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 10th-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 relative error below 10^− 10 in a couple of hours on a workstation, using 26M degrees of freedom.

提出了一种有效的直接求解器，用于求解平面内声学散射的 Lippmann-Schwinger 积分方程。对于具有N个自由度的问题，求解器在\(\mathcal {O}(N^{3/2})\)运算中构造一个近似逆，然后，给定一个入射场，可以计算 \(\mathcal {O}(N^{3/2}))\) 中的散射场(\mathcal {O}(N \log N)\)操作。该求解器基于先前发布的积分方程直接求解器，该求解器依赖于非对角块中的秩不足；具体来说，使用了所谓的 Hierarchically Block Separable 格式。此处描述的特定求解器已以提高数值稳定性和鲁棒性的方式重新制定，并利用 Lippmann-Schwinger 方程中内核的特定结构来加速近似逆的计算。求解器与规则方形网格上的 Nyström 离散化相结合，使用 Ran Duan 和 Vladimir Rokhlin 开发的求积方法，尽管积分方程的内核存在奇异性，但仍可实现高阶精度。当直接求解器以四位数的精度运行时，可以获得特别有效的求解器，并用作 GMRES 的预条件子，积分算子的每个前向应用都由 FFT 加速。提出了广泛的数值实验，说明了该方法在具有挑战性的环境中的高性能。使用 Duan-Rokhlin 正交规则的 10 阶精确版本，该方案能够解决超过 500 个波长宽且相对误差低于 10 的域上的问题⁻使用 2600 万个自由度，在工作站上几个小时内完成 10 个。