Linear complexity integral-equation based methods for large-scale electromagnetic analysis

Wenwen Chai, Purdue University

Abstract

In general, to solve problems with N parameters, the optimal computational complexity is linear complexity O( N). However, for most computational electromagnetic methods, the complexity is higher than O(N). In this work, we introduced and further developed the [special characters omitted]- and [special characters omitted]-matrix based mathematical framework to break the computational barrier of existing integral-equation (IE)-based methods for large-scale electromagnetic analysis. Our significant contributions include the first-time dense matrix inversion and LU factorization of O(N) complexity for large-scale 3-D circuit extraction and a fast direct integral equation solver that outperforms existing direct solvers for large-scale electrodynamic analysis having millions of unknowns and ∼100 wavelengths. The major contributions of this work are: (1) Direct Matrix Solution of Linear Complexity for 3-D Integrated Circuit (IC) and Package Extraction • O(N) complexity dense matrix inversion and LU factorization algorithms and their applications to capacitance extraction and impedance extraction of large-scale 3-D circuits • O(N) direct matrix solution of highly irregular matrices consisting of both dense and sparse matrix blocks arising from full-wave analysis of general 3-D circuits with lossy conductors in multiple dielectrics. (2) Fast [special characters omitted]- and [special characters omitted]-Based IE Solvers for Large-Scale Electrodynamic Analysis • theoretical proof on the error bounded low-rank representation of electrodynamic integral operators • fast [special characters omitted]-based iterative solver with O(N) computational cost and controlled accuracy from small to tens of wavelengths • fast [special characters omitted]-based direct solver with computational cost minimized based on accuracy • Findings on how to reduce the complexity of [special characters omitted]- and [special characters omitted]-based methods for electrodynamic analysis, which are also applicable to many other fast IE solvers. (3) Fast Algorithms for Accelerating [special characters omitted]- and [special characters omitted]-Based Iterative and Direct Solvers • Optimal [special characters omitted]-based representation and its applications from circuits to electrically large problems • Optimal [special characters omitted]-based representation for dense matrices arising from IE-based analysis • Iterative as well as direct solvers significantly accelerated by optimal [special characters omitted]- and [special characters omitted]-based representations. (4) Advanced Mathematical Computing • The construction of a simple [special characters omitted]-representation with Csp = 1 • Linear-time matrix-matrix multiplication with controlled accuracy. The proposed methods have successfully solved large-scale electromagnetic scattering problems having 100 wavelengths and integrated circuit problems involving millions of unknowns in fast CPU time, modest memory consumption, and without sacrificing accuracy. Comparisons with state-of-the-art solvers have demonstrated the clear advantages of the proposed methods. The proposed methods have important applications in a wide range of areas such as electromagnetics, optics, acoustics, plasmonics, etc.

Degree

Ph.D.

Advisors

Jiao, Purdue University.

Subject Area

Electromagnetics

COinS