Fast Algorithms for generating Minimal Rank H2-matrix for Electrically Large Surface Integral Operators

Computational electromagnetics (CEM) plays an important role in many aspects of today's engineering world. Among existing CEM methods, Integral Equation (IE) based solvers are popular because of their versatility, efficiency, and reliability. IE-based methods in general result in a dense system matrix. To solve this system matrix efficiently, a prevailing solution is to use a Fast Multipole Method (FMM) with an iterative solver. Recently, fast H2-matrix based direct solvers have been developed to directly invert or factorize the dense system matrix. However, how to generate an H2-representation efficiently for electrically large analysis remains an unsolved problem. Existing methods for generating an H2-matrix of electrically large integral operators are all expensive, especially for surface IE (SIE) operators.

In this work, we proposed and developed a series of fast algorithms to generate a rank-minimized H2-matrix for electrically large SIE-based analysis, the best of which has complexity as low as O(NlogN) in time and memory. Using the H2-matrix generated in this work, we can make the H2-matrix-based direct solver achieve a total complexity of O(N^1.5) in time and O(NlogN) in memory for electrically large SIE analysis. In contrast, generating an H2-matrix or inverting a dense matrix in a brute-force way both will cost O(N³) in time and O(N²) in memory. In addition to accelerating direct solvers, we significantly reduce the CPU time of a matrix-vector multiplication as well as the memory consumption because of the rank-minimized H2-representation. In addition to electromagnetic analysis, the proposed algorithms are applicable to many other disciplines, where a compact representation of dense matrices is the key to the reduction of computational complexity.