Fioralba Cakoni, David Colton
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.