The Inverse Medium Problem

  • David Colton
  • Rainer Kress
Part of the Applied Mathematical Sciences book series (AMS, volume 93)


We now turn our attention to the problem of reconstructing the refractive index from a knowledge of the far field pattern of the scattered acoustic or electromagnetic wave. We shall call this problem the inverse medium problem. We first consider the case of acoustic waves and the use of the Lippmann–Schwinger equation to reformulate the acoustic inverse medium problem as a problem in constrained optimization. Included here is a brief discussion of the use of the Born approximation to linearize the problem. We then proceed to the proof of a uniqueness theorem for the acoustic inverse medium problem. Our uniqueness result is then followed by a discussion of decomposition methods for solving the inverse medium problem for acoustic waves and the use of sampling methods and transmission eigenvalues to obtain qualitative estimates on the refractive index. We conclude by examining the use of decomposition methods to solve the inverse medium problem for electromagnetic waves followed by some numerical examples illustrating the use of decomposition methods to solve the inverse medium problem for acoustic waves.


  1. 24.
    Bao, G., and Li, P.: Inverse medium scattering for the Helmholtz equation at fixed frequency. Inverse Problems 21, 1621–1641 (2005).CrossRefMathSciNetzbMATHGoogle Scholar
  2. 26.
    Bao, G., Li, P., Lin, J., and Triki, F.: Inverse scattering problems with multi-frequencies. Inverse Problems 31, 093001 (2015).CrossRefMathSciNetzbMATHGoogle Scholar
  3. 33.
    Bleistein, N.: Mathematical Methods for Wave Phenomena. Academic Press, Orlando 1984.zbMATHGoogle Scholar
  4. 39.
    Borges, C., Gillman, A., and Greengard, L.: High resolution inverse scattering in two dimensions using recursive linearization. SIAM J. Imaging Sci. 10, 641–664 (2017).CrossRefMathSciNetzbMATHGoogle Scholar
  5. 40.
    Borges, C., and Greengard, L.: Inverse obstacle scattering in two dimensions with multiple frequency data and multiple angles of incidence. SIAM J. Imaging Sci. 8, 280–298 (2015).CrossRefMathSciNetzbMATHGoogle Scholar
  6. 45.
    Bukhgeim, A.L.: Recovering a potential from Cauchy data in the two dimensional case. Jour. Inverse Ill-Posed Problems 16, 19–33 (2008).MathSciNetzbMATHGoogle Scholar
  7. 48.
    Cakoni, F., and Colton, D.: A uniqueness theorem for an inverse electromagnetic scattering problem in inhomogeneous anisotropic media. Proc. Edinburgh Math. Soc. 46, 285–299 (2003).CrossRefMathSciNetzbMATHGoogle Scholar
  8. 51.
    Cakoni, F., and Colton, D.: Qualitative Methods in Inverse Scattering Theory. Springer, Berlin 2006.zbMATHGoogle Scholar
  9. 53.
    Cakoni, F., Colton, D., and Haddar, H.: The linear sampling method for anisotropic media. J. Comput. Appl. Math. 146, 285–299 (2002).CrossRefMathSciNetzbMATHGoogle Scholar
  10. 57.
    Cakoni, F., Colton, D., and Haddar, H.: Inverse Scattering Theory and Transmission Eigenvalues. SIAM, Philadelphia 2016.CrossRefzbMATHGoogle Scholar
  11. 59.
    Cakoni, F., Colton, D., and Monk, P.: The Linear Sampling Method in Inverse Electromagnetic Scattering. SIAM Publications, Philadelphia, 2011.CrossRefzbMATHGoogle Scholar
  12. 62.
    Cakoni, F., Gintides, D., and Haddar, H.: The existence of an infinite discrete set of transmission eigenvalues. SIAM J. Math. Anal. 42, 237–255 (2010).CrossRefMathSciNetzbMATHGoogle Scholar
  13. 72.
    Calderón, A.P.: On an inverse boundary value problem. In: Seminar on Numerical Analysis and its Applications to Continuum Mechanics. Soc. Brasileira de Matemática, Rio de Janerio, 65–73 (1980).Google Scholar
  14. 75.
    Chadan, K., Colton, D., Päivärinta, L., and Rundell, W.: An Introduction to Inverse Scattering and Inverse Spectral Problems. SIAM Publications, Philadelphia 1997.CrossRefzbMATHGoogle Scholar
  15. 79.
    Chen, Y.: Inverse scattering via Heisenberg’s uncertainty principle. Inverse Problems 13, 253–282 (1997).CrossRefMathSciNetzbMATHGoogle Scholar
  16. 81.
    Chew, W: Waves and Fields in Inhomogeneous Media. Van Nostrand Reinhold, New York 1990.Google Scholar
  17. 87.
    Colton, D., Coyle, J., and Monk, P. : Recent developments in inverse acoustic scattering theory. SIAM Review 42, 369–414 (2000).CrossRefMathSciNetzbMATHGoogle Scholar
  18. 89.
    Colton, D., and Hähner, P.: Modified far field operators in inverse scattering theory. SIAM J. Math. Anal. 24, 365–389 (1993).CrossRefMathSciNetzbMATHGoogle Scholar
  19. 92.
    Colton, D., and Kirsch, A.: An approximation problem in inverse scattering theory. Applicable Analysis 41, 23–32 (1991).CrossRefMathSciNetzbMATHGoogle Scholar
  20. 94.
    Colton, D., and Kirsch, A.: A simple method for solving inverse scattering problems in the resonance region. Inverse Problems 12, 383–393 (1996).CrossRefMathSciNetzbMATHGoogle Scholar
  21. 99.
    Colton, D., and Kress, R.: Time harmonic electromagnetic waves in an inhomogeneous medium. Proc. Royal Soc. Edinburgh 116 A, 279–293 (1990).Google Scholar
  22. 104.
    Colton, D., and Kress, R.: Integral Equation Methods in Scattering Theory. SIAM Publications, Philadelphia 2013.CrossRefzbMATHGoogle Scholar
  23. 114.
    Colton, D., and Monk, P: The inverse scattering problem for acoustic waves in an inhomogeneous medium. Quart. J. Mech. Appl. Math. 41, 97–125 (1988).CrossRefMathSciNetzbMATHGoogle Scholar
  24. 115.
    Colton, D., and Monk, P: A new method for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium. Inverse Problems 5, 1013–1026 (1989).CrossRefMathSciNetzbMATHGoogle Scholar
  25. 116.
    Colton, D., and Monk, P: A new method for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium II. Inverse Problems 6, 935–947 (1990).CrossRefMathSciNetzbMATHGoogle Scholar
  26. 117.
    Colton, D., and Monk, P: A comparison of two methods for solving the inverse scattering problem for acoustic waves in an inhomogeneous medium. J. Comp. Appl. Math. 42, 5–16 (1992).CrossRefMathSciNetzbMATHGoogle Scholar
  27. 118.
    Colton, D., and Monk, P.: On a class of integral equations of the first kind in inverse scattering theory. SIAM J. Appl. Math. 53, 847–860 (1993).CrossRefMathSciNetzbMATHGoogle Scholar
  28. 119.
    Colton, D., and Monk, P.: A modified dual space method for solving the electromagnetic inverse scattering problem for an infinite cylinder. Inverse Problems 10, 87–107 (1994).CrossRefMathSciNetzbMATHGoogle Scholar
  29. 120.
    Colton, D., and Monk, P.: A new approach to detecting leukemia: Using computational electromagnetics. Comp. Science and Engineering 2, 46–52 (1995).CrossRefGoogle Scholar
  30. 121.
    Colton, D., and Päivärinta, L.: Far field patterns and the inverse scattering problem for electromagnetic waves in an inhomogeneous medium. Math. Proc. Camb. Phil. Soc. 103, 561–575 (1988).CrossRefMathSciNetzbMATHGoogle Scholar
  31. 123.
    Colton, D. and Päivärinta, L.: The uniqueness of a solution to an inverse scattering problem for electromagnetic waves. Arch. Rational Mech. Anal. 119, 59–70 (1992).CrossRefMathSciNetzbMATHGoogle Scholar
  32. 125.
    Colton, D., Piana, M., and Potthast, R.: A simple method using Morozov’s discrepancy principle for solving inverse scattering problems. Inverse Problems 13, 1477–1493 (1997).CrossRefMathSciNetzbMATHGoogle Scholar
  33. 132.
    Devaney, A.J.: Mathematical Foundations of Imaging, Tomography and Wavefield Inversion. Cambridge University Press, Cambridge 2012.CrossRefzbMATHGoogle Scholar
  34. 152.
    Gosh Roy, D.N., and Couchman, L.S. : Inverse Problems and Inverse Scattering of Plane Waves. Academic Press, New York 2002.Google Scholar
  35. 158.
    Gutman, S., and Klibanov, M.: Regularized quasi–Newton method for inverse scattering problems. Math. Comput. Modeling 18, 5–31 (1993).CrossRefMathSciNetzbMATHGoogle Scholar
  36. 160.
    Gutman, S., and Klibanov, M.: Iterative method for multidimensional inverse scattering problems at fixed frequencies. Inverse Problems 10, 573–599 (1994).CrossRefMathSciNetzbMATHGoogle Scholar
  37. 161.
    Gylys-Colwel, F.: An inverse problem for the Helmholtz equation. Inverse Problems 12, 139–156 (1996).CrossRefMathSciNetzbMATHGoogle Scholar
  38. 166.
    Haddar, H.: The interior transmission problem for anisotropic Maxwell’s equations and its application to the inverse problem. Math. Meth. Appl. Math. 27, 2111–2129 (2004).MathSciNetzbMATHGoogle Scholar
  39. 167.
    Haddar, H.: Analysis of some qualitative methods for inverse electromagnetic scattering problems. In: Computational Electromagnetism (Bermúdez de Castro and Valli, eds). Springer, Berlin, 191–240 (2014).Google Scholar
  40. 175.
    Hähner, P.: An approximation theorem in inverse electromagnetic scattering. Math. Meth. in the Appl. Sci. 17, 293–303 (1994).CrossRefMathSciNetzbMATHGoogle Scholar
  41. 176.
    Hähner, P.: A periodic Faddeev-type solution operator. Jour. of Differential Equations 128, 300–308 (1996).CrossRefMathSciNetzbMATHGoogle Scholar
  42. 177.
    Hähner, P.: On the uniqueness of the shape of a penetrable anisotropic obstacle. J. Comput. Appl. Math. 116, 167–180 (2000).CrossRefMathSciNetzbMATHGoogle Scholar
  43. 178.
    Hähner, P.: Scattering by media. In: Scattering (Pike and Sabatier, eds). Academic Press, New York, 74–94 (2002).Google Scholar
  44. 195.
    Hohage, T.: On the numerical solution of a three-dimensional inverse medium scattering problem. Inverse Problems 17, 1743–1763 (2001).CrossRefMathSciNetzbMATHGoogle Scholar
  45. 196.
    Hohage, T.: Fast numerical solution of the electromagnetic medium scattering problem and applications to the inverse problem. J. Comput. Phys. 214, 224–238 (2006).CrossRefMathSciNetzbMATHGoogle Scholar
  46. 197.
    Hohage, T., and Langer, S.: Acceleration techniques for regularized Newton methods applied to electromagnetic inverse medium scattering problems. Inverse Problems 26, 074011 (2010).CrossRefMathSciNetzbMATHGoogle Scholar
  47. 203.
    Isakov, V.: On uniqueness in the inverse transmission scattering problem. Comm. Part. Diff. Equa. 15, 1565–1587 (1990).CrossRefMathSciNetzbMATHGoogle Scholar
  48. 235.
    Kirsch, A.: Remarks on some notions of weak solutions for the Helmholtz equation. Applicable Analysis 47, 7–24 (1992).CrossRefMathSciNetzbMATHGoogle Scholar
  49. 238.
    Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems. 2nd ed, Springer, Berlin 2011.CrossRefzbMATHGoogle Scholar
  50. 240.
    Kirsch, A.: Factorization of the far field operator for the inhomogeneous medium case and an application to inverse scattering theory. Inverse Problems 15, 413–429 (1999).CrossRefMathSciNetzbMATHGoogle Scholar
  51. 242.
    Kirsch, A.: Remarks on the Born approximation and the factorization method. Appl. Anal. 96, 70–84 (2017).CrossRefMathSciNetzbMATHGoogle Scholar
  52. 243.
    Kirsch, A., and Grinberg, N.: The Factorization Method for Inverse Problems. Oxford University Press, Oxford, 2008.zbMATHGoogle Scholar
  53. 254.
    Kleinman, R., and van den Berg, P.: A modified gradient method for two dimensional problems in tomography. J. Comp. Appl. Math. 42, 17–35 (1992).CrossRefMathSciNetzbMATHGoogle Scholar
  54. 255.
    Kleinman, R., and van den Berg, P.: An extended range modified gradient technique for profile inversion. Radio Science 28, 877–884 (1993).CrossRefGoogle Scholar
  55. 283.
    Kusiak, S. and Sylvester, J.: The scattering support. Comm. Pure Applied Math. 56, 1525–1548 (2003).CrossRefMathSciNetzbMATHGoogle Scholar
  56. 289.
    Langenberg, K.J.: Applied inverse problems for acoustic, electromagnetic and elastic wave scattering. In: Basic Methods of Tomography and Inverse Problems (Sabatier, ed). Adam Hilger, Bristol and Philadelphia, 127–467 (1987).Google Scholar
  57. 327.
    Moskow, S. and Schotland, J.: Convergence and stability of the inverse Born series for diffuse waves. Inverse Problems 24, 065004 (2008).CrossRefMathSciNetzbMATHGoogle Scholar
  58. 333.
    Nachman, A.: Reconstructions from boundary measurements. Annals of Math. 128, 531–576 (1988).CrossRefMathSciNetzbMATHGoogle Scholar
  59. 336.
    Natterer, F., and Wübbeling, F.: A propagation-backpropagation method for ultrasound tomography. Inverse Problems 11, 1225–1232 (1995).CrossRefMathSciNetzbMATHGoogle Scholar
  60. 340.
    Novikov, R.: Multidimensional inverse spectral problems for the equation − Δψ + (v(x) − E u(x)) ψ = 0. Translations in Func. Anal. and its Appl. 22, 263–272 (1988).CrossRefMathSciNetGoogle Scholar
  61. 341.
    Ola, P., Päivärinta, L., and Somersalo, E.: An inverse boundary value problem in electrodynamics. Duke Math. Jour. 70, 617–653 (1993).CrossRefMathSciNetzbMATHGoogle Scholar
  62. 342.
    Ola, P., and Somersalo, E.: Electromagnetic inverse problems and generalized Sommerfeld potentials. SIAM J. Appl. Math. 56, 1129–1145 (1996).CrossRefMathSciNetzbMATHGoogle Scholar
  63. 369.
    Ramm, A.G.: On completeness of the products of harmonic functions. Proc. Amer. Math. Soc. 98, 253–256 (1986).CrossRefMathSciNetzbMATHGoogle Scholar
  64. 370.
    Ramm, A.G.: Recovery of the potential from fixed energy scattering data. Inverse Problems 4, 877–886 (1988).CrossRefMathSciNetzbMATHGoogle Scholar
  65. 372.
    Ramm, A.G.: Multidimensional Inverse Scattering Problems. Longman–Wiley, New York 1992.zbMATHGoogle Scholar
  66. 402.
    Sun, Z., and Uhlmann, G.: An inverse boundary value problem for Maxwell’s equations. Arch. Rational. Mech. Anal. 119, 71–93 (1992).CrossRefMathSciNetzbMATHGoogle Scholar
  67. 403.
    Sylvester, J.: Notions of support for far fields. Inverse Problems 22, 1273–1288 (2006).CrossRefMathSciNetzbMATHGoogle Scholar
  68. 404.
    Sylvester, J. and Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem. Ann. of Math. 125, 153–169 (1987).CrossRefMathSciNetzbMATHGoogle Scholar
  69. 412.
    van den Berg, R. and Kleinman, R.: A contrast source inversion method. Inverse Problems 13, 1607–1620 (1997).CrossRefMathSciNetzbMATHGoogle Scholar
  70. 417.
    Vögeler, M.: Reconstruction of the three-dimensional refractive index in electromagnetic scattering using a propagation-backpropagation method. Inverse Problems 19, 739–753 (2003).CrossRefMathSciNetzbMATHGoogle Scholar
  71. 431.
    Wloka, J.: Partial Differential Equations. University Press, Cambridge 1987.CrossRefzbMATHGoogle Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  • David Colton
    • 1
  • Rainer Kress
    • 2
  1. 1.Department of Mathematical SciencesUniversity of DelawareNewarkUSA
  2. 2.Institut für Numerische und Angewandte MathematikGeorg-August-Universität GöttingenGöttingenGermany

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