新的瞄准代数快速多极子方法迭代求解LS方程，使用HODLR预处理器加速 (A new Directional Algebraic Fast Multipole Method based iterative solver for the Lippmann-Schwinger equation accelerated with HODLR preconditioner)

We present a fast iterative solver for scattering problems in 2D, where a penetrable object with compact support is considered. By representing the scattered field as a volume potential in terms of the Green's function, we arrive at the Lippmann-Schwinger equation in integral form, which is then discretized using an appropriate quadrature technique. The discretized linear system is then solved using an iterative solver accelerated by Directional Algebraic Fast Multipole Method (DAFMM). The DAFMM presented here relies on the directional admissibility condition of the 2D Helmholtz kernel. And the construction of low-rank factorizations of the appropriate low-rank matrix sub-blocks is based on our new Nested Cross Approximation (NCA)~\cite{ arXiv:2203.14832 [math.NA]}. The advantage of our new NCA is that the search space of so-called far-field pivots is smaller than that of the existing NCAs. Another significant contribution of this work is the use of HODLR based direct solver as a preconditioner to further accelerate the iterative solver. In one of our numerical experiments, the iterative solver does not converge without a preconditioner. We show that the HODLR preconditioner is capable of solving problems that the iterative solver can not. Another noteworthy contribution of this article is that we perform a comparative study of the HODLR based fast direct solver, DAFMM based fast iterative solver, and HODLR preconditioned DAFMM based fast iterative solver for the discretized Lippmann-Schwinger problem. To the best of our knowledge, this work is one of the first to provide a systematic study and comparison of these different solvers for various problem sizes and contrast functions. In the spirit of reproducible computational science, the implementation of the algorithms developed in this article is made available at \url{https://github.com/vaishna77/Lippmann_Schwinger_Solver}.

翻译：我们提出了一种快速迭代求解2D散射问题的求解器，该问题考虑支持紧凑的渗透对象。通过将散射场表示为基于格林函数的体积势，我们得到了积分形式的Lippmann-Schwinger方程，然后使用适当的积分技术对其进行离散化。离散化的线性系统随后使用一个以瞄准代数快速多极子方法（DAFMM）为加速器的迭代求解器求解。我们所提出的DAFMM基于2D Helmholtz核的方向可接受性条件。在议题适当的低秩矩阵子块的低秩因子化方面，基于我们的新的Nested Cross Approximation（NCA）\cite{arXiv:2203.14832 [math.NA]}。我们的新NCA的优点在于，所谓远场枢轴的搜索空间比现有的NCA要小。本文的另一个重要贡献是使用HODLR基于直接求解器作为预处理器，以进一步加速迭代求解器。在我们的一个数值实验中，迭代求解器在没有预处理器的情况下无法收敛。我们表明，HODLR预处理器能够解决迭代求解器无法解决的问题。本文另一个值得注意的贡献是对HODLR快速直接求解器、DAFMM快速迭代求解器和HODLR预处理的DAFMM快速迭代求解器在离散化的Lippmann-Schwinger问题上进行比较研究。据我们所知，本文是首个对这些不同求解器在各种问题大小和对比度函数下进行系统研究和比较的论文。本文的算法实现已在\url{https://github.com/vaishna77/Lippmann_Schwinger_Solver} 中提供，倡导可重复的计算科学。