Fioralba Cakoni, David Colton

On the Mathematical Basis of the Linear Sampling Method

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.