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Sparsifying Preconditioner for the Lippmann-Schwinger Equation

The Lippmann--Schwinger equation is an integral equation formulation for acoustic and electromagnetic scattering from an inhomogeneous medium and quantum scattering from a localized potential. We present the sparsifying preconditioner for accelerating the iterative solution of the Lippmann--Schwinger equation. This new preconditioner transforms the discretized Lippmann--Schwinger equation into sparse form and leverages the efficient sparse linear algebra algorithms for computing an approximate inverse. This preconditioner is efficient and easy to implement. When combined with standard iterative methods, it results in almost frequency-independent iteration counts. We provide two- and three-dimensional numerical results to demonstrate the effectiveness of this new preconditioner.