一类特殊的广义鞍点问题的下三角迭代方法 Alternative Title Low triangular splitting iteration method for a class of special generalized saddle point problem 郑重 Thesis Advisor 张国凤 2013-05-19 Degree Grantor 兰州大学 Place of Conferral 兰州 Degree Name 硕士 Keyword 广义鞍点问题 PMHSS方法 LTS方法 收敛性 最优迭代参数 PDE约束优化问题 GMRES Abstract 鞍点问题在科学研究与工程计算的很多领域都有广泛的应用, 如约束加权最小二乘估计,约束最优化问题,计算流体力学,经济学,图像处理,椭圆偏微分方程的混合有限元近似问题, 优化控制等. 由于这个问题具有广泛的应用资源和价值, 所以研究快速而有效的方法是具有重要的理论意义和广泛的应用价值. 本文针对一类应用到复线性方程组及离散控制问题等价转化到具有特殊形式块2×2广义鞍点问题基于预处理修正对称与反对称分裂(PMHSS)迭代方法的稳定性质构造下三角分裂(LTS)迭代方法, 并对该方法的收敛性进行分析, 给出LTS迭代方法的收敛条件并进一步研究了在算法收敛情况下的最优迭代参数及其相应的最优收敛因子. 最后将该方法分别应用到复线性方程组及离散控制问题中去, 数值试验结果说明了LTS迭代方法选择适当的参数去求解这类特殊的广义鞍点问题比PMHSS及Krylov子空间方法如GMRES(♯)具有更好的收敛性质. 表明了该迭代算法去求解这类特殊的广义鞍点问题的可行性和有效性. Other Abstract Saddle point problems are widely involved in many areas of scientific research and engineering computations, such as constrained and weighted least squares estimation, constrained optimization, computational fluid dynamics, economics, imageregistration, mixed finite element approximation of elli ptic partial differential equations, optimal control and so on. It is interesting to develop fast and efficient methods as these problems have such a wide application source and value. Based on the preconditioned MHSS (PMHSS) method, we construct a lower-triangular splitting (LTS) iteration method scheme for solving a class of block two-by-two linear systems and apply it to the complex linear systems and the distributed control problem. Under suitable restrictions on the iteration parameters, we prove the convergence of the LTS iteration method, moreover determine its optimal iteration parameters and the corresponding optimal convergence factor. Numerical implementations show that the resulting of LTS iteration method leads to faster convergence rate than the PMHSS iteration method and Krylov subspace iteration method such as GMRES and its restarted variants, which imply the feasibility of the new iteration method. URL 查看原文 Language 中文 Document Type 学位论文 Identifier https://ir.lzu.edu.cn/handle/262010/224446 Collection 数学与统计学院 Recommended CitationGB/T 7714 郑重. 一类特殊的广义鞍点问题的下三角迭代方法[D]. 兰州. 兰州大学,2013.
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