特殊矩阵数值分析和鞍点问题迭代求解预处理技术
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摘要
科学与工程的很多领域如高阶微分方程求解,计算电磁学,流体力学,油藏模拟和最优化问题等都离不开大型线性代数方程组的求解.大型线性代数方程组的求解研究是大规模科学与工程计算的核心,具有重要的理论价值和应用价值.本文对与大型线性代数方程组迭代求解有关的特殊矩阵和数值特征进行了深入的研究,特别系统地研究了矩阵分裂迭代法的收敛性和比较理论及鞍点问题迭代求解预处理技术.全文共六章,分四个部分:
第一部分(第二章)研究了两类特殊矩阵:非奇H-矩阵和广义H-矩阵.论文基于矩阵α-对角占优给出了非奇H-矩阵简捷判据,为非奇H-矩阵判据研究提供了新的思路.还得到了广义H-矩阵若干等价命题,充分或必要条件,对广义H-矩阵进行了进一步推广,该推广部分回答了著名计算数学专家Nabben提出的公开问题.
第二部分(第三章)给出了矩阵数值特征估计.论文给出了一类包含C-矩阵的非奇异矩阵(MC-矩阵),利用该类矩阵性质得到了实矩阵实特征值的排除区间,进而得到了随机矩阵实特征值的界.同时也得到了实矩阵特征值实部的包含区间,具有非负非对角元的实矩阵的实特征值的简单上下界.还获得了矩阵数值半径新的等价公式,并由此得到了矩阵数值半径新下界.最后给出了非奇M-矩阵和逆M-矩阵Hadamard积最小特征值下界,其中一些下界仅依赖于矩阵元素.本部分得到的结果优于近期的相关结果.
第三部分(第四章)研究了矩阵分裂的收敛性和比较理论.得到了非Hermitian正定矩阵单分裂的收敛条件.利用Hermitian正定矩阵和非负矩阵理论得到了Hermitian正定矩阵和单调矩阵双分裂的收敛性理论,基于该理论获得了非奇H-矩阵Jacobi和Gauss-Seidel双SOR方法的收敛区域.首次研究了矩阵双分裂的比较理论,得到了并行混沌多分裂的比较理论,这些理论为迭代法的选择提供了一些理论依据.
第四部分(第五章)研究了鞍点问题迭代求解预处理技术.首先给出使得非对称鞍点矩阵具有实正特征值且可对角化的充分条件,该条件比已有著名条件更弱.其次,深入研究PBP预处理子特别是正则化预处理子的谱性质,给出了预处理矩阵实特征值和非实特征值的包含区域,指出若满足一定条件,PBP预处理矩阵仅有正实特征值,可使用非标准的共轭梯度算法,用Stokes方程和Maxwell方程做数值试验测试了正则化预处理子的性能.还研究了广义鞍点问题PSS预处理子的谱性质,克服了已有的研究只针对(2,2)块是0的鞍点问题的缺陷,证明了当迭代因子趋于0时,PSS预处理矩阵的特征值聚集在(0,0)和(2,0)附近,从而理论上说明了最优迭代因子应比较小,并用Stokes方程和Oseen方程做数值试验表明迭代因子一般选择在0到1之间.最后对混合型时谐Maxwell方程,首次提出了免增广和免Schur余块三角预处理技术,理论分析说明其构造及应用代价和已有的免增广和免Schur余块对角预处理子相当,但有更好的特征值聚集性质,数值试验也说明其性能大大优于免增广和免Schur余块对角预处理技术.
Large-scale linear systems arise widely in domains of science and engineering such as solutions of PDEs with high orders, computational electromagnetics, fluid mechanics, reservoir modeling and optimization problems. Solving large-scale linear systems plays a key role in scientific and engineering computing. Some special matrices and numerical characteristics related to iteration solutions of linear systems, convergence and comparison theorems of matrix splittings, preconditioning techniques for iteration solutions of saddle point problems are deeply studied in this thesis. This thesis consists of four parts with six chapters.
Part one (Chapter two) is to study two classes of special matrices: nonsingular H-matrices and generalized H-matrices. Based onα-diagonal dominance of matrices, we derive a new brief criterion for nonsingular H-matrices, which gives a new way to study the new criteria in future. Several equivalent propositions, sufficient or necessary conditions for generalized .H-matrices are obtained. Meanwhile, we give a new construction of generalized .H-matrices, which partly answers Nabben's open problem.
Part two (Chapter three) contributes to estimates of numerical characteristics of matrices. We first present a new class of real nonsingular matrix (MC-matrices) containing C-matrices. As an application of M C-matrices, we obtain an exclusion interval of real eigenvalues of a real matrix, which, furthermore, is used to localize the real eigenvalues of stochastic matrices precisely. Moreover, we achieve inclusion intervals for real parts of eigenvalues of real matrices and brief bounds of real eigenvalues of matrices whose off-diagonal elements are all nonnegative. Then, some lower bounds of numerical radius of matrices and the smallest eigenvalues of the Hadamard product of nonsingular M-matrices and inverse M-matrices are derived. All results presented in this part are tighter than the corresponding existing ones.
Part three (Chapter four) is devoted to investigating convergence and comparison theorems of splittings of matrices. We firstly give some sufficient conditions for the convergence of single splittings of non-Hermitian positive definite matrices. Secondly, with the help of nonnegative matrix and Hermitian positive definite matrix theories, convergence theorems of double splittings of Hermitian positive definite matrices and monotone matrices are presented. Furthermore, convergence regions of Jcaobi and Gauss-Seidel double SOR methods for a nonsingular H-matrix are established. Finally, comparison theorems for double splittings and parallel chaotic multisplittings of matrices are also obtained, which provide theoretical base for the choice of iteration methods.
Part four (Chapter five) contributes to preconditioning techniques for iteration solutions of saddle point problems. A sufficient condition is firstly established such that a nonsymmetric saddle point matrix is diagonalizable with real and positive eigenvalues. This condition is weaker than the corresponding earlier conditions. Secondly, we deeply study the PBP preconditioners, in particular, the regularized preconditioners. The regions containing real and non-real eigenvalues of the PBP preconditioned matrix are obtained. All eigenvalues of the PBP preconditioned matrix are real positive and the non-standard Conjugate Gradient algorithm can be used if certain conditions are satisfied. The model problems of Stokes equations and Maxwell equations show that regularized preconditioners are robust. Thirdly, we give some spectral properties of PSS preconditioners for generalized saddle point problems, which overcome the defect that the earlier results are only concerned with the saddle point problems with zero (2,2) blocks. It is shown that all eigenvalues of the PSS preconditioned matrix form two tight clusters, one is near (0,0) and the other is near (2,0) when the iteration parameter approaches to zero from above. The model problems of Stokes equations and Oseen equations show that the 'optimal' iteration parameter is usually between 0 and 1. Finally, we present block triangular augmentation-free and Schur complement-free preconditioners for the discretization of time-harmonic Maxwell equations in mixed form. The cost of their construction and application is almost the same to that of block diagonal augmentation-free and Schur complement-free preconditioners. However, theoretical analysis and numerical experiments show that the quality of the presented block triangular preconditioners is better than that of the corresponding block diagonal preconditioners.
第一部分(第二章)研究了两类特殊矩阵:非奇H-矩阵和广义H-矩阵.论文基于矩阵α-对角占优给出了非奇H-矩阵简捷判据,为非奇H-矩阵判据研究提供了新的思路.还得到了广义H-矩阵若干等价命题,充分或必要条件,对广义H-矩阵进行了进一步推广,该推广部分回答了著名计算数学专家Nabben提出的公开问题.
第二部分(第三章)给出了矩阵数值特征估计.论文给出了一类包含C-矩阵的非奇异矩阵(MC-矩阵),利用该类矩阵性质得到了实矩阵实特征值的排除区间,进而得到了随机矩阵实特征值的界.同时也得到了实矩阵特征值实部的包含区间,具有非负非对角元的实矩阵的实特征值的简单上下界.还获得了矩阵数值半径新的等价公式,并由此得到了矩阵数值半径新下界.最后给出了非奇M-矩阵和逆M-矩阵Hadamard积最小特征值下界,其中一些下界仅依赖于矩阵元素.本部分得到的结果优于近期的相关结果.
第三部分(第四章)研究了矩阵分裂的收敛性和比较理论.得到了非Hermitian正定矩阵单分裂的收敛条件.利用Hermitian正定矩阵和非负矩阵理论得到了Hermitian正定矩阵和单调矩阵双分裂的收敛性理论,基于该理论获得了非奇H-矩阵Jacobi和Gauss-Seidel双SOR方法的收敛区域.首次研究了矩阵双分裂的比较理论,得到了并行混沌多分裂的比较理论,这些理论为迭代法的选择提供了一些理论依据.
第四部分(第五章)研究了鞍点问题迭代求解预处理技术.首先给出使得非对称鞍点矩阵具有实正特征值且可对角化的充分条件,该条件比已有著名条件更弱.其次,深入研究PBP预处理子特别是正则化预处理子的谱性质,给出了预处理矩阵实特征值和非实特征值的包含区域,指出若满足一定条件,PBP预处理矩阵仅有正实特征值,可使用非标准的共轭梯度算法,用Stokes方程和Maxwell方程做数值试验测试了正则化预处理子的性能.还研究了广义鞍点问题PSS预处理子的谱性质,克服了已有的研究只针对(2,2)块是0的鞍点问题的缺陷,证明了当迭代因子趋于0时,PSS预处理矩阵的特征值聚集在(0,0)和(2,0)附近,从而理论上说明了最优迭代因子应比较小,并用Stokes方程和Oseen方程做数值试验表明迭代因子一般选择在0到1之间.最后对混合型时谐Maxwell方程,首次提出了免增广和免Schur余块三角预处理技术,理论分析说明其构造及应用代价和已有的免增广和免Schur余块对角预处理子相当,但有更好的特征值聚集性质,数值试验也说明其性能大大优于免增广和免Schur余块对角预处理技术.
Large-scale linear systems arise widely in domains of science and engineering such as solutions of PDEs with high orders, computational electromagnetics, fluid mechanics, reservoir modeling and optimization problems. Solving large-scale linear systems plays a key role in scientific and engineering computing. Some special matrices and numerical characteristics related to iteration solutions of linear systems, convergence and comparison theorems of matrix splittings, preconditioning techniques for iteration solutions of saddle point problems are deeply studied in this thesis. This thesis consists of four parts with six chapters.
Part one (Chapter two) is to study two classes of special matrices: nonsingular H-matrices and generalized H-matrices. Based onα-diagonal dominance of matrices, we derive a new brief criterion for nonsingular H-matrices, which gives a new way to study the new criteria in future. Several equivalent propositions, sufficient or necessary conditions for generalized .H-matrices are obtained. Meanwhile, we give a new construction of generalized .H-matrices, which partly answers Nabben's open problem.
Part two (Chapter three) contributes to estimates of numerical characteristics of matrices. We first present a new class of real nonsingular matrix (MC-matrices) containing C-matrices. As an application of M C-matrices, we obtain an exclusion interval of real eigenvalues of a real matrix, which, furthermore, is used to localize the real eigenvalues of stochastic matrices precisely. Moreover, we achieve inclusion intervals for real parts of eigenvalues of real matrices and brief bounds of real eigenvalues of matrices whose off-diagonal elements are all nonnegative. Then, some lower bounds of numerical radius of matrices and the smallest eigenvalues of the Hadamard product of nonsingular M-matrices and inverse M-matrices are derived. All results presented in this part are tighter than the corresponding existing ones.
Part three (Chapter four) is devoted to investigating convergence and comparison theorems of splittings of matrices. We firstly give some sufficient conditions for the convergence of single splittings of non-Hermitian positive definite matrices. Secondly, with the help of nonnegative matrix and Hermitian positive definite matrix theories, convergence theorems of double splittings of Hermitian positive definite matrices and monotone matrices are presented. Furthermore, convergence regions of Jcaobi and Gauss-Seidel double SOR methods for a nonsingular H-matrix are established. Finally, comparison theorems for double splittings and parallel chaotic multisplittings of matrices are also obtained, which provide theoretical base for the choice of iteration methods.
Part four (Chapter five) contributes to preconditioning techniques for iteration solutions of saddle point problems. A sufficient condition is firstly established such that a nonsymmetric saddle point matrix is diagonalizable with real and positive eigenvalues. This condition is weaker than the corresponding earlier conditions. Secondly, we deeply study the PBP preconditioners, in particular, the regularized preconditioners. The regions containing real and non-real eigenvalues of the PBP preconditioned matrix are obtained. All eigenvalues of the PBP preconditioned matrix are real positive and the non-standard Conjugate Gradient algorithm can be used if certain conditions are satisfied. The model problems of Stokes equations and Maxwell equations show that regularized preconditioners are robust. Thirdly, we give some spectral properties of PSS preconditioners for generalized saddle point problems, which overcome the defect that the earlier results are only concerned with the saddle point problems with zero (2,2) blocks. It is shown that all eigenvalues of the PSS preconditioned matrix form two tight clusters, one is near (0,0) and the other is near (2,0) when the iteration parameter approaches to zero from above. The model problems of Stokes equations and Oseen equations show that the 'optimal' iteration parameter is usually between 0 and 1. Finally, we present block triangular augmentation-free and Schur complement-free preconditioners for the discretization of time-harmonic Maxwell equations in mixed form. The cost of their construction and application is almost the same to that of block diagonal augmentation-free and Schur complement-free preconditioners. However, theoretical analysis and numerical experiments show that the quality of the presented block triangular preconditioners is better than that of the corresponding block diagonal preconditioners.
引文
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