land mines which is the special case of interest in military applications. Our tests show that refractive indices and locations of dielectric abnormalities are accurately imaged.



The research of the authors was supported by US Army Research Laboratory and US Army Research Office grant W911NF-11-1-0399, the Swedish Research Council, the Swedish Foundation for Strategic Research (SSF) in Gothenburg Mathematical Modelling Centre (GMMC), and by the Swedish Institute, Visby Program.


  1. 1.
    N.V. Alexeenko, V.A. Burov and O.D. Rumyantseva, Solution of three-dimensional acoustical inverse problem: II. Modified Novikov algorithm, Acoust. Phys., 54, 407–419, 2008.Google Scholar
  2. 2.
    H. Ammari, E. Iakovleva, G. Perruson and D. Lesselier, Music-type electromagnetic imaging of a collection of small three dimensional inclusions, SIAM J. Sci.Comp., 29, 674–709, 2007.Google Scholar
  3. 3.
    M. Asadzadeh and L. Beilina, A posteriori error analysis in a globally convergent numerical method for a hyperbolic coefficient inverse problem, Inv. Probl., 26, 115007, 2010.MathSciNetCrossRefGoogle Scholar
  4. 4.
    L. Beilina and M.V. Klibanov, A globally convergent numerical method for a coefficient inverse problem,  SIAM J. Sci. Comp., 31, 478–509, 2008.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    L. Beilina and M.V. Klibanov, Synthesis of global convergence and adaptivity for a hyperbolic coefficient inverse problem in 3D, J. Inv. Ill-posed Probl., 18, 85–132, 2010.MathSciNetGoogle Scholar
  6. 6.
    L. Beilina and M.V. Klibanov, A posteriori error estimates for the adaptivity technique for the Tikhonov functional and global convergence for a coefficient inverse problem, Inv. Probl., 26, 045012, 2010.MathSciNetCrossRefGoogle Scholar
  7. 7.
    L. Beilina, M.V. Klibanov and A. Kuzhuget, New a posteriori error estimates for adaptivity technique and global convergence for a hyperbolic coefficient inverse problem, J. Math. Sci., 172, 449–476, 2011.MathSciNetMATHCrossRefGoogle Scholar
  8. 8.
    L. Beilina and M.V. Klibanov, Reconstruction of dielectrics from experimental data via a hybrid globally convergent/adaptive inverse algorithm, Inv. Probl., 26, 125009, 2010.MathSciNetCrossRefGoogle Scholar
  9. 9.
    L. Beilina and M.V. Klibanov, Approximate global convergence and adaptivity for Coefficient Inverse Problems, Springer, New-York, 2012.MATHCrossRefGoogle Scholar
  10. 10.
    L. Beilina, K. Samuelsson and K. Åhlander, Efficiency of a hybrid method for the wave equation. In  International Conference on Finite Element Methods, Gakuto International Series Mathematical Sciences and Applications. Gakkotosho CO., LTD, 2001.Google Scholar
  11. 11.
    V.A. Burov, S.A. Morozov and O.D. Rumyantseva, Reconstruction of fine-scale structure of acoustical scatterers on large-scale contrast background, Acoustical Imaging, 26, 231–238, 2002.CrossRefGoogle Scholar
  12. 12.
    K. Chadan and P.C. Sabatier, Inverse Problems in Quantum Scattering Theory, Springer-Verlag, New York, 1989.MATHCrossRefGoogle Scholar
  13. 13.
    Y. Chen, R. Duan and V. Rokhlin, On the inverse scattering problem in the acoustic environment. J. Comput. Phys., 228, 3209–3231, 2009.MathSciNetCrossRefGoogle Scholar
  14. 14.
    M. Cheney and D. Isaacson, Inverse problems for a perturbed dissipative half-space,  Inverse Problems, 11, 865- 888, 1995.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    D. Isaacson, J.L. Mueller, J.C. Newell and S. Siltanen, Imaging cardiac activity by the D-bar methods for electrical impedance tomography, Physiological Measurements, 27, S43-S50, 2006.CrossRefGoogle Scholar
  16. 16.
    M. V. Klibanov, Uniqueness of solutions in the ‘large’ of some multidimensional inverse problems, in Non-Classical Problems of Mathematical Physics, 101–114, 1981, published by Computing Center of the Siberian Branch of the USSR Academy of Science, Novosibirsk (in Russian).Google Scholar
  17. 17.
    M. V. Klibanov and A Timonov, A unified framework for constructing the globally convergent algorithms for multidimensional coefficient inverse problems, Applicable Analysis, 83, 933–955, 2004.MathSciNetMATHCrossRefGoogle Scholar
  18. 18.
    M. V. Klibanov and A. Timonov,  Carleman Estimates for Coefficient Inverse Problems and Numerical Applications, VSP, Utrecht, 2004.MATHCrossRefGoogle Scholar
  19. 19.
    M. V. Klibanov, M. A. Fiddy, L. Beilina, N. Pantong and J. Schenk, Picosecond scale experimental verification of a globally convergent numerical method for a coefficient inverse problem, Inverse Problems, 26, 045003, 2010.MathSciNetCrossRefGoogle Scholar
  20. 20.
    M. V. Klibanov, J. Su, N. Pantong, H. Shan and H. Liu, A globally convergent numerical method for an inverse elliptic problem of optical tomography, Applicable Analysis, 6, 861–891, 2010.MathSciNetCrossRefGoogle Scholar
  21. 21.
    A. V. Kuzhuget and M. V. Klibanov, Global convergence for a 1-D inverse problem with application to imaging of land mines, Applicable Analysis, 89, 125–157, 2010.MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    A. V. Kuzhuget, N. Pantong and M. V. Klibanov, A globally convergent numerical method for a coefficient inverse problem with backscattering data, Methods and Applications of Analysis, 18, 47–68, 2011.MathSciNetMATHGoogle Scholar
  23. 23.
    R. G. Novikov, The  − bar approach to approximate inverse scattering at fixed energy in three dimensions, Int. Math. Res. Reports, 6, 287–349, 2005.Google Scholar
  24. 24.
    L. Pestov, V. Bolgova, O. Kazarina, Numerical recovering of a density by the BC-method.  Inverse Probl. Imaging, 4, 703–712, 2010.MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    Portable, Extensible Toolkit for Scientific Computation PETSc at
  26. 26.
    V. G. Romanov 1986 Inverse Problems of Mathematical Physics (Utrecht, The Netherlands: VNU).Google Scholar
  27. 27.
  28. 28.
    J. Xin and M. V. Klibanov, Numerical solution of an inverse problem of imaging of antipersonnel land mines by the globally convergent convexification algorithm, SIAM J. Sci. Comp., 30, 3170–3196, 2008.MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    Software package WavES at

Copyright information

© Springer Science+Business Media, LLC 2013

Authors and Affiliations

  1. 1.Department of Mathematical SciencesChalmers University of Technology and Gothenburg UniversityGothenburgSweden
  2. 2.Department of Mathematics and StatisticsUniversity of North Carolina at CharlotteCharlotteUSA

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