Journal of Integral Equations and Applications

Reconstruction for cavities with impedance boundary condition

Hai-Hua Qin and Ji-Chuan Liu

Full-text: Open access


In this paper, we consider the inverse scattering problem of recovering the shape of a cavity or the surface impedance from one source and a knowledge of measurements placed on a curve inside the cavity. Based on a potential approach the inverse problem is equivalent to a system of nonlinear and ill-posed integral equations, a regularized Newton iterative approach is applied to reconstruct the boundary and the injectivity for the linearized system is established. Numerical examples are provided showing the viability of our method.

Article information

J. Integral Equations Applications, Volume 25, Number 3 (2013), 431-454.

First available in Project Euclid: 16 December 2013

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35J25: Boundary value problems for second-order elliptic equations 45Q05: Inverse problems 65R30: Improperly posed problems 78A46: Inverse scattering problems

Inverse scattering problem shape of a cavity surface impedance potential approach nonlinear integral equations


Qin, Hai-Hua; Liu, Ji-Chuan. Reconstruction for cavities with impedance boundary condition. J. Integral Equations Applications 25 (2013), no. 3, 431--454. doi:10.1216/JIE-2013-25-3-431.

Export citation


  • I. Akduman and R. Kress, Direct and inverse scattering problems for inhomogeneous impedance cylinders of arbitrary shape, Radio Sci. 38 (2003), 1055-1064.
  • T.S. Angell and A. Kirsch, Optimization methods in electromagnetic radiation, Springer Mono. Math., Springer-Verlag, New York, 2004.
  • L.M. Brekhovskikh, Waves in layered media, Appl. Math. Mech. 16, Academic Press, Inc., New York, 1960.
  • F. Cakoni and D. Colton, Qualitative methods in inverse scattering theory, in Interaction of mechanics and mathematics, Springer-Verlag, Berlin, 2006.
  • F. Cakoni and R. Kress, Integral equations for inverse problems in corrosion detection from partial Cauchy data, Inverse Prob. Imag. 1 (2007), 229-245.
  • F. Cakoni, R. Kress and C. Schuft, Integral equations for shape and impedance reconstruction in corrosion detection, Inverse Prob. 26 (2010), 095012, 1-24.
  • –––, Simultaneous reconstruction of shape and impedance in corrosion detection, Methods Appl. Anal. 17 (2010), 357-378.
  • D. Colton and A. Kirsch, A simple method for solving inverse scattering problems in the resonance region, Inverse Prob. 12 (1996), 383-393.
  • D. Colton and R. Kress, Inverse acoustic and electromagnetic scattering theory, Appl. Math. Sci. 93, Springer-Verlag, Berlin, 1998.
  • D. Colton, M. Piana and R. Potthast, A simple method using Morozov's discrepancy principle for solving inverse scattering problems, Inverse Prob. 13 (1997), 1477-1493.
  • P.C. Hansen, Regularization tools: A Matlab package for analysis and solution of discrete ill-posed problems, Numer. Algor. 6 (1994), 1-35.
  • O. Ivanyshyn and T. Johansson, Nonlinear integral equation methods for the reconstruction of an acoustically sound-soft obstacle, J. Int. Equat. Appl. 19 (2007), 289-308.
  • O. Ivanyshyn and R. Kress, Inverse scattering for planar cracks via nonlinear integral equations, Math. Meth. Appl. Sci. 31 (2008), 1221-1232.
  • O. Ivanyshyn, R. Kress and P. Serranho, Huygens' principle and iterative methods in inverse obstacle scattering, Adv. Comp. Math. 33 (2010), 413-429.
  • P. Jakubik and R. Potthast, Testing the integrity of some cavity-The Cauchy problem and the range test, Appl. Numer. Math. 58 (2008), 899–914.
  • T. Johansson and B.D. Sleeman, Reconstruction of an acoustically sound-soft obstacle from one incident field and the far-field pattern, IMA J. Appl. Math. 72 (2007), 96-112.
  • R. Kress, Linear integral equations, Appl. Math. Sci. 82, Springer-Verlag, New York, 1999.
  • R. Kress and W. Rundell, Inverse scattering for shape and impedance, Inverse Prob. 17 (2001), 1075-1085.
  • –––, Nonlinear integral equations and the iterative solution for an inverse boundary value problem, Inverse Prob. 21 (2005), 1207-1223.
  • C.L. Lawson and R.J. Hanson, Solving least squares problems, SIAM, Philadelphia, 1995.
  • K.-M. Lee, Inverse scattering via nonlinear integral equations for a Neumann crack, Inverse Prob. 22 (2006), 1989-2000.
  • W. Mclean, Strongly elliptic systems and boundary integral equations, Cambridge University Press, Cambridge, 2000.
  • R. Potthast, Fréchet differentiability of boundary integral operators in inverse acoustic scattering, Inverse Prob. 10 (1994), 431-447.
  • H.-H. Qin and F. Cakoni, Nonlinear integral equations for shape reconstruction in the inverse interior scattering problem, Inverse Prob. 27 (2011), 035005, 1-17.
  • H.-H. Qin and D. Colton, The inverse scattering problem for cavities, Appl. Numer. Math. 62 (2012), 699-708.
  • –––, The inverse scattering problem for cavities with impedance boundary condition, Adv. Comp. Math. 36 (2012), 157-174.
  • P. Serranho, A hybrid method for inverse scattering for shape and impedance, Inverse Prob. 22 (2006), 663-680.
  • F. Zeng, F. Cakoni and J. Sun, An inverse electromagnetic scattering problem for a cavity, Inverse Prob. 27 (2011), 125002, 1-17. \noindentstyle