Nested Reduction Algorithms for Generating a Rank-Minimized H2-Matrix From FMM for Electrically Large Analysis

In this work, we develop efficient algorithms to generate a rank-minimized H-2-matrix to represent electrically large surface integral operators for a prescribed accuracy. We first generate an H-2-matrix using the fast multipole method (FMM), and hence, the complexity for H-2-construction is as low as O( NlogN) for solving electrically large surface integral equations. We then develop fast algorithms to convert the FMM-based H-2-matrix whose rank is full asymptotically to a new H-2-representation, whose rank is minimized based on accuracy. The proposed algorithms cost O(k(3)) in time for each cluster in cluster basis generation and O(k(2)) in memory, where k is the minimal rank of the cluster basis required by accuracy. When the rank of the H-2-matrix is a constant, the complexity of the proposed algorithms is O( N) 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 H-2-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.