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 -