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

11 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

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Keywords

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

ASJC Scopus subject areas

  • Analysis

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