An H-matrix-based method for reducing the complexity of integral-equation-based solutions of electromagnetic problems

Abstract

Integral-equation-based methods generally lead to dense systems of linear equations. The resulting matrices, although dense, can be thought of as “data-sparse”, i.e., they can be specified by few parameters. This can be accomplished by remodeling the problem subject to underlying hierarchical dependencies such that all interactions can be constructed from a reduced set of parameters. There exists a general mathematical framework called the “Hierarchical (H) Matrix” framework [1–2], which enables a highly compact representation and efficient numerical computation of the dense matrices resulted from integral-equation-based methods. Storage requirements and matrix-vector multiplications using H-matrices have been shown to be of complexity O(NlogN). Authors in [3] later introduced H <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> -matrices, which are a specialized subclass of hierarchical matrices. It was shown that the storage requirements and matrix-vector products are of complexity O(N). H-matrix-based techniques have been applied to solve electrostatic and magneto-static problems [4–6]. Questions that remain open are: Can recent advances in hierarchical matrix framework reduce the computation of electrodynamic problems? Can the matrices encountered in full-wave electromagnetics-based analysis be approximated as hierarchical matrices? In this work, we demonstrate the feasibility of H-matrix-based techniques in integral-equation-based solutions of electrodynamic problems.