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：我们提出了一种针对二维散射问题的快速迭代求解器，其中考虑了具有紧支撑的可穿透物体。通过将散射场表示为以格林函数为核的体积势，我们得到积分形式的Lippmann-Schwinger方程，随后采用适当的求积技术对其进行离散。离散化后的线性系统通过方向性代数快速多极方法（DAFMM）加速的迭代求解器进行求解。本文提出的DAFMM依赖于二维亥姆霍兹核的方向性可容条件。而对相应低秩矩阵子块的低秩因子分解的构造，则基于我们提出的新型嵌套交叉近似（NCA）~\cite{ arXiv:2203.14832 [math.NA]}。该新型NCA的优势在于，其远场主元的搜索空间小于现有NCA的搜索空间。本文另一重要贡献是采用基于HODLR的直接求解器作为预条件子，以进一步加速迭代求解器。在其中一个数值实验中，迭代求解器在无预条件子时无法收敛。我们证明，HODLR预条件子能够解决迭代求解器无法求解的问题。本文另一值得注意的贡献在于：我们针对离散Lippmann-Schwinger问题，对基于HODLR的快速直接求解器、基于DAFMM的快速迭代求解器以及经HODLR预条件子加速的DAFMM快速迭代求解器进行了对比研究。据我们所知，本文是首次针对不同问题规模和对比度函数系统性地研究并比较这些求解器的工作之一。秉承可重复计算科学的精神，本文所开发算法的实现代码已公开于\url{https://github.com/vaishna77/Lippmann_Schwinger_Solver}。