Sparsifying Preconditioner for the Lippmann-Schwinger Equation

TL;DR

This paper introduces a sparsifying preconditioner that transforms the discretized Lippmann-Schwinger equation into a sparse form, significantly accelerating iterative solutions for acoustic, electromagnetic, and quantum scattering problems.

Contribution

The paper presents a novel sparsifying preconditioner that makes the Lippmann-Schwinger equation more computationally efficient and easy to implement, with near frequency-independent iteration counts.

Findings

Effective in 2D and 3D scattering problems

Reduces iteration counts for iterative solvers

Easy to implement and integrate with existing methods

Abstract

The Lippmann-Schwinger equation is an integral equation formulation for acoustic and electromagnetic scattering from an inhomogeneous media 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 2D and 3D numerical results to demonstrate the effectiveness of this new preconditioner.