Proxy-surface-based fast direct solver for TE-mode scattering problems on distributed memory systems

Jul 16, 2026

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An MPI/OpenMP hybrid parallelized fast direct solver targets transverse electric (TE)-mode electromagnetic scattering in transmission settings. TE-mode scattering reduces to a 2D Helmholtz problem, where hierarchically semiseparable (HSS) methods offer high parallel efficiency but suffer with high-order discretizations due to low-rank approximation accuracy loss. The proposed solver achieves O(h3) convergence using a weakly singular Burton–Miller boundary integral formulation and a Nyström method with one-point correction. A load-balancing strategy is introduced because matrix-entry evaluation dominates HSS-type boundary integral costs, enabling near-ideal strong and weak scalabilities.

LanguageEnglish

CategoryResearch & Report

FormatPDF

Pages4

Reading Time10 min

Introduction

Background: boundary integral and FMM acceleration

Limitation of HSS in high-order discretization

Proposed method

Burton–Miller formulation and Nyström one-point correction

Proxy-surface based low-rank approximation

Load balancing for parallel efficiency

arXiv :2607 .00790v1 [math .NA] 1 Jul 2026 Proxy-surface-based fast direct solver for TE-mode scattering problems on distributed memory systems Yasuhiro Matsumoto 1 and Rio Yokota2 1 Center for Information Infrastructure, Institute of Science Tokyo, Japan 2Institute of Integrated Research, Institute of Science Tokyo, Japan This paper describes an MPI/OpenMP hybrid parallelized fast direct solver for the scattering problem of transverse electric (TE)-mode electromagnetic waves. Because TE-mode scattering can be reduced to the two-dimensional Helmholtz equation, solvers based on the hierarchically semiseparable (HSS) representation are highly attractive due to their high parallel efficiency. However, as the HSS representation applies low-rank approximations to all off-diagonal blocks, it exhibits poor compatibility with high-order discretization methods. We developed a fast direct solver with O (h3 ) convergence for Helmholtz transmission problems, whereas conventional HSS solvers typically yield only O (h) convergence (where h represents intervals between the quadrature nodes). It is based on the weakly singular Burton–Miller boundary integral equation and the Nystrm method with a one-point correction. Furthermore, recognizing that matrix component calculation, rather than matrix factorization, dominates the total computational time of HSS-type boundary integral solvers, we introduced a load-balancing method to maximize parallel efficiency. Numerical results demonstrate that the direct solver achieves high-accuracy convergence and nearly ideal strong and weak scalabilities. Index Terms—Fast direct solver, Parallelization, HSS representation, Burton–Miller method, Boundary integral equation I. INTRODUCTION The numerical solution of large-scale electromagnetic scattering problems is important in both physics and engineering applications. For problems defined over bounded domains, numerical techniques such as the finite difference method and the finite element method are effective. However, to solve problems in unbounded domains, these methods become challenging due to the necessity of handling radiation conditions. Consequently, numerical solvers based on boundary integral equations are a promising alternative. The fast multipole method (FMM) is well known as an efficient acceleration technique of boundary integral solvers for scattering problems [1] . Parallelization techniques for FMMin distributed memory systems have also been well studied, for example, [2]–[4] . The FMM is typically interpreted as a method for speeding up matrix-vector multiplication, and is therefore often coupled with Krylov subspace iterative solvers like GMRES. Consequently, when solving problems involving many right-hand sides, the required computational time, in the worst-case scenario, scales as a multiple of the number of right-hand sides. The inherent drawback of using iterative solvers with FMM can be effectively handled by employing fast direct solvers (FDS) [5] . This study focuses on the transverse electric (TE)-mode electromagnetic scattering in transmission problems. Because TE-mode scattering reduces to two-dimensional Helmholtz scattering, using solvers based on the hierarchically semiseparable (HSS) representation [6] is attractive. FDSs based on the HSS representation include the Martinsson–Rokhlin solver [7], HSS-ULV factorization [8], and recursive skeletonization [9] . These HSS solvers exhibit high parallel efficiency because all off-diagonal blocks are recursively approximated as low rank, although alternative representations, such as strongly Manuscript received December 1, 2012; revised August 26, 2015 . Corresponding author: Y. Matsumoto ([email: matsumoto@cii.isct.ac.jp](email: matsumoto@cii.isct.ac.jp)). Fig. 1. Local corrections on the HSS representation. The blue band corresponding to the corrected kernel spills over into adjacent off-diagonal blocks. This figure corresponds to a coefficient matrix stemmi

FAQs

Why is a boundary integral approach used for TE-mode scattering in unbounded domains?

Radiation conditions make finite-difference and finite-element methods difficult on unbounded domains. Boundary integral equations avoid this challenge and offer an alternative framework for the scattering problem.

What is the core idea behind the solver’s O(h3) convergence?

The method uses the weakly singular Burton–Miller boundary integral equation and applies a Nyström method with one-point corrected quadrature, with only diagonal terms locally corrected to achieve higher-order convergence.

What dominates runtime in HSS-type boundary integral solvers, and how is it addressed?

Matrix component calculation dominates the total computational time rather than matrix factorization. The work adds a load-balancing method to improve parallel efficiency during these matrix-entry computations.

Jul 16, 2026

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