A complexity-reduced H-matrix based direct integral equation solver with prescribed accuracy for large-scale electrodynamic analysis

Abstract

In this work, by further developing the ℌ-matrix based mathematical framework, we achieved an efficient LU-factorization based direct IE solver of k <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ave</sub> C <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sp</sub> O(N log <sup xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">2</sup> N) time complexity and k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ave</sub> C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sp</sub> O(N log N) memory complexity, with the two parameters k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ave</sub> and C <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">sp</sub> minimized, with the prescribed accuracy satisfied, to solve large-scale electrodynamic problems. The k <sub xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">ave</sub> is a weighted average rank we introduce to derive the complexity bounds for the ℌ-matrix-based computation of electrodynamic problems. It is introduced based on the fact that the rank required by an electrodynamic system for a given accuracy is a variable with respect to tree levels and admissible blocks, and hence existing constant-rank based complexity analysis does not apply.