稀疏化与扫掠：Lippmann-Schwinger方程的一种高效预处理子
Sparsify and sweep: an efficient preconditioner for the Lippmann-Schwinger equation
AI总结：
本文提出一种结合稀疏化与扫掠预处理思想的高效预处理子，通过稀疏化将Lippmann-Schwinger方程的稠密线性系统转换为近稀疏系统，并设计新的完美匹配层（PML）Stencil以提高精度，实现高频率下的近线性求解。
摘要：
本文提出了一种高效的预处理子，用于Lippmann-Schwinger方程，结合了稀疏化与扫掠预处理的思想。首先通过稀疏化预处理将稠密线性系统转换为近稀疏系统，其关键创新是设计了新的完美匹配层（PML）Stencil用于边界自由度。所得到的稀疏系统能够产生较为精确的解，因此可视为Helmholtz方程的准确离散化。新的PMLStencil也为应用移动PML扫掠预处理子来近似求解所得到的稀疏系统铺平了道路。当与标准GMRES求解器结合时，该预处理子在2D和3D问题中仅需几次迭代即可收敛，迭代次数几乎与频率无关。据我们所知，这是首次在高频情况下实现近线性成本求解3D Lippmann-Schwinger方程的方法。
英文摘要：
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 2D and 3D problems, where the iteration numbers are almost independent of the frequency. To the best of our knowledge, this is the first method that achieves near-linear cost to solve the 3D Lippmann-Schwinger equation in high frequency cases.
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