Journal of Integral Equations and Applications

Fast Methods for Three-dimensional Inverse Obstacle Scattering Problems

H. Harbrecht and T. Hohage

Full-text: Open access

Article information

J. Integral Equations Applications Volume 19, Number 3 (2007), 237-260.

First available in Project Euclid: 27 September 2007

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier


Harbrecht, H.; Hohage, T. Fast Methods for Three-dimensional Inverse Obstacle Scattering Problems. J. Integral Equations Applications 19 (2007), no. 3, 237--260. doi:10.1216/jiea/1190905486.

Export citation


  • A.B. Bakushinskii, The problem of the convergence of the iteratively regularized Gauss-Newton method, Comput. Math. Math. Phys. 32 (1992), 1353-1359.
  • B. Blaschke, A. Neubauer and O. Scherzer, On convergence rates for the iteratively regularized Gauss-Newton method, IMA J. Numerical Anal. 17 (1997), 421-436.
  • H. Brackhage and P. Werner, Über das Dirichletsche Außenraumproblem für die Helmholtzsche Schwingungsgleichung, Arch. Math. 16 (1965), 325-329.
  • D. Colton and R. Kress, Integral equation methods in scattering theory, in Pure and applied mathematics, Chichester, Wiley, New York, 198-3.
  • --------, Inverse acoustic and electromagnetic scattering, 2nd Edition, Springer Verlag, Berlin, 199-7.
  • W. Dahmen, H. Harbrecht and R. Schneider, Compression techniques for boundary integral equations-optimal complexity estimates, SIAM J. Numer. Anal. 43 (2006), 2251-2271.
  • W. Dahmen, A. Kunoth and K. Urban, Biorthogonal spline-wavelets on the interval-stability and moment conditions, Appl. Comp. Harm. Anal. 6 (1999), 259-302.
  • C. Farhat, R. Tezaur and R. Djellouli, On the solution of three-dimensional inverse obstacle acoustic scattering problems by a regularized Newton method, Inverse Problems 18 (2002), 1229-1246.
  • K. Giebermann, Schnelle Summationsverfahren zur numerischen Lösung von Integralgleichungen für Streuprobleme im R3, Ph.D. thesis, Universität Karlsruhe, Germany, 199-7.
  • M. Hanke, F. Hettlich and O. Scherzer, The Landweber iteration for an inverse scattering problem, in Proc. 1995 Design Engineering Technical Conferences Vol. 3 Part C, K.W. Wang et al., eds., 909-915, ASME, New York, 199-5.
  • M. Hanke, Regularizing properties of a truncated Newton-CG algorithm for nonlinear inverse problems, Numer. Funct. Anal. Optim. 18 (1997), 971-993.
  • H. Harbrecht, Wavelet Galerkin schemes for the boundary element method in three dimensions, Ph.D. thesis, Technische Universität Chemnitz, Germany, 200-1.
  • H. Harbrecht and R. Schneider, Wavelet Galerkin schemes for boundary integral equations-implementation and quadrature, SIAM J. Sci. Comput. 27 (2006), 1347-1370.
  • --------, Biorthogonal wavelet bases for the boundary element method, Math. Nach. 269-270 (2004), 167-188.
  • F. Hettlich and W. Rundell, A second degree method for nonlinear ill-posed problems, SIAM J. Numer. Anal. 37 (2000), 587-620.
  • T. Hohage, Logarithmic convergence rates of the iteratively regularized Gauss-Newton method for an inverse potential and an inverse scattering problem, Inverse Problems 13 (1997), 1279-1299.
  • --------, On the numerical solution of a three-dimensional inverse medium scattering problem, Inverse Problems 17 (2001), 1743-1763.
  • --------, Iterative methods in inverse obstacle scattering: Regularization theory of linear and nonlinear exponentially ill-posed problems, Ph.D. thesis, Johannes-Kepler-Universität Linz, Austria, 199-9.
  • D. Huybrechs, J. Simoens and S. Vandewalle, A note on wave number dependence of wavelet matrix compression for integral equations with oscillatory kernel, J. Comput. Appl. Math. 172 (2004), 233-246.
  • B. Kaltenbacher, A. Neubauer, and O. Scherzer, Iterative regularization methods for nonlinear ill-posed problems. Kluwer, Dordrecht, 200-5. to appear.
  • H. Kersten, Grenz- und Sprungrelationen für Potentiale mit quadrat-summierbarer Dichte, Resultate d. Math. 3 (1980), 17-24.
  • A. Kirsch, Surface gradients and continuity properties for some integral operators in classical scattering theory, Math. Meth. Appl. Sci. 11 (1989), 789-804.
  • --------, Properties of far field operators in acoustic scattering, Math. Meth. Appl. Sci. 11 (1989), 773-787.
  • --------, The domain derivative and two applications in inverse scattering theory, Inverse Problems 9 (1993), 81-96.
  • R. Kress, Minimizing the condition number of boundary integral operators in acoustic and electromagnetic scattering, Quart. J. Mech. Appl. Math. 38 (1985), 323-341.
  • --------, A Newton method in inverse obstacle scattering, in Inverse problems in engineering mechanics, Bui et al., eds., Balkema, Rotterdam, 1994.
  • R.D. Murch, D.G.H. Tan and D.J.N. Wall, Newton-Kantorovich method applied to two-dimensional inverse scattering for an exterior Helmholtz problem, Inverse Problems 4 (1988), 1117-1128.
  • R. Schneider, Multiskalen- und Wavelet-Matrixkompression: Analysisbasierte Methoden zur Lösung großer vollbesetzter Gleichungssyteme, B.G. Teubner, Stuttgart, 199-8.
  • W. Tobocman, Inverse acoustic wave scattering in two dimensions from impenetrable targets, Inverse Problems 5 (1989), 1131-1144.