  We present a fast iterative solver for scattering problems in 2D, where a
penetrable object with compact support is considered. By representing the
scattered field as a volume potential in terms of the Green's function, we
arrive at the Lippmann-Schwinger equation in integral form, which is then
discretized using an appropriate quadrature technique. The discretized linear
system is then solved using an iterative solver accelerated by Directional
Algebraic Fast Multipole Method (DAFMM). The DAFMM presented here relies on the
directional admissibility condition of the 2D Helmholtz kernel. And the
construction of low-rank factorizations of the appropriate low-rank matrix
sub-blocks is based on our new Nested Cross Approximation (NCA)~\cite{
arXiv:2203.14832 [math.NA]}. The advantage of our new NCA is that the search
space of so-called far-field pivots is smaller than that of the existing NCAs.
Another significant contribution of this work is the use of HODLR based direct
solver as a preconditioner to further accelerate the iterative solver. In one
of our numerical experiments, the iterative solver does not converge without a
preconditioner. We show that the HODLR preconditioner is capable of solving
problems that the iterative solver can not. Another noteworthy contribution of
this article is that we perform a comparative study of the HODLR based fast
direct solver, DAFMM based fast iterative solver, and HODLR preconditioned
DAFMM based fast iterative solver for the discretized Lippmann-Schwinger
problem. To the best of our knowledge, this work is one of the first to provide
a systematic study and comparison of these different solvers for various
problem sizes and contrast functions. In the spirit of reproducible
computational science, the implementation of the algorithms developed in this
article is made available at
\url{https://github.com/vaishna77/Lippmann_Schwinger_Solver}.
