On the Mathematical Basis of the Linear Sampling Method

Abstract The linear sampling method is an algorithm for solving the inverse scattering problem for acoustic and electromagnetic waves. The method is based on showing that a linear integral equation of first kind has a solution that becomes unbounded as a parameter 𝑧 approaches the boundary of the scatterer 𝐷 from inside 𝐷. However, except for the case of the transmission problem, the case where z is in the exterior of 𝐷 is unresolved. Since for the inverse scattering problem 𝐷 is unknown, this step is crucial for the mathematical justification of the linear sampling method. In this paper we give a mathematical justification of the linear sampling method for arbitrary 𝑧 by using the theory of integral equations of first kind with singular kernels.

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

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

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

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

[5]  Andrew N. Norris,et al.  A direct inverse scattering method for imaging obstacles with unknown surface conditions , 1998 .

[6]  D. Colton,et al.  The direct and inverse scattering problems for partially coated obstacles , 2001 .

[7]  Michele Piana,et al.  Numerical validation of the linear sampling method , 2002 .

[8]  Yongzhi Xu,et al.  Generalized dual space indicator method for underwater imaging , 2000 .

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

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

[11]  Fioralba Cakoni,et al.  The linear sampling method for anisotropic media , 2002 .

[12]  W. McLean.  Strongly Elliptic Systems and Boundary Integral Equations , 2000 .

[13]  David Colton,et al.  AN APPROXIMATION PROPERTY OF IMPORTANCE IN INVERSE SCATTERING THEORY , 2001, Proceedings of the Edinburgh Mathematical Society.

[14]  A. Lanterman,et al.  A comparison of the Colton?Kirsch inverse scattering methods with linearized tomographic inverse scattering , 2001 .

[15]  Y. You,et al.  A fast method for acoustic imaging of multiple three-dimensional objects. , 2000, The Journal of the Acoustical Society of America.