Newton-type method with nonequivalence deflation for nonlinear eigenvalue problems arising in photonic crystal modeling

Tsung Ming Huang, Wen Wei Lin, Volker Mehrmann

Research output: Contribution to journalArticle

3 Citations (Scopus)

Abstract

The numerical simulation of the band structure of three-dimensional dispersive metallic photonic crystals with face-centered cubic lattices leads to large-scale nonlinear eigenvalue problems, which are very challenging due to a high-dimensional subspace associated with the eigenvalue zero and the fact that the desired eigenvalues (with smallest real part) cluster and are close to the zero eigenvalues. For the solution of the nonlinear eigenvalue problem, a Newton-type iterative method is proposed and the nullspace-free method is applied to exclude the zero eigenvalues from the associated generalized eigenvalue problem. To find the successive eigenvalue/eigenvector pairs, we propose a new nonequivalence deflation method to transform converged eigenvalues to infinity, while all other eigenvalues remain unchanged. The deflated problem is then solved by the same Newtontype method, which is used as a hybrid method that combines with the Jacobi-Davidson and the nonlinear Arnoldi methods to compute the clustering eigenvalues. Numerical results illustrate that the proposed method is robust even for the case of computing many clustering eigenvalues in very large problems.

Original languageEnglish
Pages (from-to)B191-B218
JournalSIAM Journal on Scientific Computing
Volume38
Issue number2
DOIs
Publication statusPublished - 2016 Jan 1

Fingerprint

Deflation
Newton-type Methods
Nonlinear Eigenvalue Problem
Photonic crystals
Iterative methods
Photonic Crystal
Eigenvalues and eigenfunctions
Crystal lattices
Band structure
Eigenvalue
Computer simulation
Modeling
Zero
Jacobi-Davidson
Clustering
Arnoldi Method
Generalized Eigenvalue Problem
Band Structure
Hybrid Method
Eigenvector

Keywords

  • Dispersive metallic photonic crystals
  • Jacobi-Davidson method
  • Maxwell equation
  • Newton-type method
  • Nonequivalence deflation
  • Nonlinear Arnoldi method
  • Nonlinear eigenvalue problem
  • Shift-invert residual Arnoldi method

ASJC Scopus subject areas

  • Computational Mathematics
  • Applied Mathematics

Cite this

Newton-type method with nonequivalence deflation for nonlinear eigenvalue problems arising in photonic crystal modeling. / Huang, Tsung Ming; Lin, Wen Wei; Mehrmann, Volker.

In: SIAM Journal on Scientific Computing, Vol. 38, No. 2, 01.01.2016, p. B191-B218.

Research output: Contribution to journalArticle

@article{1ddbde71e3bf457dafcb1220789c0492,
title = "Newton-type method with nonequivalence deflation for nonlinear eigenvalue problems arising in photonic crystal modeling",
abstract = "The numerical simulation of the band structure of three-dimensional dispersive metallic photonic crystals with face-centered cubic lattices leads to large-scale nonlinear eigenvalue problems, which are very challenging due to a high-dimensional subspace associated with the eigenvalue zero and the fact that the desired eigenvalues (with smallest real part) cluster and are close to the zero eigenvalues. For the solution of the nonlinear eigenvalue problem, a Newton-type iterative method is proposed and the nullspace-free method is applied to exclude the zero eigenvalues from the associated generalized eigenvalue problem. To find the successive eigenvalue/eigenvector pairs, we propose a new nonequivalence deflation method to transform converged eigenvalues to infinity, while all other eigenvalues remain unchanged. The deflated problem is then solved by the same Newtontype method, which is used as a hybrid method that combines with the Jacobi-Davidson and the nonlinear Arnoldi methods to compute the clustering eigenvalues. Numerical results illustrate that the proposed method is robust even for the case of computing many clustering eigenvalues in very large problems.",
keywords = "Dispersive metallic photonic crystals, Jacobi-Davidson method, Maxwell equation, Newton-type method, Nonequivalence deflation, Nonlinear Arnoldi method, Nonlinear eigenvalue problem, Shift-invert residual Arnoldi method",
author = "Huang, {Tsung Ming} and Lin, {Wen Wei} and Volker Mehrmann",
year = "2016",
month = "1",
day = "1",
doi = "10.1137/151004823",
language = "English",
volume = "38",
pages = "B191--B218",
journal = "SIAM Journal of Scientific Computing",
issn = "0036-1445",
publisher = "Society for Industrial and Applied Mathematics Publications",
number = "2",

}

TY - JOUR

T1 - Newton-type method with nonequivalence deflation for nonlinear eigenvalue problems arising in photonic crystal modeling

AU - Huang, Tsung Ming

AU - Lin, Wen Wei

AU - Mehrmann, Volker

PY - 2016/1/1

Y1 - 2016/1/1

N2 - The numerical simulation of the band structure of three-dimensional dispersive metallic photonic crystals with face-centered cubic lattices leads to large-scale nonlinear eigenvalue problems, which are very challenging due to a high-dimensional subspace associated with the eigenvalue zero and the fact that the desired eigenvalues (with smallest real part) cluster and are close to the zero eigenvalues. For the solution of the nonlinear eigenvalue problem, a Newton-type iterative method is proposed and the nullspace-free method is applied to exclude the zero eigenvalues from the associated generalized eigenvalue problem. To find the successive eigenvalue/eigenvector pairs, we propose a new nonequivalence deflation method to transform converged eigenvalues to infinity, while all other eigenvalues remain unchanged. The deflated problem is then solved by the same Newtontype method, which is used as a hybrid method that combines with the Jacobi-Davidson and the nonlinear Arnoldi methods to compute the clustering eigenvalues. Numerical results illustrate that the proposed method is robust even for the case of computing many clustering eigenvalues in very large problems.

AB - The numerical simulation of the band structure of three-dimensional dispersive metallic photonic crystals with face-centered cubic lattices leads to large-scale nonlinear eigenvalue problems, which are very challenging due to a high-dimensional subspace associated with the eigenvalue zero and the fact that the desired eigenvalues (with smallest real part) cluster and are close to the zero eigenvalues. For the solution of the nonlinear eigenvalue problem, a Newton-type iterative method is proposed and the nullspace-free method is applied to exclude the zero eigenvalues from the associated generalized eigenvalue problem. To find the successive eigenvalue/eigenvector pairs, we propose a new nonequivalence deflation method to transform converged eigenvalues to infinity, while all other eigenvalues remain unchanged. The deflated problem is then solved by the same Newtontype method, which is used as a hybrid method that combines with the Jacobi-Davidson and the nonlinear Arnoldi methods to compute the clustering eigenvalues. Numerical results illustrate that the proposed method is robust even for the case of computing many clustering eigenvalues in very large problems.

KW - Dispersive metallic photonic crystals

KW - Jacobi-Davidson method

KW - Maxwell equation

KW - Newton-type method

KW - Nonequivalence deflation

KW - Nonlinear Arnoldi method

KW - Nonlinear eigenvalue problem

KW - Shift-invert residual Arnoldi method

UR - http://www.scopus.com/inward/record.url?scp=84964858147&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=84964858147&partnerID=8YFLogxK

U2 - 10.1137/151004823

DO - 10.1137/151004823

M3 - Article

AN - SCOPUS:84964858147

VL - 38

SP - B191-B218

JO - SIAM Journal of Scientific Computing

JF - SIAM Journal of Scientific Computing

SN - 0036-1445

IS - 2

ER -