| 关于奇异线性系统和矩阵方程若干问题的数值解法研究 | |
| Alternative Title | Numerical solutions of some problems on singular linear systems and matrix equations |
| 郝悦 | |
| Subtype | 博士 |
| Thesis Advisor | 伍渝江 |
| 2021-05-24 | |
| Degree Grantor | 兰州大学 |
| Place of Conferral | 兰州 |
| Degree Name | 理学博士 |
| Degree Discipline | 计算数学 |
| Keyword | 奇异线性方程组; 鞍点问题; 块二乘二线性方程组; 矩阵方程; 偏微分方 程 最小范数最小二乘解 迭代方法 预处理 Schwarz-Christoffel 映射 Sherman-Morrison-Woodbury 公式 |
| Abstract | 科学计算和工程应用中的诸多领域, 如流体力学, 结构力学, 约束最优化, 电磁学, 图像恢复, 信号处理, 控制理论和系统理论等, 都与线性系统的求解息息相关.因此, 根据系统的结构和特点设计出稳定, 高效的数值算法具有十分重要的理论意义和实际应用价值. 本论文主要研究了两类线性系统的求解, 一类是具有奇异, 非 Hermitian 半正定系数矩阵的大型稀疏线性方程组的数值求解, 另一类是线性矩阵方程的数值求解及应用. 首先, 我们研究了系数矩阵为奇异, 非 Hermitian 半正定矩阵的线性方程组的数值求解问题. 通过为 Hermitian 和反 Hermitian (HSS) 迭代方法引入一类奇异预处理子, 我们提出了一种广义的预处理 HSS (GPHSS) 迭代方法. 并证明了在一定条件下, 该方法收敛到相容和不相容的奇异线性方程组的最小范数最小二乘解,并且该收敛不依赖于初始估计向量的选取. 同时, 我们还分析了 GPHSS 预处理的 GMRES 方法的收敛性质. 此外, 我们还证明了 GPHSS 迭代方法可以无条件收敛到奇异鞍点问题的最小范数最小二乘解. 最后, 我们用数值例子验证了 GPHSS 迭代方法和相应的预处理子的可行性和有效性. 其次, 我们将求解一类非奇异块二乘二线性方程组的 PMHSS 迭代方法推广到了奇异的情况, 提出了广义的 PMHSS (GPMHSS) 迭代方法. 收敛性分析表明, GPMHSS 迭代方法可以无条件收敛到相容和不相容的奇异块二乘二线性方程组的最小范数最小二乘解. 同时, 对应的预处理 GMRES 方法也可以收敛到相容的奇异块二乘二线性方程组的最小范数最小二乘解. 数值例子验证了 GPMHSS 迭代方法和 GPMHSS 预处理子的稳定性和高效性. 然后, 我们探讨了如何数值求解多边形区域上的椭圆偏微分方程. 借助于 Schwarz-Christoffel (SC) 共形映射和 Matlab 中的 SC 工具包, 我们将定义在多边形区域上的椭圆偏微分方程转化到矩形区域上, 并利用有限差分方法离散得到一类特殊的矩阵方程. 因为转化后的偏微分方程的复杂性, 使得相应的矩阵方程涉及到了 Hadamard 乘积, 故常见的用于求解一般广义矩阵方程的方法都不完全适用, 因此我们详细地讨论了如何数值求解该类矩阵方程. 此外, 我们还分析了如何选取 Schwarz-Christoffel映射中的控制顶点. 数值实验结果验证了该方法的可行性和有效性. 最后, 为了解决向量形式的 Sherman-Morrison-Woodbury (SMW) 算法在求解矩阵方程时的计算弊端, 我们探讨了如何利用矩阵形式的 Sherman-Morrison-Woodbury 算法来数值求解小规模和中等规模的广义线性矩阵方程, 并简单分析了该方法的稳定性. 然后, 通过将其与投影方法相结合, 我们将矩阵形式的 SMW 算法应用到了大规模广义矩阵方程的求解中. 此外, 基于矩阵形式的 SMW 算法, 我们还给出了一种求解线性张量方程的方法. 最后, 我们用数值例子验证了这些算法的有效性和稳定性. |
| Other Abstract | Many fields in scientific computing and engineering applications, such as fluid dynamics, structure dynamics, constrained optimization,electromagnetism, image restoration, signal processing, control and systems theory, are closely related to the solving of linear systems. Thus, it is of great theoretical significance and practical application value to design a kind of robust and efficient numerical algorithm based on the structure and characteristic of problems. This thesis mainly focuses on the solving of two kinds of linear systems, one is numerical solutions of large, sparse system of linearequations with singular and non-Hermitian positive semi-definite coefficient matrix, and another is numerical solutions and applicationsof linear matrix equations. Firstly, we study numerical solutions of the system of linear equations with singular and non-Hermitian positive semi-definite coefficient matrix. By introducing a class of singular preconditioner for the Hermitian and skew-Hermitian splitting (HSS) iteration method, we propose a generalized preconditioned HSS (GPHSS) iteration method. Then, it is proved that under some condition, the GPHSS iteration method converges to the minimum norm least squares solution of both the consistent and inconsistent singular system of linear equations, and the convergence does not depend on the initial guess vector. Meantime, we also analyse the convergence property of the GPHSS preconditioned GMRES method. Besides, it is also proved that the GPHSS iteration method converges unconditionally to the minimum norm least squaressolution of the singular saddle point problem. Numerical experiments are presented to show the feasibility and the effectiveness of theGPHSS iteration method and the corresponding preconditioner. Next, we extend the PMHSS iteration method for solving a class of nonsingular block two-by-two system of linear equations to the singularcase, named as generalized PMHSS (GPMHSS) iteration method. Theoretical analyses show that the GPMHSS iteration method converges unconditionally to the minimum norm least squares solution of both the consistent and inconsistent singular block two-by-two system oflinear equations. Moreover, the corresponding preconditioned GMRES method also determines the minimum norm least squares solution of the consistent singular block two-by-two system of linear equations at breakdown. Numerical experiments are used to verify the robustness and the effectiveness of the GPMHSS iteration method and the GPMHSS preconditioner. Then, we analyse how to solve the elliptic partial differential equations defined in a polygonal domain numerically. By using the Schwarz-Christoffel (SC) mappings and the SC toolbox in Matlab, we transform the partial differential equations from a polygon domain toa rectangle domain. Then, discretizing the transformed equation by the finite difference method leads to a special kind of matrix equations. Since the complexity of the transformed partial differential equation, the corresponding matrix equation involves Hadamard product, suchthat most of the methods used to solve generalized matrix equations are not applicable. Therefore, we discuss how to solve this kind of matrix equations numerically in detail. Besides, we also analyse the selection of the control vertices of the Schwarz-Christoffel mappings. Numerical experiments verify the feasibility and the effectiveness of our method. Finally, in order to solve the computational drawback of the vectorized Sherman-Morrison-Woodbury (SMW) algorithm for solving matrixequations, we explore how to use the matrix-oriented Sherman-Morrison-Woodbury algorithm to solve small and medium size generalized linear matrix equations, and simply analyse the stability of this method. Then, by combining it with the projection methods, we apply the matrix -oriented SMW algorithm to the solving of large-scale generalized matrix equations. Moreover, based on the matrix---oriented SMW algorithm,a kind of method for solving linear tensor equations is discussed. Numerical experiments are given to illustrate the feasibility andeffectiveness of our strategies. |
| Pages | 136 |
| URL | 查看原文 |
| Language | 中文 |
| Document Type | 学位论文 |
| Identifier | https://ir.lzu.edu.cn/handle/262010/459575 |
| Collection | 数学与统计学院 |
| Affiliation | 数学与统计学院 |
| First Author Affilication | School of Mathematics and Statistics |
| Recommended Citation GB/T 7714 | 郝悦. 关于奇异线性系统和矩阵方程若干问题的数值解法研究[D]. 兰州. 兰州大学,2021. |
| Files in This Item: | There are no files associated with this item. | ||||
