**Fioralba Cakoni, David Colton **

## 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.