论文标题
基于HODLR预处理加速的Lippmann-Schinginger方程的新的定向代数快速多极方法的迭代求解器
A new Directional Algebraic Fast Multipole Method based iterative solver for the Lippmann-Schwinger equation accelerated with HODLR preconditioner
论文作者
论文摘要
我们提出了一个快速的迭代求解器，用于在2D中散射问题，其中考虑了具有紧凑支撑的可穿透对象。通过将散射场作为绿色功能方面的体积潜力，我们以积分形式到达Lippmann-Schinginger方程，然后使用适当的正交技术离散化。然后，使用定向代数快速多极方法（DAFMM）加速的迭代求解器来求解离散的线性系统。这里呈现的DAFMM依赖于2D Helmholtz内核的定向可接纳条件。适当的低级矩阵子块的低级别因素的构建基于我们的新嵌套交叉近似（NCA）〜\ cite {arxiv：2203.14832 [Math.na]}。我们新的NCA的优点是，所谓的远场枢轴的搜索空间小于现有NCA的搜索空间。这项工作的另一个重要贡献是将基于HODLR的直接求解器用作进一步加速迭代求解器的预处理。在我们的数值实验之一中，没有预处理程序不会收敛。我们表明，HODLR预处理能够解决迭代求解器无法的问题。本文的另一个值得注意的贡献是，我们对基于HODLR的快速直接求解器，基于DAFMM的快速迭代求解器和HODLR预处理DAFMM的快速迭代求解器进行了比较研究，以解决离散化的Lippmann-Schwinger问题。据我们所知，这项工作是最早为各种问题大小和对比功能提供系统研究和比较这些不同求解器的研究之一。本着可重复的计算科学的精神，本文开发的算法的实现可在\ url {https://github.com/vaishna77/lippmann_schwinger_solver}提供。
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}.
