國立臺灣師範大學博碩士論文全文系統

簡易檢索 / 詳目顯示

回結果列表

研究生：	蔡欣翰 Tsai, Hsin-Han
論文名稱：	A Scalable and Ultrafast Eigensolver for Three Dimensional Photonic Crystals on GPU A Scalable and Ultrafast Eigensolver for Three Dimensional Photonic Crystals on GPU
指導教授：	黃聰明 Huang, Tsung-Ming
學位類別：	碩士 Master
系所名稱：	數學系 Department of Mathematics
論文出版年：	2017
畢業學年度：	105
語文別：	英文
論文頁數：	39
中文關鍵詞：	Maxwell equation 、band structure 、face-centered cubic lattice 、GPU 、CUDA 、MPI 、cuBLAS 、cuFFT
英文關鍵詞：	Maxwell equation, band structure, face-centered cubic lattice, GPU, CUDA, MPI, cuBLAS, cuFFT
DOI URL：	https://doi.org/10.6345/NTNU202203517
論文種類：	學術論文
相關次數：	點閱：114 下載：25
分享至:	分享至facebook 分享至twitter

查詢本校圖書館目錄查詢臺灣博碩士論文知識加值系統勘誤回報

無中文摘要

This research applies parallel computations on a GPU by CUDA for solving three dimensional Maxwell's equation with face-centered cubic (FCC) lattice. We focus on how to solve an Eigenvalue Problem more efficiently. Because of the problem we solved is Hermitian and positive definite. The algorithm of the solver is based on inverse Lanczos method for eigenvalue problems and associated conjugate gradient method for linear systems. By using cuBLAS, cuFFT, combining kernels, transpose multiple matrices simultaneously, and some skills, we can save time from computations or accessing memory. Integrating all techniques, we can solve each of a set of 5.184 million dimension eigenvalue problem for 10 smallest positive eigenvalues within 44 to 63 seconds. And we have a great scability on multiple GPU cards by MPI. All results are computed on two clusters. One is equipped two GPU cards called NVIDIA Tesla K40c, most of works are computed here. And the other is equipped a lot of GPU cards called M2070, which are used for MPI.

Contents
Introduction  3
Background  5
1 Discretize the double curl operator  5
2 Eigendecomposition of C1, C2, and C3  6
3 Eigendecomposition of doble curl  8
4 Null space Free Standard Eigenvalue Problem  9
5 Eigensolver for NFSEP  10
6 Matrix-vector multiplication for Tq and T∗p  11
Accelerate the computations of the NFSEP on GPU  14
1 Algorithms of Tq and T∗p for CUDA  14
2 Hardware implementation  17
3 First optimizing method:Access global memory coalesced  19
4 An idea of reducing memory access  21
5 Optimized Transpose  22
Numerical results  27
1 Combining kernels and not combining kernels  27
2 Three kinds of 2D transpose and three kinds of multiple 2D transposes  28
3 The better way to do multiplication and transpose  29
4 Test for cuFFT  30
5 Time spent versus different dimensions  32
6 Band structure  33
7 Time consuming versus wave vectors  34
8 breakdown analysis  35
9 Scability  36
Conclusion  37
Reference  38
                                

[1] R.-L. Chern, C. Chung Chang, C.-C. Chang, and R. Hwang, Numerical study of three dimensional photonic crystals with large band gaps, J. Phys. Soc. Japan, 73 (2004), pp. 727–737.
[2] L.-S. Chien, Matrix transpose, 2011. http://oz.nthu.edu.tw/~d947207/NVIDIA/copy3D/Matrix_transpose_post.pdf.
[3] G. H. Golub and C. F. Van Loan, Matrix Computations, Johns Hopkins Univ. Pr., 3rd ed., 1996.
[4] T.-M. Huang, W.-J. Chang, Y.-L. Huang, W.-W. Lin, W. C. Wang, and W. Wang, Preconditioning bandgap eigenvalue problems in three dimensional photonic crystals simulations, J. Comput. Phys., 229 (2010), p. 8684–8703.
[5] 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 J. Matrix
Anal. Appl., 34(2) (2013), pp. 369–391.
[6] T.-M. Huang, H.-E. Hsieh, W.-W. Lin, and W. Wang, Matrix representation of the double-curl operator for simulating three dimensional photonic crystals., Math. Comput. Model., 58(1-2) (2013), pp. 379–392.
[7] T.-M. Huang, H.-E. Hsieh, W.-W. Lin, and W. Wang, Eigenvalue solvers for three dimensional photonic crystals with face-centered cubic lattice, Journal of Computational and Applied Mathematics., 272 (2014), pp. 350–361.
[8] K. Inoue and K. Ohtaka, Photonic crystals: physics, fabrication and applications, vol. 94, Springer, 2004.
[9] J. D. Joannopoulos, S. G. Johnson, J. N. Winn, and R. D. Meade, Photonic Crystals: Modeling the Flow of Light., Princeton University Press, 2008.
[10] C. Kittel, Introduction to solid state physics., Wiley, New York, 2005.
[11] C. NVIDIA, Next generation CUDA Compute Architecture: Kepler GK110. White paper.
[12] , The NVIDIA CUDA Basic Linear Algebre Subroutines (cuBLAS) library.
[13] , The NVIDIA CUDA Fast Fourier Transform library (cuFFT).
[14] , The NVIDIA CUDA Sparse library (cuSPARSE).
[15] , Cuda c programming guide, 2016. http://www.jstor.org/stable/853365.
[16] M. Reed and B. Simon, Methods of modern mathematical physics, in Analysis of Operators IV, Academic Press, San Diego, CA, 1978.
[17] C. M. Soukoulis, Photonic crystals and light localization in the 21st century., vol. 563, Springer, 2001.
[18] K. Yee, Numerical solution of initial boundary value problems involving maxwell’s equations in isotropic media, IEEE Trans. Antennas Propag., 14 (1996), pp. 302–307.

簡易檢索 / 詳目顯示

相關論文