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

Sparsify and sweep: an efficient preconditioner for the Lippmann-Schwinger equation