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.
On the Mathematical Basis of the Linear Sampling Method