  In this paper we present a hybrid approach to numerically solve
two-dimensional electromagnetic inverse scattering problems, whereby the
unknown scatterer is hosted by a possibly inhomogeneous background. The
approach is `hybrid' in that it merges a qualitative and a quantitative method
to optimize the way of exploiting the a priori information on the background
within the inversion procedure, thus improving the quality of the
reconstruction and reducing the data amount necessary for a satisfactory
result. In the qualitative step, this a priori knowledge is utilized to
implement the linear sampling method in its near-field formulation for an
inhomogeneous background, in order to identify the region where the scatterer
is located. On the other hand, the same a priori information is also encoded in
the quantitative step by extending and applying the contrast source inversion
method to what we call the `inhomogeneous Lippmann-Schwinger equation': the
latter is a generalization of the classical Lippmann-Schwinger equation to the
case of an inhomogeneous background, and in our paper is deduced from the
differential formulation of the direct scattering problem to provide the
reconstruction algorithm with an appropriate theoretical basis. Then, the point
values of the refractive index are computed only in the region identified by
the linear sampling method at the previous step. The effectiveness of this
hybrid approach is supported by numerical simulations presented at the end of
the paper.
