Eigendecomposition of the discrete double-curl operator with application to fast eigensolver for three-dimensional photonic crystals

Tsung Ming Huang, Han En Hsieh, Wen Wei Lin, Weichung Wang

Research output: Contribution to journalArticle

10 Citations (Scopus)

Abstract

This article focuses on the discrete double-curl operator arising in the Maxwell equation that models three-dimensional photonic crystals with face-centered cubic lattice. The discrete double-curl operator is the degenerate coefficient matrix of the generalized eigenvalue problems (GEVP) due to the Maxwell equation. We derive an eigendecomposition of the degenerate coefficient matrix and explore an explicit form of orthogonal basis for the range and null spaces of this matrix. To solve the GEVP, we apply these theoretical results to project the GEVP to a standard eigenvalue problem (SEVP), which involves only the eigenspace associated with the nonzero eigenvalues of the GEVP, and therefore the zero eigenvalues are excluded and will not degrade the computational efficiency. This projected SEVP can be solved efficiently by the inverse Lanczos method. The linear systems within the inverse Lanczos method are well-conditioned and can be solved efficiently by the conjugate gradient method without using a preconditioner. We also demonstrate how two forms of matrix-vector multiplications, which are the most costly part of the inverse Lanczos method, can be computed by fast Fourier transformation due to the eigendecomposition to significantly reduce the computation cost. Integrating all of these findings and techniques, we obtain a fast eigenvalue solver. The solver has been implemented by MATLAB and successfully solves each of a set of 5.184 million dimension eigenvalue problems within 50 to 104 minutes on a workstation with two Intel Quad-Core Xeon X5687 3.6 GHz CPUs.

Original languageEnglish
Pages (from-to)369-391
Number of pages23
JournalSIAM Journal on Matrix Analysis and Applications
Volume34
Issue number2
DOIs
Publication statusPublished - 2013 Jul 29

Fingerprint

Generalized Eigenvalue Problem
Curl
Photonic Crystal
Lanczos Method
Inverse Method
Eigenvalue Problem
Three-dimensional
Operator
Eigenvalue
Maxwell's equations
Matrix-vector multiplication
Orthogonal Basis
Fourier Transformation
Null Space
Eigenspace
Conjugate Gradient Method
Coefficient
Preconditioner
Computational Efficiency
MATLAB

Keywords

  • Discrete double-curl operator
  • Eigendecomposition
  • Face centered cubic lattice
  • Fast Fourier transform
  • Photonic crystals
  • The Maxwell equation

ASJC Scopus subject areas

  • Analysis

Cite this

Eigendecomposition of the discrete double-curl operator with application to fast eigensolver for three-dimensional photonic crystals. / Huang, Tsung Ming; Hsieh, Han En; Lin, Wen Wei; Wang, Weichung.

In: SIAM Journal on Matrix Analysis and Applications, Vol. 34, No. 2, 29.07.2013, p. 369-391.

Research output: Contribution to journalArticle

@article{e401ed7fe6b245478fd529a88c976a59,
title = "Eigendecomposition of the discrete double-curl operator with application to fast eigensolver for three-dimensional photonic crystals",
abstract = "This article focuses on the discrete double-curl operator arising in the Maxwell equation that models three-dimensional photonic crystals with face-centered cubic lattice. The discrete double-curl operator is the degenerate coefficient matrix of the generalized eigenvalue problems (GEVP) due to the Maxwell equation. We derive an eigendecomposition of the degenerate coefficient matrix and explore an explicit form of orthogonal basis for the range and null spaces of this matrix. To solve the GEVP, we apply these theoretical results to project the GEVP to a standard eigenvalue problem (SEVP), which involves only the eigenspace associated with the nonzero eigenvalues of the GEVP, and therefore the zero eigenvalues are excluded and will not degrade the computational efficiency. This projected SEVP can be solved efficiently by the inverse Lanczos method. The linear systems within the inverse Lanczos method are well-conditioned and can be solved efficiently by the conjugate gradient method without using a preconditioner. We also demonstrate how two forms of matrix-vector multiplications, which are the most costly part of the inverse Lanczos method, can be computed by fast Fourier transformation due to the eigendecomposition to significantly reduce the computation cost. Integrating all of these findings and techniques, we obtain a fast eigenvalue solver. The solver has been implemented by MATLAB and successfully solves each of a set of 5.184 million dimension eigenvalue problems within 50 to 104 minutes on a workstation with two Intel Quad-Core Xeon X5687 3.6 GHz CPUs.",
keywords = "Discrete double-curl operator, Eigendecomposition, Face centered cubic lattice, Fast Fourier transform, Photonic crystals, The Maxwell equation",
author = "Huang, {Tsung Ming} and Hsieh, {Han En} and Lin, {Wen Wei} and Weichung Wang",
year = "2013",
month = "7",
day = "29",
doi = "10.1137/120872486",
language = "English",
volume = "34",
pages = "369--391",
journal = "SIAM Journal on Matrix Analysis and Applications",
issn = "0895-4798",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "2",

}

TY - JOUR

T1 - Eigendecomposition of the discrete double-curl operator with application to fast eigensolver for three-dimensional photonic crystals

AU - Huang, Tsung Ming

AU - Hsieh, Han En

AU - Lin, Wen Wei

AU - Wang, Weichung

PY - 2013/7/29

Y1 - 2013/7/29

N2 - This article focuses on the discrete double-curl operator arising in the Maxwell equation that models three-dimensional photonic crystals with face-centered cubic lattice. The discrete double-curl operator is the degenerate coefficient matrix of the generalized eigenvalue problems (GEVP) due to the Maxwell equation. We derive an eigendecomposition of the degenerate coefficient matrix and explore an explicit form of orthogonal basis for the range and null spaces of this matrix. To solve the GEVP, we apply these theoretical results to project the GEVP to a standard eigenvalue problem (SEVP), which involves only the eigenspace associated with the nonzero eigenvalues of the GEVP, and therefore the zero eigenvalues are excluded and will not degrade the computational efficiency. This projected SEVP can be solved efficiently by the inverse Lanczos method. The linear systems within the inverse Lanczos method are well-conditioned and can be solved efficiently by the conjugate gradient method without using a preconditioner. We also demonstrate how two forms of matrix-vector multiplications, which are the most costly part of the inverse Lanczos method, can be computed by fast Fourier transformation due to the eigendecomposition to significantly reduce the computation cost. Integrating all of these findings and techniques, we obtain a fast eigenvalue solver. The solver has been implemented by MATLAB and successfully solves each of a set of 5.184 million dimension eigenvalue problems within 50 to 104 minutes on a workstation with two Intel Quad-Core Xeon X5687 3.6 GHz CPUs.

AB - This article focuses on the discrete double-curl operator arising in the Maxwell equation that models three-dimensional photonic crystals with face-centered cubic lattice. The discrete double-curl operator is the degenerate coefficient matrix of the generalized eigenvalue problems (GEVP) due to the Maxwell equation. We derive an eigendecomposition of the degenerate coefficient matrix and explore an explicit form of orthogonal basis for the range and null spaces of this matrix. To solve the GEVP, we apply these theoretical results to project the GEVP to a standard eigenvalue problem (SEVP), which involves only the eigenspace associated with the nonzero eigenvalues of the GEVP, and therefore the zero eigenvalues are excluded and will not degrade the computational efficiency. This projected SEVP can be solved efficiently by the inverse Lanczos method. The linear systems within the inverse Lanczos method are well-conditioned and can be solved efficiently by the conjugate gradient method without using a preconditioner. We also demonstrate how two forms of matrix-vector multiplications, which are the most costly part of the inverse Lanczos method, can be computed by fast Fourier transformation due to the eigendecomposition to significantly reduce the computation cost. Integrating all of these findings and techniques, we obtain a fast eigenvalue solver. The solver has been implemented by MATLAB and successfully solves each of a set of 5.184 million dimension eigenvalue problems within 50 to 104 minutes on a workstation with two Intel Quad-Core Xeon X5687 3.6 GHz CPUs.

KW - Discrete double-curl operator

KW - Eigendecomposition

KW - Face centered cubic lattice

KW - Fast Fourier transform

KW - Photonic crystals

KW - The Maxwell equation

UR - http://www.scopus.com/inward/record.url?scp=84880556548&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84880556548&partnerID=8YFLogxK

U2 - 10.1137/120872486

DO - 10.1137/120872486

M3 - Article

AN - SCOPUS:84880556548

VL - 34

SP - 369

EP - 391

JO - SIAM Journal on Matrix Analysis and Applications

JF - SIAM Journal on Matrix Analysis and Applications

SN - 0895-4798

IS - 2

ER -