預條件疊代法求解大型稀疏線性系統及在 PVM 環境下的平行計算實務

Abstract

本論文目的在研究目前所提出的一些稀疏矩陣疊代解法以及常用的預條件 技術. 這些疊代解法包括古典的共軛梯度法, BCG, CGS, 以及 BiCGSTAB 等方法. 我們將應用這些方法在由最小平方有限元素法, 有限 析與比較. 而後將在以 PVM 為核心所組成的網路工作站環境之下, 以主 從架構的模式實際將這些疊代法加以平行化. 並將分析平行演算法之有效 性與可延展性, 最後將予討論所得到之數值結果. In this thesis, we focus on implementing general purpose sparse linear solvers based on the preconditioned conjugate gradient like methods. The iterative methods we investigate include the conjugate gradient (CG), bi-conjugate gradient (BCG), conjugate gradient square (CGS), and stabilized bi-conjugate gradient (BiCGSTAB) method. Applications of these methods to least squares finite element method, finite volume method, and semiconductor device equations from SPICE software package are considered. Numerical results and performance comparison are reported. We also emphasize parallel implementation of these CG- like iterative methods in a networked workstations' environment using PVM 3.0. The distributed parallel master/slave mode is adopted in our implementation. Efficiency and scalability of the parallel algorithms are analyzed. Numerical experiments are discussed in details.