Nested Reduction Algorithms for Generating a Rank-Minimized ℋ 2 -Matrix From FMM for Electrically Large Analysis

• Published in: IEEE transactions on antennas and propagation Vol. 69; no. 7; pp. 3945 - 3956
• Main Authors: Yang, Chang, Jiao, Dan
• Format: Journal Article
• Language: English
• Published: New York The Institute of Electrical and Electronics Engineers, Inc. (IEEE) 01.07.2021
• Subjects: Accuracy; Algorithms; Clusters; Complexity; Formulas (mathematics); Integral equations; Iterative methods; Multipoles; Operators (mathematics)
• ISSN: 0018-926X; 1558-2221
• DOI: 10.1109/TAP.2020.3044584

In this work, we develop efficient algorithms to generate a rank-minimized [Formula Omitted]-matrix to represent electrically large surface integral operators for a prescribed accuracy. We first generate an [Formula Omitted]-matrix using the fast multipole method (FMM), and hence, the complexity for [Formula Omitted]-construction is as low as [Formula Omitted] for solving electrically large surface integral equations. We then develop fast algorithms to convert the FMM-based [Formula Omitted]-matrix whose rank is full asymptotically to a new [Formula Omitted]-representation, whose rank is minimized based on accuracy. The proposed algorithms cost [Formula Omitted] in time for each cluster in cluster basis generation and [Formula Omitted] in memory, where [Formula Omitted] is the minimal rank of the cluster basis required by accuracy. When the rank of the [Formula Omitted]-matrix is a constant, the complexity of the proposed algorithms is [Formula Omitted] in both time and memory consumption. When the rank is a variable dependent on electrical size, the total complexity can be evaluated based on the rank’s behavior. The resultant rank-minimized [Formula Omitted]-matrix has been employed to accelerate both iterative and direct solutions. Numerical experiments on large-scale surface integral equation-based scattering analysis have demonstrated its accuracy and efficiency.