鞍点线性系统的矩阵分裂迭代方法和预处理技术研究

发布时间：2018-02-24 11:06

本文关键词： Navier-Stokes方程鞍点问题迭代方法矩阵分裂预处理子　出处：《兰州大学》2016年博士论文　论文类型：学位论文

【摘要】：当今,很多工程和物理应用问题,如计算流体动力学,计算电磁学,约束优化问题等,最后都会归为线性方程组的求解.一些微分方程,例如Navier-Stokes方程,求它们的解析解是非常困难的,此时研究它们的数值解就变得尤为重要.一般采用有限差分或有限元等方法去离散这些微分方程为大型稀疏的线性方程组,这样就把微分方程的数值解问题最后都转化为对应的线性方程组求解问题.因此,研究这些线性方程组有效的解法具有非常重要的理论意义和应用价值.本学位论文主要研究了鞍点线性系统的矩阵分裂迭代方法和预处理技术;还研究了基于双分裂的并行多分裂迭代方法.首先,关于非对称鞍点问题,提出了一类修正的位移分裂(MSS)预处理子,同时MSS预处理子对应的MSS迭代方法的收敛性质会被讨论;另外,进一步提出了局部的MSS(LMSS)预处理子,也讨论了LMSS预处理子对应的LMSS迭代方法的收敛性质;接着,讨论了MSS和LMSS预处理子的最优参数的选取方法;数值实验验证了MSS预处理子和LMSS预处理子的有效性.其次,提出了广义鞍点问题的正则的埃尔米特和反埃尔米特分裂(RHSS)迭代法和RHSS预处理子,且研究了RHSS迭代方法的收敛性质;接着,推出了修正的RHSS(MRHSS)预处理子,并分析了MRHSS预处理的广义鞍点矩阵的谱性质;此外,分别研究了RHSS和MRHSS预处理子最优参数的选取方法;数值实验验证了RHSS迭代方法的优势,以及RHSS预处理子和MRHSS预处理子的预处理效果.再次,为了克服修正的维数分裂(MDS)预处理子和广义的松弛分裂(GRS)预处理子的不足,给出了松弛的块三角分裂(RBTS)预处理子.因为RBTS预处理子有更简单的块结构,所以这个新的预处理子比MDS和GRS预处理子更容易实施;接着,推导了RBTS预处理的鞍点矩阵的谱分布和最小多项式次数的上界;另外,提出了RBTS预处理子最优参数的选取方法.数值实验证实了RBTS预处理子的有效性.然后,关于广义鞍点问题,给出一类修正的GRS(MGRS)预处理子和一类修正的块三角分裂(MBTS)预处理子;接着,分别研究了MGRS和MBTS预处理的鞍点矩阵的谱性质及它们的最小多项式次数的上界;进而,分别讨论了MGRS和MBTS预处理子最优参数的选取方法;另外,应用这两类新的预处理子到三维线性化的Navier-Stokes方程,并分别讨论了对应的MGRS和MBTS预处理子最优参数的选取;最后,通过数值实验来验证了两类新预处理子的有效性.最后,基于系数矩阵的双分裂提出了并行多分裂迭代法和并行多分裂两阶段迭代法.当系数矩阵为单调矩阵或H矩阵时,研究了新方法的收敛性,也进一步讨论了新方法的比较结果.此外,提出了鞍点线性系统的基于双分裂的多分裂迭代法.
[Abstract]:Nowadays, many engineering and physical application problems, such as computational fluid dynamics, computational electromagnetics, constrained optimization problems and so on, will be classified as the solution of linear equations, some differential equations, such as Navier-Stokes equations, It is very difficult to find their analytical solutions, so it is very important to study their numerical solutions. The finite difference method or finite element method is usually used to discretize these differential equations for large sparse linear equations. In this way, the numerical solutions of the differential equations are transformed into the corresponding linear equations. It is of great theoretical significance and practical value to study the effective solutions of these linear equations. In this dissertation, the matrix splitting iterative method and preprocessing technique for saddle point linear systems are studied. The parallel multi-splitting iterative method based on double splitting is also studied. Firstly, for the asymmetric saddle point problem, a modified displacement-splitting MSS) preprocessor is proposed, and the convergence property of the MSS iterative method corresponding to the MSS preprocessor is discussed. In addition, the local MSSS-LMSS) preprocessor is proposed, and the convergence property of the LMSS iteration method corresponding to the LMSS preprocessor is also discussed, and then the methods of selecting the optimal parameters of the MSS and LMSS preconditioners are discussed. Numerical experiments show the validity of MSS preprocessor and LMSS preprocessor. Secondly, regular Hermitian and anti-Hermitian splitting RHSS iterative method and RHSS preprocessor for generalized saddle point problem are proposed, and the convergence properties of RHSS iterative method are studied. Then, the modified RHSS-MRHSS preprocessor is proposed, and the spectral properties of the generalized saddle point matrix of MRHSS preprocessing are analyzed. In addition, the methods of selecting optimal parameters of RHSS and MRHSS preconditioners are studied, and the advantages of RHSS iterative method are verified by numerical experiments. Thirdly, in order to overcome the shortcomings of modified dimension splitters and generalized relaxation splitters, the effects of RHSS preconditioners and MRHSS preconditioners are also discussed. This new preprocessor is easier to implement than the MDS and GRS preprocessor because the RBTS preprocessor has a simpler block structure. The spectral distribution and upper bound of minimum polynomial degree of saddle point matrix of RBTS preprocessing are derived. In addition, a method for selecting optimal parameters of RBTS preprocessor is proposed. Numerical experiments show the validity of RBTS preprocessor. For the generalized saddle point problem, a class of modified GRS MGRS preconditioners and a class of modified block triangulation splitters (MBTS) preconditioners are given, and then the spectral properties of the saddle point matrices of MGRS and MBTS preprocessing and the upper bounds of their minimum polynomial degree are studied, respectively. Furthermore, the methods of selecting the optimal parameters of MGRS and MBTS preprocessor are discussed respectively. In addition, the two new preconditioners are applied to the three-dimensional linearized Navier-Stokes equation, and the optimal parameters of the corresponding MGRS and MBTS preconditioners are discussed respectively. The validity of two kinds of new preprocessor is verified by numerical experiments. Finally, the parallel multi-splitting iterative method and the parallel multi-splitting two-stage iterative method are proposed based on the coefficient matrix. When the coefficient matrix is monotone matrix or H matrix, In this paper, the convergence of the new method is studied, and the comparison results of the new method are also discussed. In addition, a double splitting iterative method based on double splitting is proposed for saddle point linear systems.
【学位授予单位】：兰州大学
【学位级别】：博士
【学位授予年份】：2016
【分类号】：O241.6

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