三維光子晶體的數值模擬之平行計算

Abstract

本論文的主要目地是平行 Krylov-Schur 方法並結合 GMRES 演算法搭配 FFT 預處理解大型稀疏矩陣的廣義特徵值問題。 我們的問題來自三維光子晶體的馬克斯威爾方程，利用 Yee 的離散方法將馬克斯威爾方程離散成廣義特徵值問題來做數值模擬。 利用 Krylov-Schur 方法結合 GMRES 演算法以及搭配 FFT 預處理解廣義特徵值問題的效能非常好，並可加速廣義特徵值問題中線性系統內迭代的收斂速度。 我們的實驗中利用 PETSc, SLEPc, Intel MKL 等數學軟體，平行的數值實驗可執行於多核心電腦以及叢集電腦。

This dissertation aims to parallel the Krylov-Schur method with applying generalized minimal residual (GMRES) algorithm, which was combined with the fast Fourier transform (FFT) technique to solve large sparse matrix generalized eigenvalue problem derived from the governing Maxwell equations. The eigenvalue problems will be derived by Yee's scheme, the eigenvalue solver is based on Krylov-Schur method, associated with GMRES algorithm and a fast Fourier transform based preconditioned, which is very e cient and used to accelerate the inner iteration. The code are implemented by using PETSc and MKL. Numerical experiments are performed in multicore CPUs and parallel clusters.