An accelerated, high-order accurate direct solver for the Lippmann–Schwinger equation for acoustic scattering in the plane
An accelerated, high-order accurate direct solver for the Lippmann–Schwinger equation for acoustic scattering in the plane
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用于平面声散射的 Lippmann Schwinger 方程的加速、高阶精确直接求解器
DOI：
10.1007/s10444-022-09963-1
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发表时间：
2022
影响因子：
1.7
通讯作者：
Martinsson, Per-Gunnar
中科院分区：
文献类型：
--
作者：
Gopal, Abinand;Martinsson, Per-Gunnar
本文提出了一种求解平面声散射Lippmann-Schwinger积分方程的有效直接解法。对于N个自由度的问题，求解器在运算中构造一个近似逆，然后，给定一个入射场，在运算中计算散射场。该求解器基于先前发布的积分方程直接求解器，该积分方程依赖于非对角块中的秩亏;具体而言，使用所谓的分层块可分离格式。这里描述的特定求解器已经以提高数值稳定性和鲁棒性的方式重新制定，并利用李普曼-施温格方程中内核的特定结构来加速近似逆的计算。该求解器与规则正方形网格上的Nyström离散化相结合，使用由Ran Duan和弗拉基米尔罗克林开发的求积方法，尽管积分方程的内核中存在奇异性，但该方法仍能达到高阶精度。当直接求解器以四位数的精度运行时，获得特别有效的求解器，并且被用作GMRES的预处理器，每个前向应用由FFT加速的积分算子。大量的数值实验表明，该方法在具有挑战性的环境中的高性能。使用Duan-Rokhlin求积规则的10阶精确版本，该方案能够在工作站上使用26 M自由度在几个小时内解决超过500个波长的相对误差低于10− 10的域上的问题。
An efficient direct solver for solving the Lippmann–Schwinger integral equation modeling acoustic scattering in the plane is presented. For a problem withNdegrees of freedom, the solver constructs an approximate inverse inoperations and then, given an incident field, can compute the scattered field inoperations. 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− 10in a couple of hours on a workstation, using 26M degrees of freedom.
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DOI：
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发表时间：
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期刊：
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影响因子：
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