## Abstract

In this paper, we develop an H-matrix-based fast direct integral equation solver that has a significantly reduced computational complexity, with prescribed accuracy satisfied, to solve large-scale electrodynamic problems. In light of the fact that the cost of an H-matrix-based computation of high-frequency problems is not only determined by the block rank that increases with electric size, but also determined by the H-matrix partition, we propose a new parameter, average partition rank kave, to derive the storage units and operation counts of the H-matrix based computation of electrodynamic problems. Different from block rank, the partition rank kave contains the information of the H-matrix partition. We show that the computational cost of an H-matrix-based computation of electrodynamic problems can be significantly reduced without sacrificing accuracy, by minimizing the rank of each admissible block based on accuracy requirements; and by optimizing the H-partition to reduce the number of admissible blocks at each tree level for a prescribed accuracy and for each frequency point. To minimize the rank for a given accuracy, we develop an efficient matrix algebra based method to determine the minimal rank for each admissible block. The algebraic method has a linear complexity, and hence the computational overhead is negligible. To minimize the number of admissible blocks at each tree level for a given accuracy, we develop a new H-partition method that is frequency dependent, and also controlled by accuracy requirements. With the proposed cost reduction methods, we develop a fast LU factorization for directly solving the dense system matrix resulting from an IE-based analysis of large-scale electrodynamic problems. The proposed solver successfully factorizes dense matrices that involve more than 1 million unknowns associated with electrodynamic problems of 96 wavelengths in fast CPU time, modest memory consumption, and with the prescribed accuracy satisfied. As an algebraic method, the underlying fast technique is kernel independent.

## Keywords

Terms—Integral-equation-based methods, electromagnetic analysis, direct solution, H matrix, large-scale analysis

## Date of this Version

2-15-2011