本文研究如何求解具有完美電導體邊界條件的馬克斯威爾方程組之中的特徵值問題。
我們主要的工作是推導出離散旋度算子的奇異值分解。此外，我們使用此奇異值分解
將廣義特徵值問題簡化為具有更多計算優勢的零空間消去標準特徵值問題。然而，奇
異值分解的矩陣向量乘法是昂貴的。在這裡，我們建立了奇異值分解與快速傅立葉變
換之間的關係，快速傅立葉變換是用於計算離散傅立葉變換的O(n log n) 演算法。在
實作上，GPU 幫助我們在電腦上進行並行計算。我們選擇CUDA 作為程式語言，並
使用cuFFT, cuBLAS 和LAPACK 等軟件包來實現我們提出的方法。數值結果顯
示，GPU 加快了昂貴的矩陣向量乘法的速度，從而提高求解零空間消去標準特徵值問
題的效率。

This paper studies how to solve the eigenvalue problem in Maxwell’s
equations with the perfect electric conductor (PEC) boundary condition.
Our main work is to derive the singular value decomposition (SVD) of the
discrete curl operator. Furthermore, we use the SVD to reduce the generalized eigenvalue problem (GEP) to the null space free standard eigenvalue problem (NFSEP), which has more computational advantages. However, the matrix-vector multiplications of the SVD are expensive. Here, we establish the relation between the SVD and the fast Fourier transform (FFT), which is a O(n log n) algorithm for computing the discrete Fourier transform. In the implementation, GPU helps us perform the parallel computing on computer. We choose CUDA as our programming language and use some packages such as cuFFT, cuBLAS, and LAPACK to realize our proposed method. The numerical results show that GPU speedup the expensive matrix-vector multiplications, thereby improving the efficiency in solving the NFSEP.

摘要 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . ii
誌謝 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iii
1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.1 Maxwell’s Equations . . . . . . . . . . . . . . . . . . . . . . 2
1.2 Perfect Electric Conductor Boundary Condition . . . . . . . . . 2
1.3 Discretization for Maxwell’s Equations . . . . . . . . . . . . . 3
1.3.1 Yee’s Scheme . . . . . . . . . . . . . . . . . . . . . . . . . 3
1.3.2 Matrix Representation of Curl Operator . . . . . . . . . . . . 6
1.4 Generalized Eigenvalue Problem . . . . . . . . . . . . . . . . . 10
1.5 Null Space Free Method . . . . . . . . . . . . . . . . . . . . . 10
2 1D Discrete Partial Differential Operator . . . . . . . . . . . . 12
2.1 Singular Value Decomposition of 1D Discrete Partial Differential
Operator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
2.2 Discrete Cosine Transform and Discrete Sine Transform via Discrete
Fourier Transform . . . . . . . . . . . . . . . . . . . . . . . . . 13
3 Discrete Curl Operator . . . . . . . . . . . . . . . . . . . . . 16
3.1 Factorization of Discrete Curl Operator . . . . . . . . . . . . 17
3.2 Singular Value Decomposition of Discrete Curl Operator . . . . 18
4 Fast Matrix-Vector Multiplication for Qr . . . . . . . . . . . . 24
5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . 27
5.1 Matrix-Vector Multiplication of Qr . . . . . . . . . . . . . . . 27
5.2 Solving the Null Space Free Standard Eigenvalue Problem . . . . 29
6 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
References . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32

[1] C. Calderón-Ramón, J. Gómez-Aguilar, M. Rodríguez-Achach, L. Morales-
Mendoza, J. Laguna-Camacho, M. Benavides-Cruz, M. Cruz-Orduna,
M. González-Lee, H. Pérez-Meana, M. Enciso-Aguilar, et al. Use of the
perfect electric conductor boundary conditions to discretize a diffractor in
fdtd/pml environment. Revista mexicana de física, 61(5):344–350, 2015.
[2] R.-L. Chern, H.-E. Hsieh, T.-M. Huang, W.-W. Lin, and W. Wang. Singular
value decompositions for single-curl operators in three-dimensional maxwell’s
equations for complex media. SIAM Journal on Matrix Analysis and Applications,
36(1):203–224, 2015.
[3] V. Fuka. Poisfft–a free parallel fast poisson solver. Applied Mathematics and
Computation, 267:356–364, 2015.
[4] T.-M. Huang, H.-E. Hsieh, W.-W. Lin, and W. Wang. Eigendecomposition
of the discrete double-curl operator with application to fast eigensolver for
three-dimensional photonic crystals. SIAM Journal on Matrix Analysis and
Applications, 34(2):369–391, 2013.
[5] I. V. Lindell and A. Sihvola. Electromagnetic boundaries with pec/pmc equivalence.
arXiv preprint arXiv:1608.05519, 2016.
[6] K. T. McDonald. Electromagnetic fields inside a perfect conductor.
[7] W. Shin. 3D finite-difference frequency-domain method for plasmonics and
nanophotonics. PhD thesis, Citeseer, 2013.
[8] G. Strang. The discrete cosine transform. SIAM review, 41(1):135–147, 1999.
[9] I. A. Sukhoivanov and I. V. Guryev. Photonic crystals: physics and practical
modeling, volume 152. Springer, 2009.
[10] K. Yee. Numerical solution of initial boundary value problems involving
maxwell’s equations in isotropic media. IEEE Transactions on antennas and
propagation, 14(3):302–307, 1966.