利用对称边界积分方程加速透射散射中的分层低秩压缩

中文摘要 AI 辅助

我们提出了一种利用矩阵对称性来加速波透射散射中分层低秩压缩的方法。针对亥姆霍兹透射问题，我们引入了一种对称的类Müller边界积分方程公式。尽管该公式不是第二类积分方程，但它不会给直接求解器带来困难。在Nyström离散化中采用的一点修正求积法保持了离散系统的复对称性。该公式被集成到一种分层半可分离直接求解器的变体中。在此框架内，通过利用左、右系数之间的对称关系，代理方法中低秩近似的计算工作量减半。此外，我们将这种利用对称性的方法扩展到弹性透射问题，采用对称Galerkin边界元法。在多节点计算集群上的数值实验表明，用于亥姆霍兹散射的对称类Müller公式相比非对称基线平均实现了1.41倍的加速，同时保持三阶收敛性。此外，用于弹性散射的对称感知求解器速度提高了1.5倍以上。

英文摘要

We propose an approach using matrix symmetry to accelerate hierarchical low-rank compression for wave transmission scattering. For Helmholtz transmission problems, we introduce a symmetric Müller-like boundary integral equation formulation. Although not of the second kind, this formulation creates no difficulties for direct solvers. A one-point corrected quadrature in a Nyström discretization preserves the complex symmetry of the discretized system. This formulation is integrated into a variant of a hierarchical semiseparable direct solver. Within this framework, the computational workload of the low-rank approximation in the proxy method is halved by exploiting the symmetric relationship between the left and right coefficients. Furthermore, we extend this symmetry-exploiting approach to elastic transmission problems by using a symmetric Galerkin boundary element method. Numerical experiments on a multi-node computing cluster demonstrate that the symmetric Müller-like formulation for Helmholtz scattering achieves on average a 1.41-fold speedup compared to the non-symmetric baseline while maintaining third-order convergence. Furthermore, the symmetry-aware solver for elastic scattering is more than 1.5 times faster.

发表机构

• Institute of Science Tokyo（东京科学大学）
• RIKEN Center for Computational Science（理化学研究所计算科学中心）

机构由 AI 辅助整理，请以论文原文为准。