论文标题
lippmann-Schwinger方程的加速，高阶精确求解器，用于平面的声学散射
An accelerated, high-order accurate direct solver for the Lippmann-Schwinger equation for acoustic scattering in the plane
论文作者
论文摘要
提出了一种有效的直接求解器，用于解决平面中的Lippmann-Schwinger积分方程模型。对于$ n $ n $自由度的问题，求解器在$ \ MATHCAL {O}（n^{3/2}）$操作中构建一个大约反向的逆，然后给定一个事件字段，可以计算$ \ Mathcal {o}（o}（n \ log n）$ Operations的散射字段。该求解器基于先前发布的直接求解器，用于积分方程，该方程依赖于非对角线块中的等级缺乏。具体而言，使用了所谓的分层可分离格式。此处描述的特定求解器以提高数值稳定性和鲁棒性的方式进行了重新进行重新重新构建，并利用了Lippmann-Schwinger方程中内核的特定结构，以加速近似逆的计算。求解器与常规平方网上的NyStröm离散化结合，使用RAN Duan和Vladimir Rokhlin开发的正交方法，尽管积分方程的内核具有奇异性，该方法仍达到高阶精度。当直接求解器以四位精度运行并用作GMRE的预处理时，将获得一个特别有效的求解器，并且积分运算符的每个正向应用都通过FFT加速。提出了广泛的数值实验，以说明该方法在具有挑战性的环境中的高性能。使用$ 10^{\ rm th} $ - 订购duan-rokhlin正交规则的订购准确版本，该方案能够在使用26m毫米的自由度上，在几个小时内使用26mmenter的自由度来解决500多个波长宽至$ 10^{-10} $以下的域的问题。
An efficient direct solver for solving the Lippmann-Schwinger integral equation modeling acoustic scattering in the plane is presented. For a problem with $N$ degrees of freedom, the solver constructs an approximate inverse in $\mathcal{O}(N^{3/2})$ operations and then, given an incident field, can compute the scattered field in $\mathcal{O}(N \log N)$ operations. The solver is based on a previously published direct solver for integral equations that relies on rank-deficiencies in the off-diagonal blocks; specifically, the so-called Hierarchically Block Separable format is used. The particular solver described here has been reformulated in a way that improves numerical stability and robustness, and exploits the particular structure of the kernel in the Lippmann-Schwinger equation to accelerate the computation of an approximate inverse. The solver is coupled with a Nyström discretization on a regular square grid, using a quadrature method developed by Ran Duan and Vladimir Rokhlin that attains high-order accuracy despite the singularity in the kernel of the integral equation. A particularly efficient solver is obtained when the direct solver is run at four digits of accuracy, and is used as a preconditioner to GMRES, with each forwards application of the integral operators accelerated by the FFT. Extensive numerical experiments are presented that illustrate the high performance of the method in challenging environments. Using the $10^{\rm th}$-order accurate version of the Duan-Rokhlin quadrature rule, the scheme is capable of solving problems on domains that are over 500 wavelengths wide to residual error below $10^{-10}$ in a couple of hours on a workstation, using 26M degrees of freedom.
