以投影Lanczos法解三維光子晶體的特徵值問題

Abstract

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。

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.