三維光子晶體數值模擬中快速傅立葉轉換預處理元的效益

Abstract

本篇論文的目標是在多核心處理器的電腦上實作一個特徵值系統的解法。我們的問題來自三維鑽石光子晶體的馬克斯威爾方程式(Maxwell equations)，離散出的矩陣A是一個不規則的大型稀疏矩陣，如何快速且有效率的解此特徵值系統是我們要面對的挑戰。因為使用shift-and-invert的特徵值解法，所以我們需要解線性 系統。我們提供了一個非常有力的預處理元來加速解此線性系統。一個適當的預 處理元能夠加速線性系統的收斂並且保持線性系統的穩定性。我們的程式使用了 一些函式庫如MKL、PETSc和SLEPc。在最後，我們會展示其數值結果並且分析多核心處理器所帶來的效應。

This thesis aims to implement the solver of the eigenvalue system Ax = λBx on computer with multi-core CPUs. The problem derived from the Maxwell equations which is 3D diamond photonic crystals. The matrix A is a irregularly large-scale sparse matrix, so how to solve the eigenvalue system more efficient is a challenge. Because using the shift and invert eigenvalue solver, we need to solve the linear system. We provide a very powerful preconditioning scheme to accelerate the linear system. A suitable preconditioner makes the linear system converge quickly and more stable. The code are implemented by using some packages such as Intel Math Kernel Library (MKL), PETCs and SLEPc. Finally, we will show our numerical results and analysis the effect on multi-core CPUs.