求解二乘二分块实线性方程组的块分裂预处理方法 Alternative Title Splitting-Based Block Preconditioning Methods for Block Two-by-Two Matrices of Real Square Blocks 严辉银 Thesis Advisor 黄玉梅 2015-05-29 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 硕士 Keyword 二乘二分块矩阵 分裂迭代方法 正交矩阵 谱性质 分裂预处理子 Abstract 许多科学计算和工程应用中的计算问题会涉及到求解 系数矩阵为二乘二分块矩阵的大型稀疏线性方程组。 因此，二乘二分块线性方程组的快速求解方法得到了广泛研究并出现了 一些有效方法。 预处理Krylov子空间方法就是其中最重要的一种。 本文先利用正交矩阵， 得到了一个与原二乘二分块线性方程组等价的新线性方程组。 然后对新线性方程组构建了分块Jacobi和分块Gauss-Seidel分 裂迭代方法，并证明了它们的收敛性。最后，利用所构造的Jacobi和 Gauss-Seidel分裂方法作为预处理矩阵，采用预处理Krylov子空间方法求解原线性方程组，并进行了谱分析。 预处理矩阵的谱性质以及数值实验都表明， 在用Krylov子空间迭代方法（如GMRES方法）求解二乘二分块线性 方程组时，我们构造的分块Jacobi和分块Gauss-Seidel分裂 预处理子比现有预处理子能更有效地提高收敛速度。 Other Abstract Many practical problems from scientific computing and engineering applications require the solution of a block two-by-two linear system. Thus, solving block two-by-two linear system has attracted much more attention and lots of efficient solvers could be found in the literature. Of which the the preconditioned Krylov subspace methods are the most important solvers. In this paper, we establish a new equivalent linear system to the original linear system by an orthogonal matrix. We construct block Jacobi and block Gauss-Seidel splitting iteration methods based on the coefficient matrix of the new linear system. The convergence of these splitting iterations is also demonstrated. Then, by utilizing the proposed block Jacobi and block Gauss-Seidel splittings, we put forward block splitting preconditioners. Spectral distributions of these preconditioned matrices and numerical experiments show that the proposed splitting-based block preconditioners can be quite competitive with the existed preconditioners when they are used to accelerate Krylov subspace iteration methods such as GMRES for solving the block two-by-two liner systems. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/224803 Collection 数学与统计学院 Recommended CitationGB/T 7714 严辉银. 求解二乘二分块实线性方程组的块分裂预处理方法[D]. 兰州. 兰州大学,2015.
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