Preconditioning bandgap eigenvalue problems in three-dimensional photonic crystals simulations

Huang, Tsung-Ming; Chang, Wei‐Jen; Huang, Yin-Liang; Lin, Wen‐Wei; Wang, Weicheng; Wang, Weichung

doi:10.1016/j.jcp.2010.08.003

Cited by 11 publications

(7 citation statements)

References 42 publications

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“…We summarize the JD and SIRA methods for solving the SEP (3.11) in Algorithm 4. Note that, in practice, a restart with search subspace contraction [16] is used in Algorithm 4. Compute W j = KV j and M j = V * j W j .…”

Section: Shift-invert Residual Arnoldi Methods For Thementioning

confidence: 99%

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A Newton-Type Method with Nonequivalence Deflation for Nonlinear Eigenvalue Problems Arising in Photonic Crystal Modeling

Huang¹,

Lin²,

Mehrmann

2016

SIAM J. Sci. Comput.

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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.

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Section: Shift-invert Residual Arnoldi Methods For Thementioning

confidence: 99%

“…The resulting matrix A arising from the discretized double-curl operator using the Yee scheme [51] on a primitive cell is then of the form [16,17,18]…”

Section: Introductionmentioning

confidence: 99%

A Newton-Type Method with Nonequivalence Deflation for Nonlinear Eigenvalue Problems Arising in Photonic Crystal Modeling

Huang¹,

Lin²,

Mehrmann

2016

SIAM J. Sci. Comput.

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“…In contrast, the GEVP contains n zero eigenvalues and 2n positive eigenvalues. This large null space usually causes numerical inefficiency [17].…”

Section: Inverse Projective Lanczos Methodmentioning

confidence: 99%

“…As we are interested in finding a few of the smallest positive eigenvalues, the large dimension of the null space leads to several numerical difficulties [8,16]. Second, the eigenvectors of A associated with the SC lattice are mutually independent of the 3D grid point indices i, j, and k. Consequently, the standard FFT can be applied to compute the associated photonic band gap in the SC lattice [11,17]. However, the FCC case has no such luxury.…”

Section: Introductionmentioning

confidence: 99%

Eigendecomposition of the Discrete Double-Curl Operator with Application to Fast Eigensolver for Three-Dimensional Photonic Crystals

Huang¹,

Hsieh²,

Lin³

et al. 2013

SIAM J. Matrix Anal. & Appl.

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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.

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“…Frequency domain solvers assume a time-harmonic variation of the fields and result in the need to solve a large generalised eigenvalue problem [1,2]. The application to complex three dimensional cavities can result in prohibitively expensive computations due to the large memory requirements or due to the lack of a preconditioners suitable for the large sparse linear systems that are encountered [3,4,5]. In addition, the performance of the solver is strongly problem dependent.…”

Section: Introductionmentioning

confidence: 99%

The application of a high-order discontinuous Galerkin time-domain method for the computation of electromagnetic resonant modes

Dawson

Sevilla

Morgan

2018

Applied Mathematical Modelling

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Preconditioning bandgap eigenvalue problems in three-dimensional photonic crystals simulations

Cited by 11 publications

References 42 publications

A Newton-Type Method with Nonequivalence Deflation for Nonlinear Eigenvalue Problems Arising in Photonic Crystal Modeling

A Newton-Type Method with Nonequivalence Deflation for Nonlinear Eigenvalue Problems Arising in Photonic Crystal Modeling

Eigendecomposition of the Discrete Double-Curl Operator with Application to Fast Eigensolver for Three-Dimensional Photonic Crystals

The application of a high-order discontinuous Galerkin time-domain method for the computation of electromagnetic resonant modes

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