Skip to main content

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

  • 5150 Accesses

Abstract

Up until now, we have considered only the direct and inverse obstacle scattering problem for time-harmonic acoustic waves. In the following two chapters, we want to extend these results to obstacle scattering for time-harmonic electromagnetic waves. As in our analysis on acoustic scattering, we begin with an outline of the solution of the direct problem.

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

Access this chapter

Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Hardcover Book
USD 169.99
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, Jour. Comp. Phys. 195, 236–262 (2004).

    Google Scholar 

  2. Adams, R.A.: Sobolev Spaces. Academic Press, New York 1975.

    Google Scholar 

  3. Akduman, I., and Kress, R.: Direct and inverse scattering problems for inhomogeneous impedance cylinders of arbitrary shape. Radio Science 38, 1055–1064 (2003).

    Google Scholar 

  4. Aktosun, T., Gintides, G., and Papanicolaou, V.: The uniqueness in the inverse problem for transmission eigenvalues for the spherically symmetric variable-speed wave equation. Inverse Problems 27, 115004 (2011).

    Google Scholar 

  5. Alessandrini, G., and Rondi, L.: Determining a sound–soft polyhedral scatterer by a single far–field measurement. Proc. Amer. Math. Soc. 133, 1685–1691 (2005).

    Google Scholar 

  6. Alves, C.J.S., and Ha-Duong, T.: On inverse scattering by screens. Inverse Problems 13, 1161–1176 (1997).

    Google Scholar 

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

    Google Scholar 

  8. Angell, T.S., Colton, D., and Kress, R.: Far field patterns and inverse scattering problems for imperfectly conducting obstacles. Math. Proc. Camb. Phil. Soc. 106, 553–569 (1989).

    Google Scholar 

  9. Angell, T.S., and Kirsch, A.: The conductive boundary condition for Maxwell’s equations. SIAM J. Appl. Math. 52, 1597–1610 (1992).

    Google Scholar 

  10. Angell, T.S., Kleinman, R.E., and Hettlich, F.: The resistive and conductive problems for the exterior Helmholtz equation. SIAM J. Appl. Math. 50, 1607–1622 (1990).

    Google Scholar 

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

    Google Scholar 

  12. Aramini, R., Caviglia, G., Masa, A., and Piana, M.: The linear sampling method and energy conservation. Inverse Problems 26, 05504 (2010).

    Google Scholar 

  13. Arens, T.: Why linear sampling works. Inverse Problems 20, 163–173 (2004).

    Google Scholar 

  14. Arens, T., and Lechleiter, A.: The linear sampling method revisited. Jour. Integral Equations and Applications 21, 179–202 (2009).

    Google Scholar 

  15. Atkinson, K.E.: The numerical solution of Laplace’s equation in three dimensions. SIAM J. Numer. Anal. 19, 263–274 (1982).

    Google Scholar 

  16. Atkinson, K.E.: The Numerical Solution of Integral Equations of the Second Kind. Cambridge Univ. Press, Cambridge 1997.

    Google Scholar 

  17. Bakushinskii, A.B.: The problem of the convergence of the iteratively regularized Gauss–Newton method. Comput. Maths. Maths. Phys. 32, 1353–1359 (1992).

    Google Scholar 

  18. Baumeister, J.: Stable Solution of Inverse Problems. Vieweg, Braunschweig 1986.

    Google Scholar 

  19. Ben Hassen, F., Erhard, K., and Potthast, R.: The point source method for 3d reconstructions for the Helmholtz and Maxwell equations. Inverse Problems 22, 331–353 (2006).

    Google Scholar 

  20. Bers, L., John, F., and Schechter, M.: Partial Differential Equations. John Wiley, New York 1964.

    Google Scholar 

  21. Blaschke, B., Neubauer, A., and Scherzer, O: On convergence rates for the iteratively regularized Gauss–Newton method. IMA J. Numerical Anal. 17, 421–436 (1997).

    Google Scholar 

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

    Google Scholar 

  23. 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).

    Google Scholar 

  24. Bojarski, N.N.: Three dimensional electromagnetic short pulse inverse scattering. Spec. Proj. Lab. Rep. Syracuse Univ. Res. Corp., Syracuse 1967.

    Google Scholar 

  25. Bojarski, N.N.: A survey of the physical optics inverse scattering identity. IEEE Trans. Ant. Prop. AP-20, 980–989 (1982).

    Google Scholar 

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

    Google Scholar 

  27. Bukhgeim, A.L.: Recovering a potential from Cauchy data in the two dimensional case. Jour. Inverse Ill-Posed Problems 16, 19–33 (2008).

    Google Scholar 

  28. Burger, M., Kaltenbacher, B., and Neubauer, A.: Iterative solution methods. In: Handbook of Mathematical Methods in Imaging (Scherzer, ed.) Springer, Berlin 345–384 (2011).

    Google Scholar 

  29. Cakoni, F.: Recent developments in the qualitative approach to inverse electromagnetic scattering theory. J. Comp. Appl. Math. 204, 242–253 (2007).

    Google Scholar 

  30. Cakoni, F., and Colton, D.: Combined far field operators in electromagnetic inverse scattering theory. Math. Methods Appl. Sci. 26, 293–314 (2003).

    Google Scholar 

  31. Cakoni, F., and Colton, D.: The determination of the surface impedance of a partially coated obstacle from far field data. SIAM J. Appl. Math. 64, 709–723 (2004).

    Google Scholar 

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

    Google Scholar 

  33. Cakoni, F., Colton, D., and Gintides, D.: The interior transmission eigenvalue problem,. SIAM J. Math. Anal. 42, 2912–2921 (2010).

    Google Scholar 

  34. 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 

  35. Cakoni, F., Colton, D. and Haddar, H.: The interior transmission problem for regions with cavities. SIAM J. Math. Anal. 42, 145–162 (2010).

    Google Scholar 

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

    Google Scholar 

  37. Cakoni, F., Colton, D., Monk, P., and Sun, J.: The inverse electromagnetic scattering problem for anisotropic media. Inverse Problems 26, 07404 (2010).

    Google Scholar 

  38. Cakoni, F., and Gintides, D.: New results on transmission eigenvalues. Inverse Problems and Imaging 4, 39–48 (2010).

    Google Scholar 

  39. 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).

    Google Scholar 

  40. Cakoni, F., and Haddar, H.: On the existence of transmission eigenvalues in an inhomogeneous medium. Applicable Analysis 88, 475–493 (2009).

    Google Scholar 

  41. Cakoni, F., and Kirsch, A.: On the interior transmission eigenvalue problem. Int. Jour. Comp. Sci. Math. 3, 142–16 (2010).

    Google Scholar 

  42. Calderón, A.P.: The multipole expansions of radiation fields. J. Rat. Mech. Anal. 3, 523–537 (1954).

    Google Scholar 

  43. 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 

  44. Catapano, I., Crocco, L., and Isernia, T.: On simple methods for shape reconstruction of unknown scatterers. IEEE Trans. Antennas Prop. 55, 1431–1436 (2007).

    Google Scholar 

  45. Chadan, K., Colton, D., Päivärinta, L., and Rundell, W.: An Introduction to Inverse Scattering and Inverse Spectral Problems. SIAM Publications, Philadelphia 1997.

    Google Scholar 

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

    Google Scholar 

  47. Chandler-Wilde, S. N., Graham, I. G., Langdon, S., and Lindner, M.: Condition number estimates for combined potential boundary integral operators in acoustic scattering. Jour. Integral Equations and Appl. 21, 229–279 (2009).

    Google Scholar 

  48. Chavent, G., Papanicolaou, G., Sacks, P., and Symes, W.: Inverse Problems in Wave Propagation. Springer, Berlin1997.

    Google Scholar 

  49. Cheng, J., and Yamamoto, M.: Uniqueness in an inverse scattering problem within non-trapping polygonal obstacles with at most two incoming waves. Inverse Problems 19, 1361–1384 (2003).

    Google Scholar 

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

    Google Scholar 

  51. Collino, C., Fares, M., and Haddar, H.: Numerical and analytical studies of the linear sampling method in electromagnetic inverse scattering problems. Inverse Problems 19, 1279–1298 (2003).

    Google Scholar 

  52. Colton, D.: Partial Differential Equations. Dover Publications, New York 2004.

    Google Scholar 

  53. Colton, D., Coyle, J., and Monk, P. : Recent developments in inverse acoustic scattering theory. SIAM Review 42, 369–414 (2000).

    Google Scholar 

  54. Colton, D., and Erbe, C.: Spectral theory of the magnetic far field operator in an orthotropic medium. In: Nonlinear Problems in Applied Mathematics (Angell et al, eds). SIAM, Philadelphia, 96–103 (1995).

    Google Scholar 

  55. Colton, D., Haddar, H., and Monk, P.: The linear sampling method for solving the electromagnetic inverse scattering problem. SIAM J. Sci. Comput. 24, 719–731 (2002).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  59. Colton, D., and Kirsch, A.: An approximation problem in inverse scattering theory. Applicable Analysis 41, 23–32 (1991).

    Google Scholar 

  60. Colton, D., and Kirsch, A.: The use of polarization effects in electromagnetic inverse scattering problems. Math. Meth. in the Appl. Sci. 15, 1–10 (1992).

    Google Scholar 

  61. Colton, D., and Kirsch, A.: A simple method for solving inverse scattering problems in the resonance region. Inverse Problems 12, 383–393 (1996).

    Google Scholar 

  62. 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).

    Google Scholar 

  63. Colton, D., and Kress, R.: The impedance boundary value problem for the time harmonic Maxwell equations. Math. Meth. in the Appl. Sci. 3, 475–487 (1981).

    Google Scholar 

  64. Colton, D., and Kress, R.: Integral Equation Methods in Scattering Theory. Wiley-Interscience Publication, New York 1983.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  69. 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).

    Google Scholar 

  70. Colton, D., and Kress, R.: On the denseness of Herglotz wave functions and electromagnetic Herglotz pairs in Sobolev spaces. Math. Methods Applied Science 24, 1289–1303 (2001).

    Google Scholar 

  71. Colton, D., and Kress, R.: Using fundamental solutions in inverse scattering. Inverse Problems 22, R49–R66 (2006).

    Google Scholar 

  72. 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).

    Google Scholar 

  73. Colton, D., and Monk, P.: A novel method for solving the inverse scattering problem for time-harmonic acoustic waves in the resonance region II. SIAM J. Appl. Math. 46, 506–523 (1986).

    Google Scholar 

  74. 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).

    Google Scholar 

  75. 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).

    Google Scholar 

  76. 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).

    Google Scholar 

  77. 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).

    Google Scholar 

  78. 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).

    Google Scholar 

  79. 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).

    Google Scholar 

  80. 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).

    Google Scholar 

  81. 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).

    Google Scholar 

  82. Colton, D., and Monk, P.: A new approach to detecting leukemia: Using computational electromagnetics. Comp. Science and Engineering 2, 46–52 (1995).

    Google Scholar 

  83. 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).

    Google Scholar 

  84. 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 

  85. 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).

    Google Scholar 

  86. Colton, D., Päivärinta, L., and Sylvester, J.: The interior transmission problem. Inverse Problems and Imaging 1, 13–28 (2007).

    Google Scholar 

  87. 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).

    Google Scholar 

  88. Colton, D., and Sleeman, B.D.: Uniqueness theorems for the inverse problem of acoustic scattering. IMA J. Appl. Math. 31, 253–259 (1983).

    Google Scholar 

  89. Colton, D., and Sleeman, B.D.: An approximation property of importance in inverse scattering theory. Proc. Edinburgh Math. Soc. 44, 449–454 (2001).

    Google Scholar 

  90. Davis, P.J.: Interpolation and Approximation. Blaisdell Publishing Company, Waltham 1963.

    Google Scholar 

  91. Davis, P.J., and Rabinowitz, P.: Methods of Numerical Integration. Academic Press, New York 1975.

    Google Scholar 

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

    Google Scholar 

  93. 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 

  94. Elliott, D.: Sigmoidal transformations and the trapezoidal rule. ANZIAM Jour. B 40, E77–E137 (1998).

    Google Scholar 

  95. Elliott, D. and Prössdorf, S.: An algorithm for the approximate solution of integral equations of Mellin type. Numer. Math. 70, 427–452 (1995).

    Google Scholar 

  96. Engl, H.W., Hanke, M. and Neubauer, A.: Regularization of Inverse Problems. Kluwer Academic Publisher, Dordrecht 1996.

    Google Scholar 

  97. Erdélyi, A.: Asymptotic Expansions. Dover Publications, New York 1956.

    Google Scholar 

  98. Farhat, C., Tezaur, R., and Djellouli, R.: On the solution of three-dimensional inverse obstacle acoustic scattering problems by a regularized Newton method. Inverse Problems 18, 1229–1246 (2002).

    Google Scholar 

  99. Ganesh, M., and Graham, I. G.: A high-order algorithm for obstacle scattering in three dimensions. J. Comput. Phys. 198, 211–242 (2004).

    Google Scholar 

  100. Ganesh, M., and Hawkins, S. C.: A spectrally accurate algorithm for electromagnetic scattering in three dimensions. Numer. Algorithms 43, 25–60 (2006).

    Google Scholar 

  101. Ganesh, M., and Hawkins, S. C.: An efficient surface integral equation method for the time-harmonic Maxwell equations. ANZIAM J. 48, C17–C33 (2007).

    Google Scholar 

  102. Ganesh, M., and Hawkins, S. C.: A high-order tangential basis algorithm for electromagnetic scattering by curved surfaces. J. Comput. Phys. 227, 4543–4562 (2008).

    Google Scholar 

  103. Gerlach, T. and Kress, R.: Uniqueness in inverse obstacle scattering with conductive boundary condition. Inverse Problems 12, 619–625 (1996).

    Google Scholar 

  104. Gieseke, B.: Zum Dirichletschen Prinzip für selbstadjungierte elliptische Differentialoperatoren. Math. Z. 68, 54–62 (1964).

    Google Scholar 

  105. Gilbarg, D., and Trudinger, N.S.: Elliptic Partial Differential Equations of Second Order. Springer, Berlin 1977.

    Google Scholar 

  106. Gintides, D.: Local uniqueness for the inverse scattering problem in acoustics via the Faber–Krahn inequality. Inverse Problems 21, 1195–1205 (2005).

    Google Scholar 

  107. Goldstein, C.I.: The finite element method with non-uniform mesh sizes applied to the exterior Helmholtz problem. Numer. Math. 38, 61–82 (1981).

    Google Scholar 

  108. Gosh Roy, D.N., and Couchman, L.S. : Inverse Problems and Inverse Scattering of Plane Waves. Academic Press, New York 2002.

    Google Scholar 

  109. Graham, I. G., and Sloan, I. H.: Fully discrete spectral boundary integral methods for Helmholtz problems on smooth closed surfaces in I​R3. Numer. Math. 92, 289–323 (2002).

    Google Scholar 

  110. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston 1985.

    Google Scholar 

  111. Groetsch, C.W.: The Theory of Tikhonov Regularization for Fredholm Equations of the First Kind. Pitman, Boston 1984.

    Google Scholar 

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

    Google Scholar 

  113. Gutman, S., and Klibanov, M.: Two versions of quasi–Newton method for multidimensional inverse scattering problem. J. Comput. Acoust. 1, 197–228 (1993).

    Google Scholar 

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

    Google Scholar 

  115. Haas, M., and Lehner, G.: Inverse 2D obstacle scattering by adaptive iteration. IEEE Transactions on Magnetics 33, 1958–1961 (1997)

    Google Scholar 

  116. 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 

  117. Hackbusch, W.: Multi-grid Methods and Applications. Springer, Berlin 1985.

    Google Scholar 

  118. Hadamard, J.: Lectures on Cauchy’s Problem in Linear Partial Differential Equations. Yale University Press, New Haven 1923.

    Google Scholar 

  119. Haddar, H., and Kress. R.: On the Fréchet derivative for obstacle scattering with an impedance boundary condition. SIAM J. Appl. Math. 65, 194–208 (2004).

    Google Scholar 

  120. Hähner, P.: Abbildungseigenschaften der Randwertoperatoren bei Randwertaufgaben für die Maxwellschen Gleichungen und die vektorielle Helmholtzgleichung in Hölder- und L 2–Räumen mit einer Anwendung auf vollständige Flächenfeldsysteme. Diplomarbeit, Göttingen 1987.

    Google Scholar 

  121. Hähner, P.: An exterior boundary-value problem for the Maxwell equations with boundary data in a Sobolev space. Proc. Roy. Soc. Edinburgh 109A, 213–224 (1988).

    Google Scholar 

  122. Hähner, P.: Eindeutigkeits- und Regularitätssätze für Randwertprobleme bei der skalaren und vektoriellen Helmholtzgleichung. Dissertation, Göttingen 1990.

    Google Scholar 

  123. Hähner, P.: A uniqueness theorem for the Maxwell equations with L 2 Dirichlet boundary conditions. Meth. Verf. Math. Phys. 37, 85–96 (1991).

    Google Scholar 

  124. Hähner, P.: A uniqueness theorem for a transmission problem in inverse electromagnetic scattering. Inverse Problems 9, 667–678 (1993).

    Google Scholar 

  125. Hähner, P.: An approximation theorem in inverse electromagnetic scattering. Math. Meth. in the Appl. Sci. 17, 293-303 (1994).

    Google Scholar 

  126. Hähner, P.: A periodic Faddeev-type solution operator. Jour. of Differential Equations 128, 300–308 (1996).

    Google Scholar 

  127. Hähner, P.: Scattering by media. In: Scattering (Pike and Sabatier, eds). Academic Press, New York, 74–94, 2002.

    Google Scholar 

  128. Hanke, M.: A regularization Levenberg–Marquardt scheme, with applications to inverse groundwater filtration problems. Inverse Problems 13, 75–95 (1997).

    Google Scholar 

  129. Hanke, M.: Why linear sampling really seems to work. Inverse Problems and Imaging 2, 373–395 (2008).

    Google Scholar 

  130. 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). The American Society of Mechanical Engineers, New York, 909–915 (1995).

    Google Scholar 

  131. Hanke, M., Neubauer, A., and Scherzer, O.: A convergence analysis for the Landweber iteration for nonlinear ill-posed problems. Numer. Math. 72, 21–37 (1995).

    Google Scholar 

  132. Harbrecht, H., and Hohage. T.: Fast methods for three-dimensional inverse obstacle scattering problems. Jour. Integral Equations and Appl. 19, 237–260 (2007).

    Google Scholar 

  133. Hartman, P., and Wilcox, C.: On solutions of the Helmholtz equation in exterior domains. Math. Z. 75, 228–255 (1961).

    Google Scholar 

  134. Hellwig, G.: Partial Differential Equations. Blaisdell Publishing, New York 1964.

    Google Scholar 

  135. Hettlich, F.: Die Integralgleichungsmethode bei Streuung an Körpern mit einer dünnen Schicht. Diplomarbeit, Göttingen 1989.

    Google Scholar 

  136. Hettlich, F.: On the uniqueness of the inverse conductive scattering problem for the Helmholtz equation. Inverse Problems 10, 129–144 (1994).

    Google Scholar 

  137. Hettlich, F.: Fréchet derivatives in inverse obstacle scattering. Inverse Problems 11, 371–382 (1995).

    Google Scholar 

  138. 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 

  139. Hettlich, F., and Rundell, W.: A second degree method for nonlinear inverse problem. SIAM J. Numer. Anal. 37, 587–620 (2000).

    Google Scholar 

  140. Hitrik, M., Krupchyk, K., Ola, P., and Päivärinta, L.: Transmission eigenvalues for operators with constant coefficients. SIAM J. Math. Anal. 42, 2965–2986 (2010).

    Google Scholar 

  141. Hitrik, M., Krupchyk, K., Ola, P., and Päivärinta, L.: Transmission eigenvalues for elliptic operators. SIAM J. Math. Anal. 43, 2630–2639 (2011).

    Google Scholar 

  142. 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).

    Google Scholar 

  143. Hohage, T.: Iterative Methods in Inverse Obstacle Scattering: Regularization Theory of Linear and Nonlinear Exponentially Ill-Posed Problems. Dissertation, Linz 1999.

    Google Scholar 

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

    Google Scholar 

  145. Hohage, T.: Fast numerical solution of the electromagnetic medium scattering problem and applications to the inverse problem. J. Comp. Phys. 214, 224–238 (2006).

    Google Scholar 

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

    Google Scholar 

  147. Hsiao, G.C.: The coupling of boundary element and finite element methods. Z. Angew. Math. Mech. 70, T493–T503 (1990).

    Google Scholar 

  148. Hsiao, G.C., and Wendland, W. L.: Boundary Integral Equations. Springer, Berlin 2008.

    Google Scholar 

  149. Ikehata, M.: Reconstruction of the shape of an obstacle from the scattering amplitude at a fixed frequency. Inverse Problems 14, 949–954 (1998).

    Google Scholar 

  150. Ikehata, M.: Reconstruction of obstacle from boundary measurements. Wave Motion 30, 205–223 (1999).

    Google Scholar 

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

    Google Scholar 

  152. Isakov, V.: On uniqueness in the inverse transmission scattering problem. Comm. Part. Diff. Equa. 15, 1565–1587 (1990).

    Google Scholar 

  153. Isakov, V.: Inverse Problems for Partial Differential Equations. 2nd ed, Springer, Berlin 2006.

    Google Scholar 

  154. Ivanov, K.V.: Integral equations of the first kind and an approximate solution for the inverse problem of potential. Soviet Math. Doklady 3, 210–212 (1962) (English translation).

    Google Scholar 

  155. Ivanov, K.V.: On linear problems which are not well-posed. Soviet Math. Doklady 3, 981–983 (1962) (English translation).

    Google Scholar 

  156. Ivanyshyn, O.: Shape reconstruction of acoustic obstacles from the modulus of the far field pattern. Inverse Problems and Imaging 1, 609–622 (2007).

    Google Scholar 

  157. Ivanyshyn, O.: Nonlinear Boundary Integral Equations in Inverse Scattering. Dissertation, Göttingen, 2007.

    Google Scholar 

  158. Ivanyshyn, O., and Johansson, T.: Nonlinear integral equation methods for the reconstruction of an acoustically sound-soft obstacle. J. Integral Equations Appl. 19, 289–308 (2007).

    Google Scholar 

  159. Ivanyshyn, O., and Johansson, T.: A coupled boundary integral equation method for inverse sound-soft scattering. In: Proceedings of waves 2007. The 8th international conference on mathematical and numerical aspects of waves, pp. 153–155 University of Reading 2007.

    Google Scholar 

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

    Google Scholar 

  161. Ivanyshyn, O., and Kress, R.: Inverse scattering for planar cracks via nonlinear integral equations. Math. Meth. Appl. Sciences 31, 1221–1232 (2007).

    Google Scholar 

  162. Ivanyshyn, O., and Kress, R.: Identification of sound-soft 3D obstacles from phaseless data. Inverse Problems and Imaging 4, 131–149 (2010).

    Google Scholar 

  163. Ivanyshyn, O., and Kress, R.: Inverse scattering for surface impedance from phase-less far field data. J. Comp. Phys. 230, 3443–3452 (2011).

    Google Scholar 

  164. Ivanyshyn, O., Kress, R., and Serranho, P.: Huygens’ principle and iterative methods in inverse obstacle scattering. Adv. Comput. Math. 33, 413–429 (2010).

    Google Scholar 

  165. Jeon, Y.: A Nyström method for boundary integral equations in domains with a piecewise smooth boundary. Jour. Integral Equations Appl. 5, 221–242 (1993).

    Google Scholar 

  166. 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).

    Google Scholar 

  167. Jones, D.S.: Methods in Electromagnetic Wave Propagation. Clarendon Press, Oxford 1979.

    Google Scholar 

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

    Google Scholar 

  169. Jörgens, K.: Lineare Integraloperatoren. Teubner–Verlag, Stuttgart 1970.

    Google Scholar 

  170. Kabanikhin, S.I.: Inverse and Ill-posed Problems: Theory and Applications. de Gruyter, Berlin-Boston 2011

    Google Scholar 

  171. Kaltenbacher, B., Neubauer, A., and Scherzer, O.: Iterative Regularization Methods for Nonlinear Ill-Posed Problems. de Gruyter, Berlin, 2008.

    Google Scholar 

  172. 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 

  173. Kedzierawski, A.: The inverse scattering problem for time-harmonic acoustic waves in an inhomogeneous medium with complex refraction index. J. Comp. Appl. Math. 47, 83-100 (1993).

    Google Scholar 

  174. Kellogg, O.D.: Foundations of Potential Theory. Springer, Berlin 1929.

    Google Scholar 

  175. Kersten, H.: Grenz- und Sprungrelationen für Potentiale mit quadratsummierbarer Dichte. Resultate d. Math. 3, 17–24 (1980).

    Google Scholar 

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

    Google Scholar 

  177. Kirsch, A.: Properties of far field operators in acoustic scattering. Math. Meth. in the Appl. Sci. 11, 773–787 (1989).

    Google Scholar 

  178. Kirsch, A.: Surface gradients and continuity properties for some integral operators in classical scattering theory. Math. Meth. in the Appl. Sci. 11, 789–804 (1989).

    Google Scholar 

  179. Kirsch, A.: Remarks on some notions of weak solutions for the Helmholtz equation. Applicable Analysis 47, 7–24 (1992).

    Google Scholar 

  180. Kirsch, A.: The domain derivative and two applications in inverse scattering. Inverse Problems 9, 81–96 (1993).

    Google Scholar 

  181. 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 

  182. Kirsch, A.: An Introduction to the Mathematical Theory of Inverse Problems. 2nd ed, Springer, Berlin 2011.

    Google Scholar 

  183. Kirsch, A.: Characterization of the shape of the scattering obstacle by the spectral data of the far field operator. Inverse Problems 14, 1489–1512 (1998).

    Google Scholar 

  184. 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).

    Google Scholar 

  185. Kirsch, A.: On the existence of transmission eigenvalues. Inverse Problems and Imaging 3, 155–172 (2009).

    Google Scholar 

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

    Google Scholar 

  187. 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 

  188. Kirsch, A., and Kress, R.: A numerical method for an inverse scattering problem. In: Inverse Problems (Engl and Groetsch, eds). Academic Press, Orlando, 279–290 (1987).

    Google Scholar 

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

    Google Scholar 

  190. Kirsch, A., and Kress, R.: Uniqueness in inverse obstacle scattering. Inverse Problems 9, 285–299 (1993).

    Google Scholar 

  191. Kirsch, A., Kress, R., Monk, P., and Zinn, A.: Two methods for solving the inverse acoustic scattering problem. Inverse Problems 4, 749–770 (1988).

    Google Scholar 

  192. Kirsch, A., and Monk, P.: An analysis of the coupling of finite element and Nyström methods in acoustic scattering. IMA J. Numerical Anal. 14, 523–544 (1994).

    Google Scholar 

  193. 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).

    Google Scholar 

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

    Google Scholar 

  195. 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).

    Google Scholar 

  196. Kress, R.: Ein ableitungsfreies Restglied für die trigonometrische Interpolation periodischer analytischer Funktionen. Numer. Math. 16, 389–396 (1971).

    Google Scholar 

  197. Kress, R.: Minimizing the condition number of boundary integral operators in acoustic and electromagnetic scattering. Q. Jl. Mech. appl. Math. 38, 323–341 (1985).

    Google Scholar 

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

    Google Scholar 

  199. Kress, R.: On the low wave number asymptotics for the two-dimensional exterior Dirichlet problem for the reduced wave equation. Math. Meth. in the Appl. Sci. 9, 335–341 (1987).

    Google Scholar 

  200. Kress, R.: A Nyström method for boundary integral equations in domains with corners. Numer. Math. 58, 145–161 (1990).

    Google Scholar 

  201. Kress, R.: Boundary integral equations in time-harmonic acoustic scattering. Mathl. Comput. Modelling 15, 229–243 (1991).

    Google Scholar 

  202. Kress, R.: On the numerical solution of a hypersingular integral equation in scattering theory. J. Comp. Appl. Math. 61, 345–360 (1995).

    Google Scholar 

  203. Kress, R.: Inverse scattering from an open arc. Math. Meth. in the Appl. Sci. 18, 267–293 (1995).

    Google Scholar 

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

    Google Scholar 

  205. Kress, R.: Linear Integral Equations. 2nd ed, Springer, Berlin 1999.

    Google Scholar 

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

    Google Scholar 

  207. Kress, R., and Päivärinta, L.: On the far field in obstacle scattering. SIAM J. Appl. Math. 59, 1413–1426 (1999).

    Google Scholar 

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

    Google Scholar 

  209. Kress, R., and Rundell, W.: Inverse obstacle scattering with modulus of the far field pattern as data. In: Inverse Problems in Medical Imaging and Nondestructive Testing (Engl et al, eds). Springer, Wien 75–92 (1997).

    Google Scholar 

  210. Kress, R., and Rundell, W.: Inverse obstacle scattering using reduced data. SIAM J. Appl. Math. 59, 442–454 (1999).

    Google Scholar 

  211. Kress, R., and Rundell, W.: Inverse scattering for shape and impedance. Inverse Problems 17, 1075–1085 (2001).

    Google Scholar 

  212. Kress, R., and Rundell, W.: Nonlinear integral equations and the iterative solution for an inverse boundary value problem. Inverse Problems 21, 1207–1223 (2005).

    Google Scholar 

  213. Kress, R., and Serranho, P.: A hybrid method for two-dimensional crack reconstruction. Inverse Problems 21, 773–784 (2005)

    Google Scholar 

  214. Kress, R., and Serranho, P.: A hybrid method for sound-hard obstacle reconstruction. J. Comput. Appl. Math. 24, 418–427 (2007).

    Google Scholar 

  215. Kress, R., Tezel, N., and Yaman, F.: A second order Newton method for sound soft inverse obstacle scattering. Jour. Inverse and Ill-Posed Problems 17, 173–185 (2009).

    Google Scholar 

  216. 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).

    Google Scholar 

  217. Kussmaul, R.: Ein numerisches Verfahren zur Lösung des Neumannschen Aussenraumproblems für die Helmholtzsche Schwingungsgleichung. Computing 4, 246–273 (1969).

    Google Scholar 

  218. 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 

  219. Lax, P.D.: Symmetrizable linear transformations. Comm. Pure Appl. Math. 7, 633–647 (1954).

    Google Scholar 

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

    Google Scholar 

  221. Lebedev, N.N.: Special Functions and Their Applications. Prentice-Hall, Englewood Cliffs 1965.

    Google Scholar 

  222. Lee, K.M.: Inverse scattering via nonlinear integral equations for a Neumann crack. Inverse Problems 22, 1989–2000 ( 2006).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  225. Le Louër, F.: Fast methods for the shape identification problem of a perfectly conducting obstacle. In preparation.

    Google Scholar 

  226. Leung, Y.J., and Colton,D.: Complex transmission eigenvalues for spherically stratified media. Inverse Problems 28, 07505 (2012).

    Google Scholar 

  227. Levine, L.M.: A uniqueness theorem for the reduced wave equation. Comm. Pure Appl. Math. 17, 147–176 (1964).

    Google Scholar 

  228. Lin, T.C.: The numerical solution of Helmholtz’s equation for the exterior Dirichlet problem in three dimensions. SIAM J. Numer. Anal. 22, 670–686 (1985).

    Google Scholar 

  229. Liu, C.: Inverse obstacle problem: local uniqueness for rougher obstacles and the identification of a ball. Inverse Problems 13, 1063–1069 (1997).

    Google Scholar 

  230. Liu, H., Yamamoto, M. and Zou, J.: Reflection principle for the Maxwell equations and its application to inverse electromagnetic scattering. Inverse Problems 23, 2357–2366 (2007).

    Google Scholar 

  231. Liu, H., and Zou, J.: Uniqueness in an inverse acoustic obstacle scattering problem for both sound-hard and sound-soft polyhedral scatterers. Inverse Problems 22, 515–524 (2006).

    Google Scholar 

  232. Louis, A.K.: Inverse und schlecht gestellte Probleme. Teubner, Stuttgart 1989.

    Google Scholar 

  233. Magnus, W.: Fragen der Eindeutigkeit und des Verhaltens im Unendlichen für Lösungen von Δu + k 2 u = 0. Abh. Math. Sem. Hamburg 16, 77–94 (1949).

    Google Scholar 

  234. Martensen, E.: Über eine Methode zum räumlichen Neumannschen Problem mit einer Anwendung für torusartige Berandungen. Acta Math. 109, 75–135 (1963).

    Google Scholar 

  235. Martensen, E.: Potentialtheorie. Teubner-Verlag, Stuttgart 1968.

    Google Scholar 

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

    Google Scholar 

  237. Mautz, J.R., and Harrington, R.F.: A combinded-source solution for radiating and scattering from a perfectly conducting body. IEEE Trans. Ant. and Prop. AP-27, 445–454 (1979).

    Google Scholar 

  238. McLaughlin, J., and Polyakov, P.: On the uniqueness of a spherically symmetric speed of sound from transmission eigenvalues. J. Diff. Equations 107, 351–382 (1994).

    Google Scholar 

  239. McLaughlin, J., Polyakov, P. and Sacks, P.: Reconstruction of a spherically symmetric speed of sound. SIAM.J. Appl. Math. 54, 1203–1223 (1994).

    Google Scholar 

  240. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge 2000.

    Google Scholar 

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

    Google Scholar 

  242. Mikhlin, S.G.: Mathematical Physics, an Advanced Course. North-Holland, Amsterdam 1970.

    Google Scholar 

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

    Google Scholar 

  244. Mönch, L.: On the inverse acoustic scattering problem from an open arc: the sound-hard case. Inverse Problems 13, 1379–1392 (1997).

    Google Scholar 

  245. Monk, P.: Finite Element Methods for Maxwell’s Equations. Clarendon Press, Oxford, 2003.

    Google Scholar 

  246. Moré, J.J.: The Levenberg–Marquardt algorithm, implementatiion and theory. In: Numerical analysis (Watson, ed). Springer Lecture Notes in Mathematics 630, Berlin, 105–116 (1977).

    Google Scholar 

  247. Morozov, V.A.: On the solution of functional equations by the method of regularization. Soviet Math. Doklady 7, 414–417 (1966) (English translation).

    Google Scholar 

  248. Morozov, V.A.: Choice of parameter for the solution of functional equations by the regularization method. Soviet Math. Doklady 8, 1000–1003 (1967) (English translation).

    Google Scholar 

  249. Morozov, V.A.: Methods for Solving Incorrectly Posed Problems. Springer, Berlin 1984.

    Google Scholar 

  250. Morrey, C.M.: Multiple Integrals in the Calculus of Variations. Springer, Berlin 1966.

    Google Scholar 

  251. Morse, P.M., and Ingard, K.U.: Linear acoustic theory. In: Encyclopedia of Physics (Flügge, ed). Springer, Berlin 1–128 (1961).

    Google Scholar 

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

    Google Scholar 

  253. Müller, C.: Über die ganzen Lösungen der Wellengleichung. Math. Annalen 124, 235–264 (1952).

    Google Scholar 

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

    Google Scholar 

  255. Müller, C: On the behavior of solutions of the differential equation Δu = F(x, u) in the neighborhood of a point. Comm. Pure Appl. Math. 7, 505–515 (1954).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  258. Natterer, F.: The Mathematics of Computerized Tomography. Teubner, Stuttgart and Wiley, New York 1986.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  263. Novikov, R.: Multidimensional inverse spectral problems for the equation − Δψ + (v(x) − Eu(x)) ψ = 0. Translations in Func. Anal. and its Appl. 22, 263–272 (1988).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  266. Olver, F.W.J: Asymptotics and Special Functions. Academic Press, New York 1974.

    Google Scholar 

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

    Google Scholar 

  268. 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 

  269. Pieper, M.: Spektralrandintegralmethoden zur Maxwell-Gleichung. Dissertation, Göttingen 2007.

    Google Scholar 

  270. Pieper, M.: Nonlinear integral equations for an inverse electromagnetic scattering problem. Journal of Physics: Conference Series 124, 012040 (2008).

    Google Scholar 

  271. Pieper, M.: Vector hyperinterpolation on the sphere. J. Approx. Theory 156, 173–186 (2009).

    Google Scholar 

  272. Pironneau, O.: Optimal Shape Design for Elliptic Systems. Springer, Berlin 1984.

    Google Scholar 

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

    Google Scholar 

  274. Potthast, R.: Fréchet Differenzierbarkeit von Randintegraloperatoren und Randwertproblemen zur Helmholtzgleichung und den zeitharmonischen Maxwellgleichungen. Dissertation, Göttingen 1994.

    Google Scholar 

  275. Potthast, R.: Fréchet differentiability of the solution to the acoustic Neumann scattering problem with respect to the domain. Jour. on Inverse and Ill-posed Problems 4, 67–84 (1996).

    Google Scholar 

  276. Potthast, R.: Domain derivatives in electromagnetic scattering. Math. Meth. in the Appl. Sci. 19, 1157–1175 (1996).

    Google Scholar 

  277. Potthast, R.: A fast new method to solve inverse scattering problems. Inverse Problems 12, 731–742 (1996).

    Google Scholar 

  278. Potthast, R.: A point-source method for inverse acoustic and electromagnetic obstacle scattering problems. IMA J. Appl. Math 61, 119–140 (1998).

    Google Scholar 

  279. Potthast, R.: Stability estimates and reconstructions in inverse acoustic scattering using singular sources. J. Comp. Appl. Math. 114, 247–274 (2000).

    Google Scholar 

  280. Potthast, R.: On the convergence of a new Newton-type method in inverse scattering. Inverse Problems 17, 1419–1434 (2001).

    Google Scholar 

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

    Google Scholar 

  282. Potthast, R.: Sampling and probe methods – an algorithmical view. Computing 75, 215–235 (2005).

    Google Scholar 

  283. Potthast, R.: A survey on sampling and probe methods for inverse problems. Inverse Problems 22, R1–R47 (2006).

    Google Scholar 

  284. Potthast, R. and Schulz, J.: A multiwave range test for obstacle reconstructions with unknown physical properties. J. Comput. Appl. Math. 205, 53–71 (2007).

    Google Scholar 

  285. Protter, M.H.: Unique continuation for elliptic equations. Trans. Amer. Math. Soc. 95, 81–90, (1960).

    Google Scholar 

  286. Protter, M.H., and Weinberger, H.F.: Maximum Principles in Differential Equations. Prentice-Hall, Englewood Cliffs 1967.

    Google Scholar 

  287. Ramm, A.G.: Scattering by Obstacles. D. Reidel Publishing Company, Dordrecht 1986.

    Google Scholar 

  288. Ramm, A.G.: On completeness of the products of harmonic functions. Proc. Amer. Math. Soc. 98, 253–256 (1986).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  291. Ramm, A.G.: Multidimensional Inverse Scattering Problems. Longman–Wiley, New York 1992.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  294. Ringrose, J.R.: Compact Non–Self Adjoint Operators. Van Nostrand Reinhold, London 1971.

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  297. Ruland, C.: Ein Verfahren zur Lösung von (Δ + k 2)u = 0 in Aussengebieten mit Ecken. Applicable Analysis 7, 69–79 (1978).

    Google Scholar 

  298. Rynne, B.P., and Sleeman, B.D.: The interior transmission problem and inverse scattering from inhomogeneous media. SIAM J. Math. Anal. 22, 1755–1762 (1991).

    Google Scholar 

  299. Saranen, J., and Vainikko, G.: Periodic Integral and Pseudodifferential Equations with Numerical Approximation. Springer, Berlin 2002.

    Google Scholar 

  300. Sauter, S., and Schwab, C.: Boundary Element Methods. Springer, Berlin 2011.

    Google Scholar 

  301. Schatz, A.H.: An observation concerning Ritz–Galerkin methods with indefinite bilinear forms. Math. Comp. 28, 959–962 (1974).

    Google Scholar 

  302. Schechter, M.: Principles of Functional Analysis. Academic Press, New York 1971.

    Google Scholar 

  303. Schormann, C.: Analytische und numerische Untersuchungen bei inversen Transmissionsproblemen zur zeitharmonischen Wellengleichung. Dissertation, Göttingen 2000.

    Google Scholar 

  304. Serranho, P.: A hybrid method for inverse scattering for shape and impedance. Inverse Problems 22, 663–680 (2006).

    Google Scholar 

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

    Google Scholar 

  306. Silver, S.: Microwave Antenna Theory and Design. M.I.T. Radiation Laboratory Series Vol. 12, McGraw-Hill, New York 1949.

    Google Scholar 

  307. Sloan, I. H., and Womersley, R. S.: Constructive approximations on the sphere. J. Approx. Theory 103, 91–118 (2000).

    Google Scholar 

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

    Google Scholar 

  309. Stefanov, P., and Uhlmann, G.: Local uniqueness for the fixed energy fixed angle inverse problem in obstacle scattering. Proc. Amer. Math. Soc. 132, 1351–1354 (2003).

    Google Scholar 

  310. Stratton, J.A., and Chu, L.J.: Diffraction theory of electromagnetic waves. Phys. Rev. 56, 99–107 (1939).

    Google Scholar 

  311. Sun, J.: Estimation of transmission eigenvalues and the index of refraction from Cauchy data. Inverse Problems 27, 015009 (2011).

    Google Scholar 

  312. Sun, Z., and Uhlmann, G.: An inverse boundary value problem for Maxwell’s equations. Arch. Rational. Mech. Anal. 119, 71–93 (1992).

    Google Scholar 

  313. Sylvester, J. and Uhlmann, G.: A global uniqueness theorem for an inverse boundary value problem. Ann. of Math. 125, 153–169 (1987).

    Google Scholar 

  314. Tikhonov, A.N.: On the solution of incorrectly formulated problems and the regularization method. Soviet Math. Doklady 4, 1035–1038 (1963) (English translation).

    Google Scholar 

  315. Tikhonov, A.N.: Regularization of incorrectly posed problems. Soviet Math. Doklady 4, 1624–1627 (1963) (English translation).

    Google Scholar 

  316. Tikhonov, A.N., and Arsenin, V.Y.: Solutions of Ill-posed Problems. Winston and Sons, Washington 1977.

    Google Scholar 

  317. Treves, F.: Basic Linear Partial Differential Equations. Academic Press, New York 1975.

    Google Scholar 

  318. Vainikko, G.: Fast solvers of the Lippmann–Schwinger equation In: Direct and Inverse Problems of Mathematical Physics (Gilbert, Kajiwara and Xu, eds). Kluwer, Dordrecht (2000).

    Google Scholar 

  319. van Bladel, J.: Electromagnetic Fields. Hemisphere Publishing Company, Washington 1985.

    Google Scholar 

  320. van den Berg, R. and Kleinman, R.: A contrast source inversion method. Inverse Problems 13, 1607–1620 (1997).

    Google Scholar 

  321. 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 

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

    Google Scholar 

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

    Google Scholar 

  324. Wang, Y., Yagola, A.G., and Yang, C.: Optimization and Regularization for Computational Inverse Problems and Applications. Springer, Berlin 2011.

    Google Scholar 

  325. Weck, N.: Klassische Lösungen sind auch schwache Lösungen. Arch. Math. 20, 628–637 (1969).

    Google Scholar 

  326. Werner, P.: Zur mathematischen Theorie akustischer Wellenfelder. Arch. Rational Mech. Anal. 6, 231–260 (1961).

    Google Scholar 

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

    Google Scholar 

  328. Werner, P. : On the exterior boundary value problem of perfect reflection for stationary electromagnetic wave fields. J. Math. Anal. Appl. 7, 348–396 (1963).

    Google Scholar 

  329. Werner, P.: Low frequency asymptotics for the reduced wave equation in two-dimensional exterior spaces. Math. Meth. in the Appl. Sc. 8, 134–156 (1986).

    Google Scholar 

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

    Google Scholar 

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

    Google Scholar 

  332. Wienert, L.: Die numerische Approximation von Randintegraloperatoren für die Helmholtzgleichung im \({\mathrm{I\!R}}^{3}\). Dissertation, Göttingen 1990.

    Google Scholar 

  333. Wilcox, C.H.: A generalization of theorems of Rellich and Atkinson. Proc. Amer. Math. Soc. 7, 271–276 (1956).

    Google Scholar 

  334. Wilcox, C.H.: An expansion theorem for electromagnetic fields. Comm. Pure Appl. Math. 9, 115–134 (1956).

    Google Scholar 

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

    Google Scholar 

  336. Wloka, J.: Partial Differential Equations. University Press, Cambridge 1987.

    Google Scholar 

  337. Zinn, A.: On an optimisation method for the full- and limited-aperture problem in inverse acoustic scattering for a sound-soft obstacle. Inverse Problems 5, 239–253 (1989).

    Google Scholar 

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

Copyright information

© 2013 Springer Science+Business Media New York

About this chapter

Cite this chapter

Colton, D., Kress, R. (2013). The Maxwell Equations. In: Inverse Acoustic and Electromagnetic Scattering Theory. Applied Mathematical Sciences, vol 93. Springer, New York, NY. https://doi.org/10.1007/978-1-4614-4942-3_6

Download citation

Publish with us

Policies and ethics