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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 two-dimensional and three-dimensional (3D) problems, where the iteration numbers are almost independ...

Sparsify and Sweep: An Efficient Preconditioner for the Lippmann--Schwinger Equation