兰州大学机构库 >数学与统计学院
Alternative TitleResearches on four numerical methods for complex sysmmetric, linear complementarity and linear discrete ill-posed problems
Thesis Advisor伍渝江 ; 杨爱利
Degree Grantor兰州大学
Place of Conferral兰州
Degree Name理学博士
Degree Discipline计算数学
Keyword复对称线性系统 线性互补问题 线性离散不适定问题 迭代方法 收敛性
Abstract本文主要针对三类大型稀疏线性系统的数值求解问题展开研究, 这三类线性系统 分别是复对称线性方程组、线性互补问题以及线性离散不适定问题. 对这些问题构造 快速高效的数值求解方法具有重要的理论价值和实际意义. 第二章中对于一类常见的复对称线性方程组, 我们将极小残量技术与修正的 Hermitian 和反 Hermitian 分裂 (MHSS) 迭代方法相结合, 提出了一种求解上述复对称 线性方程组的新的迭代格式, 将其称为极小残量的 MHSS (MRMHSS) 迭代方法. 与 经典的 MHSS 迭代方法相比, MRMHSS 迭代格式中多了两个迭代参数, 但是它们的 值可在迭代过程中方便地确定下来. 然后, 我们详细分析了 MRMHSS 迭代方法的理 论性质. 最后, 通过四个实际应用中常见的数值算例并通过与几类现有方法进行比较 验证了 MRMHSS 迭代方法的可行性和可靠性. 第三章中对于一类大型稀疏且具有非对称正定系数矩阵的线性互补问题, 我们将 该问题转换为与之等价的隐式不动点方程组, 然后给出一种高效的模系矩阵分裂迭代 方法, 称之为 MINPS 方法. 该方法由内外迭代组成, 其中, 外迭代借助于模迭代格式, 内迭代采用非精确计算方式对每步外迭代中的模迭代方程组实行预处理矩阵分裂迭 代技巧. 详细地分析了算法的收敛性质, 亦通过数值例子比较了 MINPS 与已有迭代 方法, 获得了所论算法求解线性互补问题的有效性和可行性. 对于科学计算和工程应用中广泛存在的线性离散不适定问题, LSQR 是解决这 类问题非常有效的方法之一, 它具有存储量小、数值稳定性好等优点. 但是, 考虑到 LSQR 的迭代解具有半收敛性, 即如果迭代步数太少那么迭代解不足以包含问题的 解的足够信息, 而迭代步数太多将导致迭代解积累大量的误差, 所以如何及时地停止 LSQR 迭代过程显得至关重要. 在第四章中, 我们通过提出一种简单有效的停止准则 进一步研究了 LSQR 迭代方法, 具体来说, 就是利用 LSQR 方法和 Craig 方法所得 迭代解的残差来判定 LSQR 的正则参数. 大量数值结果表明该方法能够很好地解决 测量数据中噪音水平未知的实际问题. 第五章中我们再次考虑了上述线性离散不适定问题, 它的解对数据的扰动非常敏 感, 通常使用正则化方法来降低解的这种敏感性. 基于 Donatelli 和 Hanke (2013) 提 出的迭代 Tikhonov 正则化方法 (AIT), 该方法中用一个易于运算的近似矩阵来近似 原矩阵, 从而能够减小计算量并对一些实际问题有很好的效果. 但是, AIT 方法的收 I敛条件在实际应用中很难满足且对数据扰动较为敏感, 为此, 我们提出了一种更加稳 定的迭代方法来求解线性离散不适定问题, 将该方法称为 MAIT. 文中对该方法的理 论性质和收敛情况做了细致的分析. 通过数值实验还发现, MAIT 方法比 AIT 方法的 适用范围更广泛, 特别当测量数据中误差水平较低时, AIT 会失效, 但 MAIT 方法仍 然可以有效地求解这类问题.
Other AbstractThis dissertation is mainly focused on the numerical solutions for three kinds of large sparse linear systems, namely complex symmetric system of linear equations, linear complementarity problem and linear discrete ill-posed problem. It is of great theoretical value and practical meaning to construct fast and efficient numerical methods for solving these linear systems. In Chapter 2, for solving a class of complex symmetric system of linear equations, we combine the minimum residual technique with the modified Hermitian and skew-Hermitian splitting (MHSS) iteration scheme and propose an iteration method referred to as minimum residual MHSS (MRMHSS) iteration method. Compared with the classical MHSS method, the MRMHSS method involves two more iteration parameters, which can be automatically and easily computed in practical implementations. Then, some properties of the MRMHSS iteration method are carefully studied. Finally, we use four examples to test the feasibility and reliability of MRMHSS iteration method by comparing its numerical results with several other iteration methods. In Chapter 3, for a class of large sparse linear complementarity problem with nonsymmetric positive definite coefficient matrix, by reformulating it as equivalent implicit fixed-point equations, we then establish a highly efficient modulus-based matrix splitting iteration method which will be referred to as MINPS iteration method. This method is made up of an inner iteration and an outer iteration, of which a modulus iteration is used as the outer iteration and the inner iteration applies a preconditioned matrix splitting iteration method with its inexact variant used to solve the module equations at each outer iteration. The convergence properties of the MINPS iteration method are carefully analyzed. By comparing the numerical results of MINPS with those of other existing iteration methods, we illustrate the efficiency and feasibility of our method when used for solving linear complementarity problems. For solving linear discrete ill-posed problems arising in many areas of scientific computation and engineering application, LSQR is one of the most popular numerical schemes, which needs small storage requirement and has well numerical stability. But since the LSQR has semi-convergence, i.e., too few iterations give an approximate solution that may lack many details that can be of interest, while too many steps yield an approximate solution that suffers from a large propagated error due to the error in the data, thus it is important to terminate the iterations after a suitable number of steps. In Chapter 4, we further study the LSQR iteration method with the aid of proposing a simple but efficient stopping criterion, precisely speaking, which is based on comparing the residual errors associated with iterates generated by the LSQR and Craig iterative methods to determine the regularization parameter. A great many numerical results show that the LSQR can perform well when applied to solve the practical problems of which the variance of the errors are not known. In Chapter 5, we again consider the above-mentioned linear discrete ill-posed problems, whose computing solutions are significantly sensitive to perturbations in the data and we usually apply a regularization method to reduce the sensitivity in practice. Based on an iterated Tikhonov regularization method (AIT) proposed by Donatelli and Hanke (2013), in which the original coefficient matrix is approximated by a proper matrix that is easier to work with, in this method the amount of computations are decreased and this method works well in some practical examples. Nevertheless, its convergence condition is rarely satisfied in practice and the computed solution is sensitive to the perturbation of the measured data. Thus, we propose a more stable iteration method called MAIT for solving linear discrete ill-pose problems, moreover, the theoretical properties and convergence results are studied in detail. From numerical experiments we even find that MAIT works more widely than AIT, specifically, when the noise level of measured data is low, AIT method fails, however, MAIT method can still efficiently solve this kind of problems.
Document Type学位论文
First Author AffilicationSchool of Mathematics and Statistics
Recommended Citation
GB/T 7714
张维红. 复对称问题、线性互补问题和线性离散不适定问题的四种数值解法研究[D]. 兰州. 兰州大学,2020.
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