Skip to main content
SpringerLink
Log in
SpringerLink
Log in

Introduction

  • Chapter
  • First Online:
Inverse Acoustic and Electromagnetic Scattering Theory

Part of the book series: Applied Mathematical Sciences ((AMS,volume 93))

  • 1774 Accesses

Abstract

The purpose of this chapter is to provide a survey of our book by placing what we have to say in a historical context. We obviously cannot give a complete account of inverse scattering theory in a book of only a few hundred pages, particularly since before discussing the inverse problem we have to give the rudiments of the theory of the direct problem. Hence, instead of attempting the impossible, we have chosen to present inverse scattering theory from the perspective of our own interests and research program. This inevitably means that certain areas of scattering theory are either ignored or given only cursory attention. In view of this fact, and in fairness to the reader, we have therefore decided to provide a few words at the beginning of our book to tell the reader what we are going to do, as well as what we are not going to do, in the forthcoming chapters.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 119.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 159.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 159.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Abubakar, A., and van den Berg, P.: Iterative forward and inverse algorithms based on domain integral equations for three-dimensional electric and magnetic objects. J. Comput. Phys. 195, 236–262 (2004).

    Article  MATH  Google Scholar 

  2. Angell, T.S., Colton, D., and Kirsch, A.: The three dimensional inverse scattering problem for acoustic waves. J. Diff. Equations 46, 46–58 (1982).

    Article  MathSciNet  MATH  Google Scholar 

  3. Angell, T.S., Kleinman, R.E., and Roach, G.F.: An inverse transmission problem for the Helmholtz equation. Inverse Problems 3, 149–180 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  4. Bleistein, N.: Mathematical Methods for Wave Phenomena. Academic Press, Orlando 1984.

    MATH  Google Scholar 

  5. Blöhbaum, J.: Optimisation methods for an inverse problem with time-harmonic electromagnetic waves: an inverse problem in electromagnetic scattering. Inverse Problems 5, 463–482 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  6. Brakhage, H., and Werner, P.: Über das Dirichletsche Aussenraumproblem für die Helmholtzsche Schwingungsgleichung. Arch. Math. 16, 325–329 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  7. Cakoni, F., and Colton, D.: Qualitative Methods in Inverse Scattering Theory. Springer, Berlin 2006.

    MATH  Google Scholar 

  8. Cakoni, F., Colton, D., and Haddar, H.: On the determination of Dirichlet and transmission eigenvalues from far field data. C. R. Math. Acad. Sci. Paris, Ser. 1 348, 379–383 (2010).

    Google Scholar 

  9. Cakoni, F., Colton, D., and Monk, P.: The Linear Sampling Method in Inverse Electromagnetic Scattering. SIAM Publications, Philadelphia, 2011.

    Book  MATH  Google Scholar 

  10. 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).

    Article  MathSciNet  MATH  Google Scholar 

  11. Chadan, K., and Sabatier, P. C.: Inverse Problems in Quantum Scattering Theory. Springer, Berlin 1989.

    Book  MATH  Google Scholar 

  12. Chavent, G., Papanicolaou, G., Sacks, P., and Symes, W.: Inverse Problems in Wave Propagation. Springer, Berlin 1997.

    Book  MATH  Google Scholar 

  13. Chew, W: Waves and Fields in Inhomogeneous Media. Van Nostrand Reinhold, New York 1990.

    Google Scholar 

  14. Colton, D., and Hähner, P.: Modified far field operators in inverse scattering theory. SIAM J. Math. Anal. 24, 365–389 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  15. Colton, D., and Kirsch, A.: Dense sets and far field patterns in acoustic wave propagation. SIAM J. Math. Anal. 15, 996–1006 (1984).

    Article  MathSciNet  MATH  Google Scholar 

  16. Colton, D., and Kirsch, A.: Karp’s theorem in acoustic scattering theory. Proc. Amer. Math. Soc. 103, 783–788 (1988).

    MathSciNet  MATH  Google Scholar 

  17. Colton, D., Kirsch, A., and Päivärinta, L.: Far field patterns for acoustic waves in an inhomogeneous medium. SIAM J. Math. Anal. 20, 1472–1483 (1989).

    Article  MathSciNet  MATH  Google Scholar 

  18. Colton, D., and Kress, R.: Dense sets and far field patterns in electromagnetic wave propagation. SIAM J. Math. Anal. 16, 1049–1060 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  19. Colton, D., and Kress, R.: Karp’s theorem in electromagnetic scattering theory. Proc. Amer. Math. Soc. 104, 764–769 (1988).

    MathSciNet  MATH  Google Scholar 

  20. Colton, D., and Kress, R.: Time harmonic electromagnetic waves in an inhomogeneous medium. Proc. Royal Soc. Edinburgh 116 A, 279–293 (1990).

    Google Scholar 

  21. Colton, D., and Kress, R.: Eigenvalues of the far field operator and inverse scattering theory. SIAM J. Math. Anal. 26, 601–615 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  22. Colton, D., and Kress, R.: Eigenvalues of the far field operator for the Helmholtz equation in an absorbing medium. SIAM J. Appl. Math. 55, 1724–1735 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  23. Colton, D., and Kress, R.: Integral Equation Methods in Scattering Theory. SIAM Publications, Philadelphia 2013.

    Book  MATH  Google Scholar 

  24. Colton, D., and Kress, R.: Looking back on inverse scattering theory. SIAM Review 60, 779–807 (2018).

    Article  MathSciNet  MATH  Google Scholar 

  25. Colton, D., and Monk, P.: A novel method for solving the inverse scattering problem for time-harmonic acoustic waves in the resonance region. SIAM J. Appl. Math. 45, 1039–1053 (1985).

    Article  MathSciNet  MATH  Google Scholar 

  26. Colton, D., and Monk, P.: The numerical solution of the three dimensional inverse scattering problem for time-harmonic acoustic waves. SIAM J. Sci. Stat. Comp. 8, 278–291 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  27. Colton, D., and Monk, P: The inverse scattering problem for time harmonic acoustic waves in a penetrable medium. Quart. J. Mech. Appl. Math. 40, 189–212 (1987).

    Article  MathSciNet  MATH  Google Scholar 

  28. 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).

    Article  MathSciNet  MATH  Google Scholar 

  29. 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).

    Article  MathSciNet  MATH  Google Scholar 

  30. 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).

    Article  MathSciNet  MATH  Google Scholar 

  31. 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).

    Article  MathSciNet  MATH  Google Scholar 

  32. 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).

    Article  MathSciNet  MATH  Google Scholar 

  33. 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).

    Article  MathSciNet  MATH  Google Scholar 

  34. 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).

    Article  MathSciNet  MATH  Google Scholar 

  35. Colton, D., and L. Päivärinta, L.: Far-field patterns for electromagnetic waves in an inhomogeneous medium. SIAM J. Math. Anal. 21, 1537–1549 (1990).

    Google Scholar 

  36. 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).

    Article  MathSciNet  MATH  Google Scholar 

  37. Devaney, A.J.: Mathematical Foundations of Imaging, Tomography and Wavefield Inversion. Cambridge University Press, Cambridge 2012.

    Book  MATH  Google Scholar 

  38. Dolph, C. L.: The integral equation method in scattering theory. In: Problems in Analysis (Gunning, ed). Princeton University Press, Princeton, 201–227 (1970).

    Google Scholar 

  39. Gutman, S., and Klibanov, M.: Regularized quasi–Newton method for inverse scattering problems. Math. Comput. Modeling 18, 5–31 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  40. Gutman, S., and Klibanov, M.: Iterative method for multidimensional inverse scattering problems at fixed frequencies. Inverse Problems 10, 573–599 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  41. Haas, M., Rieger, W., Rucker, W., and Lehner, G.: Inverse 3D acoustic and electromagnetic obstacle scattering by iterative adaption. In: Inverse Problems of Wave Propagation and Diffraction (Chavent and Sabatier, eds). Springer, Berlin 1997.

    Google Scholar 

  42. Hanke, M., Hettlich, F., and Scherzer, O.: The Landweber iteration for an inverse scattering problem. In: Proceedings of the 1995 Design Engineering Technical Conferences, Vol. 3, Part C (Wang et al, eds).

    Google Scholar 

  43. Hettlich, F.: An iterative method for the inverse scattering problem from sound-hard obstacles. In: Proceedings of the ICIAM 95, Vol. II, Applied Analysis (Mahrenholz and Mennicken, eds). Akademie Verlag, Berlin (1996).

    Google Scholar 

  44. Hohage, T.: Logarithmic convergence rates of the iteratively regularized Gauss–Newton method for an inverse potential and an inverse scattering problem. Inverse Problems 13, 1279–1299 (1997).

    Article  MathSciNet  MATH  Google Scholar 

  45. Hohage, T.: On the numerical solution of a three-dimensional inverse medium scattering problem. Inverse Problems 17, 1743–1763 (2001).

    Article  MathSciNet  MATH  Google Scholar 

  46. Hohage, T., and Langer, S.: Acceleration techniques for regularized Newton methods applied to electromagnetic inverse medium scattering problems. Inverse Problems 26, 074011 (2010).

    Article  MathSciNet  MATH  Google Scholar 

  47. Imbriale, W.A., and Mittra, R.: The two-dimensional inverse scattering problem. IEEE Trans. Ant. Prop. AP-18, 633–642 (1970).

    Article  Google Scholar 

  48. Ivanyshyn, O., and Kress, R.: Nonlinear integral equations in inverse obstacle scattering. In: Mathematical Methods in Scattering Theory and Biomedical Engineering, (Fotiatis and Massalas, eds). World Scientific, Singapore, 39–50 (2006).

    Google Scholar 

  49. Johansson, T., and Sleeman, B.: Reconstruction of an acoustically sound-soft obstacle from one incident field and the far field pattern. IMA J. Appl. Math. 72, 96–112 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  50. Jones, D.S.: Acoustic and Electromagnetic Waves. Clarendon Press, Oxford 1986.

    Google Scholar 

  51. Karp, S.N.: Far field amplitudes and inverse diffraction theory. In: Electromagnetic Waves (Langer, ed). Univ. of Wisconsin Press, Madison, 291–300 (1962).

    Google Scholar 

  52. Kirsch, A.: The denseness of the far field patterns for the transmission problem. IMA J. Appl. Math. 37, 213–225 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  53. Kirsch, A.: Numerical algorithms in inverse scattering theory. In: Ordinary and Partial Differential Equations, Vol. IV, (Jarvis and Sleeman, eds). Pitman Research Notes in Mathematics 289, Longman, London, 93–111 (1993).

    Google Scholar 

  54. Kirsch, A., and Grinberg, N.: The Factorization Method for Inverse Problems. Oxford University Press, Oxford, 2008.

    MATH  Google Scholar 

  55. Kirsch, A., and Hettlich, F.: The Mathematical Theory of Time-Harmonic Maxwell’s Equations. Springer, New York 2015.

    Book  MATH  Google Scholar 

  56. Kirsch, A., and Kress, R.: On an integral equation of the first kind in inverse acoustic scattering. In: Inverse Problems (Cannon and Hornung, eds). ISNM 77, 93–102 (1986).

    Google Scholar 

  57. Kirsch, A., and Kress, R.: An optimization method in inverse acoustic scattering. In: Boundary elements IX, Vol 3. Fluid Flow and Potential Applications (Brebbia, Wendland and Kuhn, eds). Springer, Berlin, 3–18 (1987).

    Google Scholar 

  58. 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).

    Article  MathSciNet  MATH  Google Scholar 

  59. Kleinman, R., and van den Berg, P.: An extended range modified gradient technique for profile inversion. Radio Science 28, 877–884 (1993).

    Article  Google Scholar 

  60. Knauff, W., and Kress, R.: On the exterior boundary value problem for the time-harmonic Maxwell equations. J. Math. Anal. Appl. 72, 215–235 (1979).

    Article  MathSciNet  MATH  Google Scholar 

  61. Kress, R.: On the boundary operator in electromagnetic scattering. Proc. Royal Soc. Edinburgh 103A, 91–98 (1986).

    Article  MathSciNet  MATH  Google Scholar 

  62. Kress, R.: Integral equation methods in inverse acoustic and electromagnetic scattering. In: Boundary Integral Formulations for Inverse Analysis (Ingham and Wrobel, eds). Computational Mechanics Publications, Southampton, 67–92 (1997).

    Google Scholar 

  63. Kress, R.: Newton’s Method for inverse obstacle scattering meets the method of least squares. Inverse Problems 19, 91–104 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  64. Kress, R., and Rundell, W.: A quasi-Newton method in inverse obstacle scattering. Inverse Problems 10, 1145–1157 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  65. Kress, R., and Zinn, A.: On the numerical solution of the three dimensional inverse obstacle scattering problem. J. Comp. Appl. Math. 42, 49–61 (1992).

    Article  MathSciNet  MATH  Google Scholar 

  66. 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 

  67. Lax, P.D., and Phillips, R.S.: Scattering Theory. Academic Press, New York 1967.

    MATH  Google Scholar 

  68. Leis, R.: Zur Dirichletschen Randwertaufgabe des Aussenraums der Schwingungsgleichung. Math. Z. 90, 205–211 (1965).

    Article  MathSciNet  MATH  Google Scholar 

  69. Leis, R.: Initial Boundary Value Problems in Mathematical Physics. John Wiley, New York 1986.

    Book  MATH  Google Scholar 

  70. Martin, P.: Multiple Scattering: Interaction of Time-harmonic Waves with N Obstacles. Cambridge University Press, Cambridge 2006.

    Book  MATH  Google Scholar 

  71. Melrose, R.B.: Geometric Scattering Theory. Cambridge University Press, Cambridge 1995.

    MATH  Google Scholar 

  72. Mönch, L.: A Newton method for solving the inverse scattering problem for a sound-hard obstacle. Inverse Problems 12, 309–323 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  73. Müller, C.: Zur mathematischen Theorie elektromagnetischer Schwingungen. Abh. deutsch. Akad. Wiss. Berlin 3, 5–56 (1945/46).

    Google Scholar 

  74. Müller, C.: Randwertprobleme der Theorie elektromagnetischer Schwingungen. Math. Z. 56, 261–270 (1952).

    Article  MathSciNet  MATH  Google Scholar 

  75. Müller, C.: Foundations of the Mathematical Theory of Electromagnetic Waves. Springer, Berlin 1969.

    Book  MATH  Google Scholar 

  76. Nachman, A.: Reconstructions from boundary measurements. Annals of Math. 128, 531–576 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  77. Nakamura, G and Potthast, R.: Inverse Modeling. IOP Publishing, Bristol 2015.

    MATH  Google Scholar 

  78. Natterer, F., and Wübbeling, F.: A propagation-backpropagation method for ultrasound tomography. Inverse Problems 11, 1225–1232 (1995).

    Article  MathSciNet  MATH  Google Scholar 

  79. Nédélec, J.C.; Acoustic and Electromagnetic Equations. Springer, Berlin 2001.

    Book  MATH  Google Scholar 

  80. Newton, R.G.: Scattering Theory of Waves and Particles. Springer, Berlin 1982.

    Book  MATH  Google Scholar 

  81. Newton, R.G.: Inverse Schrödinger Scattering in Three Dimensions. Springer, Berlin 1989.

    Book  MATH  Google Scholar 

  82. 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).

    Article  MathSciNet  Google Scholar 

  83. Ola, P., Päivärinta, L., and Somersalo, E.: An inverse boundary value problem in electrodynamics. Duke Math. Jour. 70, 617–653 (1993).

    Article  MathSciNet  MATH  Google Scholar 

  84. Ola, P., and Somersalo, E.: Electromagnetic inverse problems and generalized Sommerfeld potentials. SIAM J. Appl. Math. 56, 1129–1145 (1996).

    Article  MathSciNet  MATH  Google Scholar 

  85. Päivärinta, L., and Sylvester, J.: Transmission eigenvalues. SIAM J. Math. Anal. 40, 738–758 (2008).

    Article  MathSciNet  MATH  Google Scholar 

  86. Panich, O.I.: On the question of the solvability of the exterior boundary-value problems for the wave equation and Maxwell’s equations. Usp. Mat. Nauk 20A, 221–226 (1965) (in Russian).

    Google Scholar 

  87. Potthast, R.: Fréchet differentiability of boundary integral operators in inverse acoustic scattering. Inverse Problems 10, 431–447 (1994).

    Article  MathSciNet  MATH  Google Scholar 

  88. Potthast, R.: Point-Sources and Multipoles in Inverse Scattering Theory. Chapman & Hall, London 2001.

    Book  MATH  Google Scholar 

  89. Ramm, A.G.: Recovery of the potential from fixed energy scattering data. Inverse Problems 4, 877–886 (1988).

    Article  MathSciNet  MATH  Google Scholar 

  90. Ramm, A.G.: Symmetry properties of scattering amplitudes and applications to inverse problems. J. Math. Anal. Appl. 156, 333–340 (1991).

    Article  MathSciNet  MATH  Google Scholar 

  91. Reed, M., and Simon, B.: Scattering Theory. Academic Press, New York 1979.

    MATH  Google Scholar 

  92. Rellich, F.: Über das asymptotische Verhalten der Lösungen von Δu + λu = 0 in unendlichen Gebieten. Jber. Deutsch. Math. Verein. 53, 57–65 (1943).

    MathSciNet  MATH  Google Scholar 

  93. Rjasanow, S., and Steinbach, O.: The Fast Solution of Boundary Integral Equations. Springer, Berlin 2007.

    MATH  Google Scholar 

  94. Roger, A.: Newton Kantorovich algorithm applied to an electromagnetic inverse problem. IEEE Trans. Ant. Prop. AP-29, 232–238 (1981).

    Article  MATH  Google Scholar 

  95. Serranho, P.: A hybrid method for sound-soft obstacles in 3D. Inverse Problems and Imaging 1, 691–712 (2007).

    Article  MathSciNet  MATH  Google Scholar 

  96. Sommerfeld, A.: Die Greensche Funktion der Schwingungsgleichung. Jber. Deutsch. Math. Verein. 21, 309–353 (1912).

    MATH  Google Scholar 

  97. Taylor, M.E.: Partial Differential Equations. 2nd ed, Springer, New York 2011.

    MATH  Google Scholar 

  98. van den Berg, R. and Kleinman, R.: Gradient methods in inverse acoustic and electromagnetic scattering. In: Large-Scale Optimization with Applications, Part I: Optimization in Inverse Problems and Design (Biegler et al, eds). The IMA Volumes in Mathematics and its Applications 92, Springer, Berlin; 173–194 (1977).

    Google Scholar 

  99. Vekua, I.N.: Metaharmonic functions. Trudy Tbilisskogo matematichesgo Instituta 12, 105–174 (1943).

    MathSciNet  MATH  Google Scholar 

  100. Vögeler, M.: Reconstruction of the three-dimensional refractive index in electromagnetic scattering using a propagation-backpropagation method. Inverse Problems 19, 739–753 (2003).

    Article  MathSciNet  MATH  Google Scholar 

  101. Werner, P.: Randwertprobleme der mathematischen Akustik. Arch. Rational Mech. Anal. 10, 29–66 (1962).

    Article  MathSciNet  MATH  Google Scholar 

  102. Weston, V.H., and Boerner, W.M.: An inverse scattering technique for electromagnetic bistatic scattering. Canadian J. Physics 47, 1177–1184 (1969).

    Article  Google Scholar 

  103. Weyl, H.: Kapazität von Strahlungsfeldern. Math. Z. 55, 187–198 (1952).

    Google Scholar 

  104. Wilcox, C.H.: Scattering Theory for the d’Alembert Equation in Exterior Domains. Springer Lecture Notes in Mathematics 442, Berlin 1975.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematical Sciences, University of Delaware, Newark, DE, USA

    David Colton

  2. Institut für Numerische und Angewandte Mathematik, Georg-August-Universität Göttingen, Göttingen, Germany

    Rainer Kress

Authors
  1. David Colton
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Rainer Kress
    View author publications

    You can also search for this author in PubMed Google Scholar

Rights and permissions

Reprints and permissions

Copyright information

© 2019 Springer Nature Switzerland AG

About this chapter

Check for updates. Verify currency and authenticity via CrossMark

Cite this chapter

Colton, D., Kress, R. (2019). Introduction. In: Inverse Acoustic and Electromagnetic Scattering Theory. Applied Mathematical Sciences, vol 93. Springer, Cham. https://doi.org/10.1007/978-3-030-30351-8_1

Download citation

Publish with us

Policies and ethics

Search

Navigation