【作者】 张理涛；

【导师】 黄廷祝；

【作者基本信息】 电子科技大学 ， 应用数学， 2009， 博士

【摘要】 大规模科学计算和工程技术中许多问题的解决，最终归结为大型稀疏线性方程组的求解，其求解时间在整个问题求解时间中占有很大的比重，有的甚至达到80％．由于现今科学研究和大型项目中各种复杂的课题对计算精度和计算速度的要求越来越高．因此，作为大规模科学计算基础的线性代数方程组的高效数值求解引起了人们的普遍关注．这种方程组的求解一般采用迭代法，所以，迭代法的收敛性和收敛速度就成为人们关注的焦点，为许多专家和学者所研究．本文对与大型稀疏线性方程组迭代求解有关的特殊矩阵迭代法进行了深入和系统的研究，特别研究了松弛型矩阵多分裂迭代法的收敛性，两种Krylov子空间方法的性能和鞍点问题的预处理技术．全文共六章，分四个部分：研究了H-矩阵的松弛型矩阵多分裂法，并给出详细的理论分析和敛散速度的比较．一方面，给出了松弛型矩阵多分裂TOR法，研究了方法的收敛性，比较了他们的敛散速度，分别进行了串行和并行试验，验证了所提方法的优越性．另一方面，给出了松弛型矩阵多分裂USAOR法，研究了方法的收敛性，给出了实现算法的例子，并用数值试验与存在的方法进行了比较．进一步研究了一些H-矩阵松弛型矩阵多分裂法新的收敛性结果．分别为非线性方程组的非定常矩阵多分裂法，线性互补问题的矩阵多分裂法，松弛型矩阵多分裂SSOR法和松弛型矩阵多分裂TOR法，构建了相应方法的收敛性理论，得到了新的更弱的收敛性条件，进行了数值试验的比较．基于多分裂法的并行性，矩阵多分裂的研究和理论分析对于多分裂预处理子的构造有一定的理论和应用价值．我们的方法选取参数的余地更大，当选取近似最优参数时，能实现更快的收敛速度和构造出更有效的预处理子．基于BiCR算法设计了求解非对称线性方程组Krylov子空间平方共轭残差(CRS)算法和适合分布式并行计算的改进的平方共轭残差(ICRS)算法，并对两种算法进行了理论分析和算法比较，串行和并行数值试验表明所提方法具有较好的收敛速度和并行性能．研究了鞍点问题特殊矩阵迭代求解预处理技术．首先，对内点优化问题产生的一类鞍点问题给出了一种预处理技术，进行了相应的理论分析和数值试验．接着，基于离散化混合型时谐Maxwell方程的块三角鞍点问题和特殊矩阵的结构，提出了带多个参数的预处理技术，并进行了理论分析和参数的理论选取．理论分析表明提出的预处理子有更好的特征值聚集性．数值试验也表明本章提出的预处理子性能大大优于免增广和免Schur余块对角预处理子．更多还原

【Abstract】 Solutions of large-scale sparse linear systems arise widely in large-scale scientific computing, and engineering disciplines. Moreover, solving large-scale sparse linear systems plays a key role in scientific and engineering computing and the corresponding time of computing always arrives at 80%, so more and more people pay attention to find effective numerical solutions of linear algebraic systems based on large-scale scientific and engineering computing. Solutions of large-scale sparse linear systems are always iterative methods, so convergence and convergence rate of iterative methods are deeply studied by many experts and scholars. The author mainly studies some iterative methods of special matrices related to iteration solutions of large-scale sparse linear systems, especially convergence of relaxed matrix multisplitting iterative methods, performance of two Krylov subspace methods, and preconditioning techniques of saddle point problems. This paper consists of four parts with six chapters.Research on relaxed matrix multisplitting iterative methods for H-matrices, and gives detailedly theoretical analysis and comparisons of convergence and divergence rates. On the one hand, we present relaxed matrix multisplitting TOR iterative method, study convergence of our method, compare the corresponding convergence and divergence rates, and carry out sequential and parallel experiments which show the validity of our method. On the other hand, we give relaxed matrix multisplitting USAOR iterative method, study convergence of our method, show the examples to put algorithms in practice, and carry out numerical experiments to compare with the corresponding existing methods.Further study new convergence of some relaxed matrix multisplitting iterative methods for H-matrices. We analyze non-stationary matrix multisplitting iterative methods for almost linear systems, matrix multisplitting iterative methods for the linear complementarity problems, relaxed matrix multisplitting SSOR iterative method, and relaxed matrix multisplitting TOR iterative method for linear systems, respectively. Moreover, this chapter studies convergence theory of the corresponding methods, finds new weaker convergence conditions, and carries out some comparisons of numerical experiments. Based on the parallelity of multisplitting, the study and theory of matrix multisplitting contributing to the structure of multisplitting preconditioners have many theoretical and applied values. Our methods have more optional parameters, so we can obtain more quick convergence rate and structure more effective preconditioners when choosing the approximately optimal relaxed parameters.Design Krylov subspace conjugate residual squared (CRS) algorithm for nonsym-metric linear systems and the improved conjugate residual squared (ICRS) algorithm for distributed parallel computing based on biconjugate residual method (BiCR) algorithm, and give theoretical analysis and comparisons of algorithms for two algorithms. In the end, sequential and parallel numerical experiments show that CRS and ICRS methods have more quick convergence rate than BiCR method and that ICRS method has better parallel performance than CRS method and so on.Research on generalized block preconditioning techniques for iteration solutions of saddle point problems which arise from the discretized time-harmonic Maxwell equations and interior point optimization method. On the one hand, new block triangular precondi-tioner for linear systems arising from interior point optimization method is obtained, and the corresponding theoretical analysis and numerical experiments are given to show the validity of our preconditioner. On the other hand, we deeply study block triangular preconditioners with multi-parameters based on block triangular saddle point linear systems and structure of special matrices arising from time-harmonic Maxwell equations, and give theoretical analysis and theoretical choice of optimal parameters. In particular, theoretical analysis shows that all eigenvalues of two generalized block triangular preconditioners are strongly clustered. Finally, numerical experiments show that the quality of the presented block triangular preconditioners is better than that of the corresponding block diagonal augmentation-free and Schur complement-free preconditioners.更多还原