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. 1. Introduction. This paper is concerned with the efficient solution of the Lippmann–Schwinger equation, which describes time-harmonic scattering from in-homogeneous media in acoustics and electromagnetics as well as time-harmonic scattering from localized potentials in quantum mechanics. The simplest form of this equation comes from inhomogeneous acoustic scattering. Let ω be the frequency of the time-harmonic wave, and denote the index of refraction by 1 − m(x). The inho-mogeneity m(x) is a function supported in a compact domain Ω ⊂ R d of size O(1), so the index of refraction is 1 outside Ω. Given an incoming wave u I (x) that satisfies the free space Helmholtz equation