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Articles

The Kirsch-Kress method for inverse scattering by infinite locally rough interfaces

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Pages 85-107 | Received 05 Feb 2016, Accepted 14 May 2016, Published online: 02 Jun 2016

References

  • Chandler-Wilde SN, Zhang B. Scattering of electromagnetic waves by rough interfaces and inhomogeneous layers. SIAM J. Math. Anal. 1999;30:559–583.
  • Chorfi L, Gaitan P. Reconstruction of the interface between two-layered media using far field measurements. Inverse Prob. 2011;27:075001.
  • Natroshvili D, Arens T, Chandler-Wilde SN. Uniqueness, existence, and integral equation formulations for interface scattering problems. Mem. Differ. Equ. Math. Phys. 2003;30:105–146.
  • Li J, Sun G, Zhang R. The numerical solution of scattering by infinite rough interfaces based on the integral equation method. Comput. Math. Appl. 2016;71:1491–1502.
  • Chandler-Widle SN, Heinemeyer E, Potthast R. Acoustic scattering by mildly rough unbounded surfaces in three dimensions. SIAM J. Appl. Math. 2006;66:1002–1026.
  • Chandler-Wilde SN, Ross CR. Scattering by rough surfaces: the Dirichlet problem for the Helmholtz equation in a non-locally perturbed half-plane. Math. Methods Appl. Sci. 1996;19:959–976.
  • Chandler-Wilde S, Zhang B. A uniqueness result for scattering by infinite rough surfaces. SIAM J. Appl. Math. 1998;58:1774–1790.
  • Chandler-Wilde SN, Ross CR, Zhang B. Scattering by infinite one-dimensional rough surfaces. Proc. R. Soc. Lond. 1999;A455:3767–3787.
  • Zhang B, Chandler-Wilde SN. Integral equation methods for scattering by infinite rough surfaces. Math. Methods Appl. Sci. 2003;26:463–488.
  • Colton D, Kress R. Inverse acoustic and electromagnetic scattering theory. 3rd ed. New York (NY):Springer; 2013.
  • Chandler-Wilde SN, Ross CR. Uniqueness results for direct and inverse scattering by infinite surfaces in a lossy medium. Inverse Prob. 1995;11:1063–1067.
  • Chandler-Widle SN, Lines C. A time-domain point source method for inverse scattering by rough surfaces. Computing. 2005;75:157–180.
  • Burkard C, Potthast R. A time-domain probe method for three-dimensional rough surface reconstructions. Inverse Prob. Imaging. 2009;3:259–274.
  • Burkard C, Potthast R. A multi-section approach for rough surface reconstruction via the Kirsch-Kress scheme. Inverse Prob. 2010;26:045007.
  • Li J, Sun G. A nonlinear integral equation method for the inverse scattering problem by sound-soft rough surfaces. Inverse Prob. Sci. Eng. 2015;23:557–577.
  • DeSanto JA, Wombell RJ. Reconstruction of rough surface profiles with the Kirchhoff approximation. J. Opt. Soc. Amer. A. 1991;8:1892–1897.
  • DeSanto JA, Wombell RJ. The reconstruction of shallow rough-surface profiles from scattered field data. Inverse Prob. 1991;7:L7–L12.
  • Bao G, Lin J. Imaging of local surface displacement on an infinite ground plane: the multiple frequency case. SIAM J. Appl. Math. 2011;71:1733–1752.
  • Kress R, Tran T. Inverse scattering for a locally perturbed half-plane. Inverse Prob. 2000;16:1541–1559.
  • Zhang H, Zhang B. A novel integral equation for scattering by locally rough surfaces and application to the inverse problem. SIAM J. Appl. Math. 2013;73:1811–1829.
  • Bao G, Li P. Near-field imaging of infinite rough surfaces. SIAM J. Appl. Math. 2013;73:2162–2187.
  • Bao G, Li P. Near-field imaging of infinite rough surfaces in dielectric media. SIAM J. Imaging Sci. 2014;7:867–899.
  • Akduman I, Kress R, Yapar A. Iterative reconstruction of dielectric rough surface profiles at fixed frequency. Inverse Prob. 2006;22:939–954.
  • Bruckner G, Elschner J. The numerical solution of an inverse periodic transmission problem. Math. Mech. Appl. Sci. 2005;28:757–778.
  • Lechleiter A. Imaging of periodic dielectrics. BIT Numer. Math. 2010;50:59–83.
  • Yang J, Zhang B, Zhang R. Reconstruction of penetrable grating profiles. Inverse Prob. Imaging. 2013;7:1393–1407.
  • Zhang R, Sun J. Efficient finite element method for grating profile reconstruction. J. Comput. Phys. 2015;302:405–419.
  • Chandler-Wilde SN, Hothersall DC. Efficient calculation of the Green function for acoustic propagation above a homogeneous impedance plane. J. Sound Vib. 1995;180:705–724.
  • Mor\’{e} JJ. The Levenberg-marquardt algorithm: implementation and theory. Watson GA, editor. Numerical analysis. Berlin: Springer; 1977. p. 105–116
  • Chandler-Widle SN, Monk P. Existence, uniqueness and variational methods for scattering by unbounded rough surfaces. SIAM J. Math. Anal. 2005;37:598–618.
  • Chandler-Wilde SN, Zhang B, Ross CR. On the solvability of second kind integral equations on the real line. J. Math. Anal. Appl. 2000;245:28–51.
  • Kirsch A, Kress R. On an integral equation of the first kind in inverse acoustic scattering. Cannon JR, Hornung U, editors. Inverse problems. Vol. 77, ISNM. Basel: Birkhauser; 1986. p. 93–102
  • Kirsch A, Kress R. A numerical method for an inverse scattering problem. Engl HW, Groetsch CW, editors. Inverse problems. Orlando: Academic Press; 1987. p. 279–290
  • Kirsch A, Kress R. An optimization method in inverse acoustic scattering. Boundary elements IX. Brebbia CA et al. editor. Fluid flow and potential applications. Vol. 3, Berlin: Springer; 1987. p. 3–18
  • Arens T, Chandler-Wilde SN, Haseloh KO. Solvability and spectral properties of integral equations on the real line: II. Lp spaces and applications. J. Integral Equ. Appl. 2003;15:1–35.

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