One-Stage O(N \log N) Algorithm for Generating Nested Rank-Minimized Representation of Electrically Large Volume Integral Equations

• Published in: IEEE journal on multiscale and multiphysics computational techniques Vol. 10; pp. 169 - 178
• Main Authors: Wang, Yifan, Jiao, Dan
• Format: Journal Article
• Language: English
• Published: IEEE 2025
• Subjects: <named-content xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:ali="http://www.niso.org/schemas/ali/1.0/" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xsi="http://www.w3.org/2001/XMLSchema-instance" content-type="math" xlink:type="simple"> <inline-formula> <tex-math notation="LaTeX"> {\mathcal {H}}^{2}</tex-math> </inline-formula> </named-content> matrix; Accuracy; Approximation algorithms; Clustering algorithms; Complexity theory; Costs; electrically large analysis; fast solvers; Indexes; Integral equations; low-rank representation; matrix compression; nested representation; rank-minimized compression; Scattering; Three-dimensional displays; Training; Volume integral equations
• ISSN: 2379-8815; 2379-8815
• DOI: 10.1109/JMMCT.2025.3544143

In this paper, we develop a new one-stage <inline-formula><tex-math notation="LaTeX"> O(N \log N)</tex-math></inline-formula> algorithm to generate a rank-minimized <inline-formula><tex-math notation="LaTeX">\mathcal {H}^{2}</tex-math></inline-formula>-representation of electrically large volume integral equations (VIEs), which significantly reduces the CPU run time of state-of-the-art algorithms for completing the same task. Unlike existing two-stage algorithms, this new algorithm requires only one stage to build nested cluster bases. The cluster basis is obtained directly from the interaction between a cluster and its admissible clusters composed of real or auxiliary ones that cover all interaction directions. Furthermore, the row and column pivots of the resultant low-rank representation are chosen from the source and observer points in an analytical way without the need for numerically finding them. This further speeds up the computation. Numerical experiments on a suite of electrically large 3D scattering problems have demonstrated the efficiency and accuracy of the proposed new algorithm.