鞍点问题及复对称线性系统迭代算法的研究

发布时间：2018-01-18 12:29

本文关键词：鞍点问题及复对称线性系统迭代算法的研究　出处：《华东师范大学》2017年博士论文　论文类型：学位论文

【摘要】：大型稀疏线性方程组的数值求解问题广泛存在于电磁学问题,最小二乘问题,约束优化问题及工程中数值模拟问题等,这些问题经过有限元或有限差分等数值离散方法得到一些具有特殊结构的大型稀疏线性方程组,如鞍点问题,复线性系统等.该论文主要针对几类具有特殊结构的大型稀疏线性方程组:奇异鞍点问题,非奇异鞍点问题和奇异复对称线性系统,给出几种有效的迭代算法和预处理子,并给出相应迭代方法的收敛性质和数值实验,具体如下:首先,针对奇异鞍点问题,提出了两类含参数的不精确Uzawa方法:广义含参数不精确Uzawa方法(GPIU)和广义预处理含参数不精确Uzawa方法(GPPIU).首先分别介绍了这两类方法的迭代格式,然后利用半收敛的定义给出其半收敛的充分条件.其次,结合Uzawa方法和SOR方法各自的优点,得到一类Uzawa-SOR迭代方法,分析给出了该方法半收敛的条件,然后通过相应的数值算例验证它的有效性.然后针对非奇异鞍点问题,利用广义SOR方法(GSOR)的特点,给出推导该方法最优参数的一个简单方法.然后,针对奇异鞍点问题,首先介绍了正则化的Hermitian和Skew-Hermitian分裂迭代方法(RHSS),然后分析得到该方法是无条件半收敛的.同时,在分析的过程中,我们发现HSS方法求解奇异鞍点问题时,也是无条件半收敛的,弱化了之前文章的结果.最后通过一系列的数值实验验证该方法的有效性和稳定性.再次,针对非奇异鞍点问题,利用矩阵分裂方法,给出两类预处理子:然后针对广义鞍点问题,也给出了两类有效的预处理子.分别对这四类预处理子给出了详细的谱分析,他们具有较好的特征值聚集性质,最后通过一系列的数值实验验证这些预处理子的谱分布的情况及实际的有效性.最后,针对奇异复对称线性系统,我们将广义修正的HSS算法(GMHSS)推广到求解奇异线性系统,详细给出了半收敛分析,并得到了半收敛的条件.最后给出了详细的数值实验结果,进一步验证该算法的半收敛性和有效性.
[Abstract]:The numerical solutions of large sparse linear equations are widely used in electromagnetic problems, least-squares problems, constrained optimization problems and numerical simulation problems in engineering and so on. Some large sparse linear equations with special structure, such as saddle point problem, are obtained by finite element method or finite difference method. This paper focuses on several kinds of large sparse linear equations with special structure: singular saddle point problem, nonsingular saddle point problem and singular complex symmetric linear system. Several effective iterative algorithms and preconditioners are given, and the convergence properties and numerical experiments of the corresponding iterative methods are given. The main results are as follows: first, for the singular saddle point problem. Two kinds of imprecise Uzawa methods with parameters are proposed: the generalized imprecise Uzawa method with parameters and the generalized preprocessing Uzawa method with imprecise parameters. First, the iterative schemes of these two methods are introduced. Then the sufficient conditions of semi-convergence are given by using the definition of semi-convergence. Secondly, combining the advantages of Uzawa method and SOR method, a class of Uzawa-SOR iterative method is obtained. The condition of semi-convergence of the method is analyzed, and the validity of the method is verified by corresponding numerical examples. Then, the generalized SOR method is used to solve the nonsingular saddle point problem. A simple method to deduce the optimal parameters of this method is given. Then, the singular saddle point problem is discussed. Firstly, the regularized Hermitian and Skew-Hermitian splitting iterative methods are introduced, and the results show that the method is unconditionally semi-convergent. In the process of analysis, we find that the HSS method is also unconditionally semi-convergent in solving singular saddle point problems. Finally, the validity and stability of the method are verified by a series of numerical experiments. Thirdly, the matrix splitting method is used to solve the nonsingular saddle point problem. Two kinds of preconditioners are given: then for the generalized saddle point problem, two kinds of effective preconditioners are also given. The spectral analysis of the four preconditioners is given in detail, and they have better characteristic of eigenvalue aggregation. Finally, a series of numerical experiments are carried out to verify the spectral distribution of these preconditioners and their effectiveness. Finally, for singular complex symmetric linear systems. We generalize the generalized modified HSS algorithm GMHSS to solve singular linear systems, give the semi-convergence analysis in detail, and obtain the conditions of semi-convergence. Finally, the numerical results are given in detail. The semi-convergence and validity of the algorithm are further verified.
【学位授予单位】：华东师范大学
【学位级别】：博士
【学位授予年份】：2017
【分类号】：O241.6

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