Spectral Filtering of 3D Integral Operators Using Modified Green's Functions

2026-06-11

AI Summary

This work addresses the computational inefficiency associated with solving the dense matrix resulting from the discretization of the three-dimensional electric field integral equation (EFIE). To this end, the authors propose a spectral truncation filtering method based on the spherical Hankel transform. By constructing a spectral representation of the Green’s function, they apply an analytical spectral filter to the integral operator, marking the first application of this technique to operator compression and regularization for the three-dimensional EFIE. The proposed approach significantly improves the spectral distribution of both continuous and discrete operators in static and dynamic regimes, thereby substantially enhancing the convergence rate and computational efficiency of both iterative and direct solvers.

Abstract

Several recent contributions have analyzed and illustrated the effectiveness of operator filtering, both in terms of regularization and compression, when handling dense matrices arising from the discretization of integral operators, e.g. the single-layer operator. Previous works have introduced different filtering strategies, ranging from Laplacian-based filters to analytically derived ones, with the goal of improving the computational efficiency of iterative and direct solvers for integral equations in the two-dimensional space, like the 2D Electric Field Integral Equation (EFIE). In this work, we propose a filtering strategy based on the spectral truncation of the kernels of integral operators associated with the 3D EFIE. The approach relies on an appropriate spectral representation of the Green's function obtained via the spherical Hankel transform, which provides an analytical foundation for the proposed approach. Finally, we provide semi-analytical and numerical evidence of the impact of this filtering technique on the spectral properties of continuous integral operators and of their discretization through boundary elements, both for the static and dynamic cases.

Problem

• spectral filtering
• 3D integral operators
• Green's functions
• EFIE
• boundary element discretization

Innovation

• spectral filtering
• 3D integral operators
• modified Green's functions
• spherical Hankel transform
• boundary element method