大型稀疏代数系统的数值求解研究
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摘要
科学与工程计算的很多领域,诸如计算流体力学、约束优化、计算电磁学、PDEs的混合有限元近似、非线性规划、中子输运理论等问题的求解,最终都可归结为大型稀疏代数系统的求解.因此,对大型稀疏代数系统的求解研究就具有非常重要的理论意义和实际应用价值.由于许多实际问题产生的大型稀疏代数系统往往具有某种特殊结构,对具有这些特殊结构的大型稀疏代数系统的数值求解研究引起了国内外众多专家和学者的关注.本文对几类特殊的大型稀疏代数系统的数值求解方法进行了深入系统地研究.特别研究了求解线性鞍点问题的迭代法和预处理技术,求解非线性鞍点问题的迭代法以及求解代数黎卡提方程的迭代法.本文共六章,分四个部分:
研究求解线性鞍点问题的Uzawa类迭代法.首先,提出了一个修正的非线性Uzawa迭代法,讨论了算法的收敛性,并给出了理论和数值比较,数值实验也验证了修正方法的有效性.其次,给出了求解(2,2)块不为零的非对称广义鞍点问题的GMLHSS迭代方法,并探讨了算法的收敛条件.最后,给出了非精确Uzawa方法、GSSOR方法和MLHSS方法求解奇异鞍点系统时的半收敛性分析.
研究求解非对称鞍点问题的预处理技术.首先,给出了含参数的广义非精确块三角预条件子,对预处理矩阵的特征对性质给出了分析,并给出了预处理矩阵的特征值扰动分析.其次,基于系数矩阵的部分HS分裂和PS分裂,提出了PHSS预条件子和PPSS预条件子,并详细研究了预处理矩阵的谱性质,指出对于一个充分小的正参数,预处理矩阵特征值聚集在两个点附近:一个是(0,0)点,另一个是(2,0)点,并且通过大量的数值实验验证了理论分析和此两类预条件子的有效性.最后,给出了SIMPLE预条件子,利用特征值理论,研究了预处理矩阵两种不同表达形式的谱之间的联系.
研究求解非线性鞍点问题的迭代法.在求解线性鞍点问题的迭代解法基础上,给出了几个求解非线性鞍点问题的迭代法,并对方法进行了收敛性分析,数值实验验证了所提算法的有效性.
研究求解代数黎卡提方程的迭代法.事实上,输运理论中的代数黎卡提方程可以写成与之等价的向量方程.首先,基于松弛思想和牛顿方法,提出了求解向量方程的松弛Newton-like方法,并给出了算法的收敛性分析.其次,利用拟牛顿思想,结合已有的牛顿类型方法,给出了两个修正牛顿方法求解向量方程,并得到了算法的收敛结果.数值实验也表明所给出的三个算法能有效改进和提高已有算法的收敛性.
Large-scale sparse algebraic systems arise in a wide variety of applications through-out computational science and engineering fields, such as computational ?uid dynam-ics、constrained optimization、computational electromagnetics、mixed finite elementapproximations of PDEs、nonlinear optimization、transport theory and so on. Hence, ithas important theoretic and practical significance to study on the numerical methods forsolving this type of algebraic systems. Many large-scale sparse algebraic systems arisingfrom practical problems often have special structures. Therefore, the experts and schol-ars from home and abroad have been attracted on the research of numerical methods forsolving large sparse algebraic systems with special structure. This thesis represents deepand systematic research on the numerical methods for solving large-scale sparse algebraicsystems which have special structures, especially on the iterative methods and precondi-tioning techniques for linear saddle point problems, iterative methods for nonlinear saddleproblems and algebraic Riccati equation. This thesis is divided into four parts, includingsix chapters.
Uzawa type iterative methods are studied for solving linear saddle point problems.First, a modified nonlinear Uzawa iterative method is proposed and convergence of themodified method is discussed, and also theoretical and numerical comparison are given.Numerical experiments are provided to show the efficiency of the modified method. Sec-ond, a generalized modified local Hermitian and Skew-Hermitian splitting method is pre-sented for the generalized saddle point problems with nonzero (2, 2) block and conver-gence of the algorithm is discussed. Last, the inexact Uzawa method, the GSSOR methodand the MLHSS method are applied to solve singular saddle point systems and the semi-convergence analysis of these three methods are provided, respectively.
Preconditioning techniques are investigated for nonsymmetric saddle point prob-lems. Firstly, inexact block triangular preconditioners which contain a parameter arepresented, and eigenpair properties of the preconditioned matrix are analyzed. Further-more, eigenvalue perturbation analysis of the preconditioned matrix is shown. Secondly,on the basis of partial HS and PS splitting of coefficient matrix, PHSS preconditioner andPPSS preconditioner are presented and analyzed for nonsymmetric saddle point prob- lems, and also the spectral properties of the preconditioned matrix are studied in detail.Moreover, it can be concluded that the eigenvalue of the preconditioned matrix will gatherinto two clusters: one is near (0, 0) and the other is near (2, 0) when the iteration param-eter becomes small enough. In addition, the theoretical results and effectiveness of thetwo preconditioners are proved by a lot of numerical experiments. Finally, SIMPLE pre-conditioner is given. Using eigenvalue theory, the relation of two different expression ofspectrum of preconditioned matrix is investigated.
Iterative methods are considered for solving nonlinear saddle point problems. Basedon the iterative methods for solving linear saddle point problems, several iterative methodsare proposed for solving nonlinear saddle point problems, and their convergence analysisare given, respectively. Numerical experiments are provided to reveal the performance ofthose proposed iterative methods.
Iterative methods are investigated for the algebraic Riccati equation. In fact, alge-braic Riccati equation arising in transport theory can be rewritten as a vector equation.Firstly, based on the relaxation technique and Newton method, a relaxed Newton-likemethod containing a relaxation parameter is proposed for solving the vector equation,and some convergence results are given. Secondly, using quasi-Newton idea and combin-ing the existing Newton type methods, two modified Newton type methods are presentedfor solving vector equation and some convergence results are obtained. The results of ex-periments show that our methods can effectively improve the convergence of the existingalgorithms.
研究求解线性鞍点问题的Uzawa类迭代法.首先,提出了一个修正的非线性Uzawa迭代法,讨论了算法的收敛性,并给出了理论和数值比较,数值实验也验证了修正方法的有效性.其次,给出了求解(2,2)块不为零的非对称广义鞍点问题的GMLHSS迭代方法,并探讨了算法的收敛条件.最后,给出了非精确Uzawa方法、GSSOR方法和MLHSS方法求解奇异鞍点系统时的半收敛性分析.
研究求解非对称鞍点问题的预处理技术.首先,给出了含参数的广义非精确块三角预条件子,对预处理矩阵的特征对性质给出了分析,并给出了预处理矩阵的特征值扰动分析.其次,基于系数矩阵的部分HS分裂和PS分裂,提出了PHSS预条件子和PPSS预条件子,并详细研究了预处理矩阵的谱性质,指出对于一个充分小的正参数,预处理矩阵特征值聚集在两个点附近:一个是(0,0)点,另一个是(2,0)点,并且通过大量的数值实验验证了理论分析和此两类预条件子的有效性.最后,给出了SIMPLE预条件子,利用特征值理论,研究了预处理矩阵两种不同表达形式的谱之间的联系.
研究求解非线性鞍点问题的迭代法.在求解线性鞍点问题的迭代解法基础上,给出了几个求解非线性鞍点问题的迭代法,并对方法进行了收敛性分析,数值实验验证了所提算法的有效性.
研究求解代数黎卡提方程的迭代法.事实上,输运理论中的代数黎卡提方程可以写成与之等价的向量方程.首先,基于松弛思想和牛顿方法,提出了求解向量方程的松弛Newton-like方法,并给出了算法的收敛性分析.其次,利用拟牛顿思想,结合已有的牛顿类型方法,给出了两个修正牛顿方法求解向量方程,并得到了算法的收敛结果.数值实验也表明所给出的三个算法能有效改进和提高已有算法的收敛性.
Large-scale sparse algebraic systems arise in a wide variety of applications through-out computational science and engineering fields, such as computational ?uid dynam-ics、constrained optimization、computational electromagnetics、mixed finite elementapproximations of PDEs、nonlinear optimization、transport theory and so on. Hence, ithas important theoretic and practical significance to study on the numerical methods forsolving this type of algebraic systems. Many large-scale sparse algebraic systems arisingfrom practical problems often have special structures. Therefore, the experts and schol-ars from home and abroad have been attracted on the research of numerical methods forsolving large sparse algebraic systems with special structure. This thesis represents deepand systematic research on the numerical methods for solving large-scale sparse algebraicsystems which have special structures, especially on the iterative methods and precondi-tioning techniques for linear saddle point problems, iterative methods for nonlinear saddleproblems and algebraic Riccati equation. This thesis is divided into four parts, includingsix chapters.
Uzawa type iterative methods are studied for solving linear saddle point problems.First, a modified nonlinear Uzawa iterative method is proposed and convergence of themodified method is discussed, and also theoretical and numerical comparison are given.Numerical experiments are provided to show the efficiency of the modified method. Sec-ond, a generalized modified local Hermitian and Skew-Hermitian splitting method is pre-sented for the generalized saddle point problems with nonzero (2, 2) block and conver-gence of the algorithm is discussed. Last, the inexact Uzawa method, the GSSOR methodand the MLHSS method are applied to solve singular saddle point systems and the semi-convergence analysis of these three methods are provided, respectively.
Preconditioning techniques are investigated for nonsymmetric saddle point prob-lems. Firstly, inexact block triangular preconditioners which contain a parameter arepresented, and eigenpair properties of the preconditioned matrix are analyzed. Further-more, eigenvalue perturbation analysis of the preconditioned matrix is shown. Secondly,on the basis of partial HS and PS splitting of coefficient matrix, PHSS preconditioner andPPSS preconditioner are presented and analyzed for nonsymmetric saddle point prob- lems, and also the spectral properties of the preconditioned matrix are studied in detail.Moreover, it can be concluded that the eigenvalue of the preconditioned matrix will gatherinto two clusters: one is near (0, 0) and the other is near (2, 0) when the iteration param-eter becomes small enough. In addition, the theoretical results and effectiveness of thetwo preconditioners are proved by a lot of numerical experiments. Finally, SIMPLE pre-conditioner is given. Using eigenvalue theory, the relation of two different expression ofspectrum of preconditioned matrix is investigated.
Iterative methods are considered for solving nonlinear saddle point problems. Basedon the iterative methods for solving linear saddle point problems, several iterative methodsare proposed for solving nonlinear saddle point problems, and their convergence analysisare given, respectively. Numerical experiments are provided to reveal the performance ofthose proposed iterative methods.
Iterative methods are investigated for the algebraic Riccati equation. In fact, alge-braic Riccati equation arising in transport theory can be rewritten as a vector equation.Firstly, based on the relaxation technique and Newton method, a relaxed Newton-likemethod containing a relaxation parameter is proposed for solving the vector equation,and some convergence results are given. Secondly, using quasi-Newton idea and combin-ing the existing Newton type methods, two modified Newton type methods are presentedfor solving vector equation and some convergence results are obtained. The results of ex-periments show that our methods can effectively improve the convergence of the existingalgorithms.
引文
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