New characterizations of solutions in inverse scattering theory

This paper introduces a new approach to characterize the shape ω of a scattering medium (either an acoustically soft obstacle or an inhomogeneous medium) by the far field data. In contrast to the Linear Sampling Method normality of the far field operator is not needed. Therefore, also scattering by limited far field data and absorbing media can be treated. While in the Linear Sampling Method the points in the interior of ω are characterized by the solution of an integral equation of the first kind, for our new method a constrained optimization problem has to be solved. Although this new approach is more time consuming some numerical experiments at the end of this paper show the practicability of the method.

[1]  A. Kirsch,et al.  A simple method for solving inverse scattering problems in the resonance region , 1996 .

[2]  J. Planchard,et al.  Une méthode variationnelle d’éléments finis pour la résolution numérique d’un problème extérieur dans $\mathbf {R}^3$ , 1973 .

[3]  Gunther Uhlmann,et al.  Recovery of singularities for formally determined inverse problems , 1993 .

[4]  A. Kirsch.  An Introduction to the Mathematical Theory of Inverse Problems , 1996, Applied Mathematical Sciences.

[5]  M. Hanke,et al.  Numerical implementation of two noniterative methods for locating inclusions by impedance tomography , 2000 .

[6]  Peter Monk,et al.  A Regularized Sampling Method for Solving Three-Dimensional Inverse Scattering Problems , 1999, SIAM J. Sci. Comput..

[7]  Jochen Werner,et al.  Optimization Theory and Applications , 1984 .

[8]  Stefan Ritter,et al.  A linear sampling method for inverse scattering from an open arc Inverse Problems , 2000 .

[9]  Martin Brühl,et al.  Explicit Characterization of Inclusions in Electrical Impedance Tomography , 2001, SIAM J. Math. Anal..

[10]  William Rundell,et al.  Surveys on solution methods for inverse problems , 2000 .

[11]  Peter Monk,et al.  The Linear Sampling Method in Inverse Scattering Theory , 2000 .

[12]  René Marklein,et al.  Applied Inversion in Nondestructive Testing , 1997 .

[13]  Rainer Kress,et al.  CORRIGENDUM: Uniqueness in inverse obstacle scattering with conductive boundary condition , 1996 .

[14]  Andreas Kirsch,et al.  Factorization of the far-field operator for the inhomogeneous medium case and an application in inverse scattering theory , 1999 .

[15]  M. G. Cote.  Automated swept-angle bistatic scattering measurements using continuous wave radar , 1991, [1991] Conference Record. IEEE Instrumentation and Measurement Technology Conference.

[16]  Richard A. Albanese,et al.  Mathematics, medicine and microwaves , 1994 .

[17]  Rainer Kress,et al.  Inverse scattering from an open arc , 1995 .

[18]  Peter Monk,et al.  Recent Developments in Inverse Acoustic Scattering Theory , 2000, SIAM Rev..

[19]  D. Colton,et al.  A simple method using Morozov's discrepancy principle for solving inverse scattering problems , 1997 .

[20]  R. Kress,et al.  Inverse Acoustic and Electromagnetic Scattering Theory , 1992 .

[21]  P. Bassanini,et al.  Elliptic Partial Differential Equations of Second Order , 1997 .

[22]  Peter Monk,et al.  Mathematical Problems in Microwave Medical Imaging , 1999 .

[23]  David Colton,et al.  The simple method for solving the electromagnetic inverse scattering problem: the case of TE polarized waves , 1998 .

[24]  R. Kress,et al.  Integral equation methods in scattering theory , 1983 .

[25]  Rainer Kress,et al.  Uniqueness in inverse obstacle scattering (acoustics) , 1993 .

[26]  P. M. Berg,et al.  A modified gradient method for two-dimensional problems in tomography , 1992 .

[27]  Andreas Kirsch,et al.  Characterization of the shape of a scattering obstacle using the spectral data of the far field operator , 1998 .

[28]  Peter Monk,et al.  A Linear Sampling Method for the Detection of Leukemia Using Microwaves , 1998, SIAM J. Appl. Math..

[29]  R. Kress.  Minimizing the condition number of boundary integral operators in acoustic and electromagnetic scattering , 1985 .