Asymptotic Spectral Insights Behind Fast Direct Solvers for High-Frequency Electromagnetic Integral Equations on Non-Canonical Geometries

2026-03-04

AI Summary

This work addresses the lack of efficient and theoretically well-founded direct solvers for high-frequency electromagnetic scattering problems on irregular geometries. The authors propose a fast direct solver that integrates preconditioning and acceleration techniques, and—building on semiclassical microlocal analysis—rigorously establish its applicability to irregular geometries in the high-frequency regime. This study not only develops an asymptotic spectral theory underpinning the solver’s performance for high frequencies and complex geometries but also demonstrates its computational efficacy. The resulting framework offers a new paradigm for solving high-frequency integral equations, combining algorithmic efficiency with strong theoretical guarantees.

Technology Category

• Machine Learning: Kernel Methods
• Search and Optimization: Mixed Discrete/Continuous Search
• Constraint Satisfaction and Optimization: Solvers and Tools

Application Category

• Graph Algorithms and Modeling for the Web: Algorithms and analysis for heterogeneous, signed, attributed, multi-relational, temporal, higher-order, and annotated Web-related graphs
• Security and Privacy: Large-scale security measurements
• Search and Retrieval-Augmented AI: Efficiency and scalability of Web search engines

Abstract

Integral-equation-based fast direct solvers for electromagnetic scattering can substantially reduce computational costs, especially in the presence of multiple excitations. We recently proposed a new high-frequency fast direct solver strategy that combines preconditioning techniques with acceleration algorithms. However, the validity of this approach applied to non-canonical geometries requires further justification. In this contribution, we collect relevant semiclassical microlocal results and use them to assess the legitimacy and effectiveness of the proposed fast direct solver in the high-frequency regime.

Problem

• high-frequency
• electromagnetic integral equations
• non-canonical geometries
• fast direct solvers
• asymptotic spectral analysis

Innovation

• fast direct solver
• high-frequency electromagnetics
• non-canonical geometries
• semiclassical microlocal analysis
• integral equations