两类二次矩阵方程的数值求解方法
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摘要
二次矩阵方程在物理学、材料学、工程学、控制理论和科学计算等诸多领域有着广泛而深刻的应用.对其解的存在性研究和相应的数值求解方法不但在理论上具有重要意义而且在实际应用中也非常有价值.尤其近十几年随着计算机的飞速发展,非线性矩阵方程的数值解在工程控制领域和计算数学领域都逐渐发展成为了一个非常热门的课题.本文主要研究来自于物理中质量一弹簧系统的一类单边二次矩阵方程的数值求解问题和来自粒子转移理论中的非对称代数Riccati矩阵方程数值求解问题.
在第2章,我们研究来自于质量-弹簧系统的一类单边二次矩阵方程的数值求解问题.我们首先提出这一方程解存在的一个充分条件；其次根据方程系数矩阵的特点,我们提出一种保M-矩阵结构的加倍算法来计算方程的极端解；在适当的条件下,我们还证明该算法的单调收敛性和局部二次收敛性.我们的数值试验说明我们提出的算法要优于带精确线性搜索的牛顿法和伯努利迭代法.
在第3章,我们研究用循环约化算法来求解过阻尼系统产生的单边二次矩阵方程.与现有的二次收敛循环约化算法不同,我们提出一种三次收敛的循环约化算法.在过阻尼条件下我们证明所提出算法的适定性和收敛性.数值试验表明该算法在方程接近于过阻尼系统的临界状态时将比原来的循环约化算法具有更快的收敛性.
在第4章,我们继续研究循环约化算法的在临界状态过阻尼系统中的收敛性.Guo, Higham和Tisseur在假设临界过阻尼系统中按绝对值大小顺序排列的第n个特征值的部分重数(partial multiplicity)为2的条件下证明了循环约化算法的线性收敛性,而且算法产生的某些矩阵序列收敛于零矩阵.我们首先给出一个例子说明当上述假设条件不满足时,循环约化算法的收敛性与Guo等的收敛结论并不完全相同,即算法产生的相应的矩阵序列可以不收敛到零矩阵；其次在不需要对第n个的特征值部分重数做任何假设的条件下,我们对一类临界状态过阻尼系统证明循环约化算法的收敛性；最后通过数值试验验证本文的收敛性结果.
在第5章,我们研究来自粒子转移理论中的非对称代数Riccati矩阵方程数值求解问题.我们重新考虑用牛顿法和不动点迭代法来求得这一方程具有物理意义的最小正解.通过注意到牛顿法子问题的特殊矩阵结构,我们基于分解的交替方向隐式(Factored Alternating Direction Implicit, FADI)迭代设计一种低记忆低复杂度的牛顿法.随后我们进一步将这一思想拓展到不动点迭代方法的子问题从而提出了两种低记忆低复杂度的不动点迭代法.同时我们还证明这些算法在迭代过程中系数矩阵特征值和迭代点列所具有的良好性质.数值试验表明我们提出的算法能非常有效的求得这一非对称代数Riccati矩阵方程的最小正解.尤其在中等规模和大规模问题中,低记忆低复杂度的牛顿法要优于Bai等提出的NBGS算法和Bini等提出的快速牛顿法.
此博士论文得到了教育部重大项目(309023)和国家自然科学基金(11071087)的资助.
此博士论文用LATEX2ε软件打印.
Quadratic matrix equation and its related problems arise from many fields, such as physics, mechanism, engineering, optimal control theory, scientific com-puting and so on. The study in the existence of the solutions and the numerical methods are very important from theoretical view point as well as applications. In recent ten years, numerical methods for solving the quadratic matrix equation had become a very hot topic in engineering and computational mathematics. In this thesis, we shall focus on the numerical methods for finding the solutions of two classes of quadratic matrix equations, the unilateral quadratic matrix equa-tion arising from damped-mass system and the nonsymmetric algebraic Riccati equation arising from transport theory of physics.
In Chapter2, we are concerned with a unilateral quadratic matrix equation arising from the damped mass-spring system. We first give a sufficient condition for the existence of the solvents to the equation. We then develop a nonsingular M-matrix structure-preserving doubling algorithm (MSD) to calculate the extreme solvents of the equation. Under appropriate conditions, we establish the quadratic convergence of the proposed method. Numerical experiments show that the pro-posed MSD algorithm outperforms Newton's method with exact line searches and Bernoulli's method.
In Chapter3, we review the cyclic reduction (CR) algorithm for the unilat-eral quadratic matrix equation arising from the overdamped mass-spring system. Unlike the original CR algorithm, we propose a cubic cyclic reduction (CCR) al-gorithm to calculate the extremal solutions of this equation. The CCR algorithm is well defined under the overdamped condition. We also establish its convergence. Our preliminary numerical results show that the CCR algorithm is more effective for finding the extremal solutions than the original CR algorithm especially for the problem near the critical case.
In Chapter4, we study the convergence of the cyclic reduction method in the critical case of overdamped system. Guo, Higham and Tisseur has obtained the linear convergence of CR method under the assumption that the partial multiplic-ities of the n-th largest eigenvalue are all equal to2. Moreover, some generated matrix sequences converge to zero matrix. We will give an example to show the convergence of CR method is not the same with Guo et. al.'s result when their assumption on the partial multiplicities is not satisfied. In other words, the related matrix sequences may not necessary converge to zero matrix. We then prove the convergence of CR method for a class of overdamped system in the critical case without the assumption that the partial multiplicities of the n-th largest eigenvalue are all equal to2. The numerical example indicates that the convergence behavior of the CR algorithm is largely dictated by the theory we established.
In Chapter5, we consider the nonsymmetric algebraic Riccati equation arising from transport theory. We look at Newton's method and the fixed-point meth-ods for finding the minimal positive solution of this matrix equation from a new view point. We first rewrite the subproblem of Newton's method into an equivalent form with some special structure. By the use of the particular structure of the sub-problems, we present a low memory and low complexity version Newton's method with a factored alternating-direction-implicit iteration. We then extend this idea to some fixed-point methods to develop two low memory fixed-point methods. We also show some nice properties for the eigenvalues of the coefficient matrices of the subproblems. We do some numerical experiments to test the proposed methods and compare their performances with the recently developed NBGS method and fast Newton method. The results show that the proposed methods are highly effi-cient to obtain the minimal positive solution. The low memory and low complexity Newton's method is particularly efficient for solving large scale Riccati equation arising from transport theory.
This dissertation is supported by the major project of the Ministry of Edu-cation of China (309023) and the National Natural Science Foundation of China (11071087).
This dissertation is typeset by software LATEX2ε.
在第2章,我们研究来自于质量-弹簧系统的一类单边二次矩阵方程的数值求解问题.我们首先提出这一方程解存在的一个充分条件；其次根据方程系数矩阵的特点,我们提出一种保M-矩阵结构的加倍算法来计算方程的极端解；在适当的条件下,我们还证明该算法的单调收敛性和局部二次收敛性.我们的数值试验说明我们提出的算法要优于带精确线性搜索的牛顿法和伯努利迭代法.
在第3章,我们研究用循环约化算法来求解过阻尼系统产生的单边二次矩阵方程.与现有的二次收敛循环约化算法不同,我们提出一种三次收敛的循环约化算法.在过阻尼条件下我们证明所提出算法的适定性和收敛性.数值试验表明该算法在方程接近于过阻尼系统的临界状态时将比原来的循环约化算法具有更快的收敛性.
在第4章,我们继续研究循环约化算法的在临界状态过阻尼系统中的收敛性.Guo, Higham和Tisseur在假设临界过阻尼系统中按绝对值大小顺序排列的第n个特征值的部分重数(partial multiplicity)为2的条件下证明了循环约化算法的线性收敛性,而且算法产生的某些矩阵序列收敛于零矩阵.我们首先给出一个例子说明当上述假设条件不满足时,循环约化算法的收敛性与Guo等的收敛结论并不完全相同,即算法产生的相应的矩阵序列可以不收敛到零矩阵；其次在不需要对第n个的特征值部分重数做任何假设的条件下,我们对一类临界状态过阻尼系统证明循环约化算法的收敛性；最后通过数值试验验证本文的收敛性结果.
在第5章,我们研究来自粒子转移理论中的非对称代数Riccati矩阵方程数值求解问题.我们重新考虑用牛顿法和不动点迭代法来求得这一方程具有物理意义的最小正解.通过注意到牛顿法子问题的特殊矩阵结构,我们基于分解的交替方向隐式(Factored Alternating Direction Implicit, FADI)迭代设计一种低记忆低复杂度的牛顿法.随后我们进一步将这一思想拓展到不动点迭代方法的子问题从而提出了两种低记忆低复杂度的不动点迭代法.同时我们还证明这些算法在迭代过程中系数矩阵特征值和迭代点列所具有的良好性质.数值试验表明我们提出的算法能非常有效的求得这一非对称代数Riccati矩阵方程的最小正解.尤其在中等规模和大规模问题中,低记忆低复杂度的牛顿法要优于Bai等提出的NBGS算法和Bini等提出的快速牛顿法.
此博士论文得到了教育部重大项目(309023)和国家自然科学基金(11071087)的资助.
此博士论文用LATEX2ε软件打印.
Quadratic matrix equation and its related problems arise from many fields, such as physics, mechanism, engineering, optimal control theory, scientific com-puting and so on. The study in the existence of the solutions and the numerical methods are very important from theoretical view point as well as applications. In recent ten years, numerical methods for solving the quadratic matrix equation had become a very hot topic in engineering and computational mathematics. In this thesis, we shall focus on the numerical methods for finding the solutions of two classes of quadratic matrix equations, the unilateral quadratic matrix equa-tion arising from damped-mass system and the nonsymmetric algebraic Riccati equation arising from transport theory of physics.
In Chapter2, we are concerned with a unilateral quadratic matrix equation arising from the damped mass-spring system. We first give a sufficient condition for the existence of the solvents to the equation. We then develop a nonsingular M-matrix structure-preserving doubling algorithm (MSD) to calculate the extreme solvents of the equation. Under appropriate conditions, we establish the quadratic convergence of the proposed method. Numerical experiments show that the pro-posed MSD algorithm outperforms Newton's method with exact line searches and Bernoulli's method.
In Chapter3, we review the cyclic reduction (CR) algorithm for the unilat-eral quadratic matrix equation arising from the overdamped mass-spring system. Unlike the original CR algorithm, we propose a cubic cyclic reduction (CCR) al-gorithm to calculate the extremal solutions of this equation. The CCR algorithm is well defined under the overdamped condition. We also establish its convergence. Our preliminary numerical results show that the CCR algorithm is more effective for finding the extremal solutions than the original CR algorithm especially for the problem near the critical case.
In Chapter4, we study the convergence of the cyclic reduction method in the critical case of overdamped system. Guo, Higham and Tisseur has obtained the linear convergence of CR method under the assumption that the partial multiplic-ities of the n-th largest eigenvalue are all equal to2. Moreover, some generated matrix sequences converge to zero matrix. We will give an example to show the convergence of CR method is not the same with Guo et. al.'s result when their assumption on the partial multiplicities is not satisfied. In other words, the related matrix sequences may not necessary converge to zero matrix. We then prove the convergence of CR method for a class of overdamped system in the critical case without the assumption that the partial multiplicities of the n-th largest eigenvalue are all equal to2. The numerical example indicates that the convergence behavior of the CR algorithm is largely dictated by the theory we established.
In Chapter5, we consider the nonsymmetric algebraic Riccati equation arising from transport theory. We look at Newton's method and the fixed-point meth-ods for finding the minimal positive solution of this matrix equation from a new view point. We first rewrite the subproblem of Newton's method into an equivalent form with some special structure. By the use of the particular structure of the sub-problems, we present a low memory and low complexity version Newton's method with a factored alternating-direction-implicit iteration. We then extend this idea to some fixed-point methods to develop two low memory fixed-point methods. We also show some nice properties for the eigenvalues of the coefficient matrices of the subproblems. We do some numerical experiments to test the proposed methods and compare their performances with the recently developed NBGS method and fast Newton method. The results show that the proposed methods are highly effi-cient to obtain the minimal positive solution. The low memory and low complexity Newton's method is particularly efficient for solving large scale Riccati equation arising from transport theory.
This dissertation is supported by the major project of the Ministry of Edu-cation of China (309023) and the National Natural Science Foundation of China (11071087).
This dissertation is typeset by software LATEX2ε.
引文
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