以投影Lanczos法解三維光子晶體的特徵值問題

解決大型特徵值問題往往所需大量的時間,尤其在迭代的過程中,矩陣乘向量更是佔大多數的時間。在本篇論文中,我們的問題來自於光子晶體的馬克斯威爾方程之鑽石結構。在以往研究中,皆是利用許多不同的方法來加速收斂。這次發展一個新的方法-投影Lanczos法來解決我們的特徵值問題,其目的是將原本的問題投影至非零的不變子空間,最後得到新的特徵值問題,而在矩陣乘向量中只剩傅立葉矩陣與對角矩陣的運算,在此配合傅立葉矩陣乘向量演算法與多線程來平行剩下的運算將可以快速地收斂到我們要的特徵值。

關鍵字

投影Lanczos法；馬克斯威爾方程；三維光子晶體；鑽石結構；特徵值問題

並列摘要

To solve large-scale eigenvalue problems often require a lot of time, especially in the iterative process, the matrix-vector multiplication accounted for most of the time.In this thesis, our problem is from the three dimension photonic crystals of Maxwell’s equa- tion which is diamond structure.From previous studies in the question, we know that the dimension of null space, and then we will develop a new method for this problem. The purpose is that the original problem is projected to the non-zero invariant space, and us- ing FFT to quickly converge to the non-zero eigenvalues which we want. Finally,we will show the numerical result based on C program.

並列關鍵字

Projective Lanczos ； Maxwell’s equations ； Eigenvalue Problem

參考文獻

[1] C. Kittel. “Introduction to solid state physics.” New York, 2005, Wiley.

[2] A. Mekis, J. C. Chen, I. Kurland, S. Fan, P. R. Villeneuve, and J. D. Joannopoulos. “ High transmission through sharp bends in photonic crystal waveguides.” Phys. Rev. Lett., 77:3787-3790, 1996.

[3] K. Yee. “ Numerical solution of initial boundary value problems involving Maxwell’s equations in isotropic media.” IEEE Trans. Antennas Propag., 14:302–307, 1966.

[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., 229:8684-8793, 2010.

[5] P. Arbenz and R. Geus. “ A comparison of solvers for large eigenvalue prob- lems occuring in the design of resonant cavities.” Numer. Linear Algebr. Appl., 6:3–16, 1999.

國際替代計量

以投影Lanczos法解三維光子晶體的特徵值問題

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