  This paper presents an efficient preconditioner for the Lippmann-Schwinger
equation that combines the ideas of the sparsifying and the sweeping
preconditioners. Following first the idea of the sparsifying preconditioner,
this new preconditioner starts by transforming the dense linear system of the
Lippmann-Schwinger equation into a nearly sparse system. The key novelty is a
newly designed perfectly matched layer (PML) stencil for the boundary degrees
of freedoms. The resulting sparse system gives rise to fairly accurate
solutions and hence can be viewed as an accurate discretization of the
Helmholtz equation. This new PML stencil also paves the way for applying the
moving PML sweeping preconditioner to invert the resulting sparse system
approximately. When combined with the standard GMRES solver, this new
preconditioner for the Lippmann-Schwinger equation takes only a few iterations
to converge for both 2D and 3D problems, where the iteration numbers are almost
independent of the frequency. To the best of our knowledge, this is the first
method that achieves near-linear cost to solve the 3D Lippmann-Schwinger
equation in high frequency cases.
