Hierarchical low-rank compression exploiting symmetric boundary integral equations in transmission scattering

Abstract

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 M\"uller-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 Nystr\"om 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 M\"uller-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.