论文总字数：12138字

摘 要

在二次优化，流体力学，线性弹性力学，加权最小二乘问题和计算科学与工程学领域中有很多问题可以被再生为鞍点问题。随着科技的高速发展以及鞍点问题在理论与实际中的广泛运用，求鞍点问题就变得越来越重要，受到众多研究者的关注与重视，本文主要探讨鞍点问题的数值算法。

对于求解鞍点问题，迭代法已经成为了最为广泛应用和最重要的一种方法。一种迭代方法的好坏是根据它的收敛速度而决定的，所以本文考虑了几种不同的迭代方法.在Uzawa方法的基础上进行迭代计算，并得出组合算法，提高收敛速度。

1. ，介绍了鞍点问题的目的和意义，详细介绍了鞍点问题的研究历史和研究成果，国内外不同的学者提出了许多的改性方法和新的算法。Uzawa经典迭代算法最早是K.Arrow,L.Hurwicz 和H.Uzawa提到的，这个方法最初是为了解决经济学的二次优化问题，由于容易用计算机来完成，很多人使用这种方法。但是它有个最大的缺点就是每一次迭代都要精确地计算逆矩阵。所以在这以后，更多的学者对这个算法进行更深层次的研究。所以需要和别的迭代法结合起来。
2. ，预备知识部分，这部分主要介绍了最主流的几种迭代方法，为了第三章做准备。通过对同一矩阵的不同分裂方式，得到不同的迭代方法，方便第三章在Uzawa方法的基础上再进行迭代。
3. ，详细的介绍了不同迭代法和Uzawa算法的组合方式，并且求出算法。着重介绍了本文的主题Uzawa-SOR算法及其比较定理，并验证了它的收敛性。这章是本文的主要部分，有具体的推导过程，和引理定理的应用和证明，其他组合算法也有详细的介绍。
4. ，这章是为了验证第三章而进行的数值实验，本文使用的是软件是MATLAB，将不同的算法的运行结果进行比较，通过分析结果来证明本文之前的理论分析。

关键词：鞍点问题 迭代方法 对称正定矩阵 Uzawa方法 SOR迭代法

A class of methods like Uzawa-SOR for solving saddle point problems

Abstract

In the second optimization, fluid mechanics, linear elastic mechanics, weighted least squares problems in computational science and engineering, many problems can be reproduced as saddle point problems. With the rapid development of science and technology and the application of saddle point problem in theory and practice, solving the saddle point problem becomes more and more important and attracts lots of attention, this paper mainly discusses the numerical algorithm for saddle point problems.

For solving the saddle point problem, iterative method has become the most widely used and most important method. An iterative approach is good or bad based on its convergence rate , so we consider several different iterative methods. On the basis of the Uzawa method, the iterative calculation is performed, and the combination algorithm is obtained to improve the convergence speed.

The purpose and significance of the saddle point are introduced in the first chapter. The detailed description of the history and achievements of the saddle point problem, different scholars have put forward a number of modified methods and new algorithms. Uzawa classical iterative algorithm was first mentioned by K.Arrow, L.Hurwicz and H.Uzawa, this method was originally intended to solve quadratic optimization problem of economics. Since it is easy to use a computer to complete , a lot of people use this method. But it has a biggest drawback is that each iteration should accurately calculate the inverse matrix. So after this, more and more scholars make a deeper study based on this algorithm. So it is necessary to combine it with other iterative methods.

The second chapter is the prior knowledge section, which describes the most popular iterative methods, this section is prepared for the third chapter. Use different ways to split the same matrix , then we can achieve different iterative methods ,its purpose is to iterate again in the Uzawa method .

The different combinations of iterative method and Uzawa algorithm are introduced in detail in third chapter, and the algorithm is obtained. This paper focuses on the Uzawa-SOR algorithm and its comparison theorem, and proves its convergence. This chapter is the main part of this paper, there is a specific derivation process, the application of lemma theorem and proof, Other combination algorithms are also described in detail.
The fourth chapter is to verify the third chapter of the numerical experiments,MATLAB has been chosen in this paper, The results of different algorithms are compared,and the theoretical analysis is proved by analyzing the results.

Keywords: iterative method; symmetric positive definite matrix; Uzawa method; SOR iterative method

目录

摘 要 I

Abstract II

第一章 引言 1

1.1 研究目的及意义 1

1.2 研究相关背景 1

1.3 本文的研究方法及创新点 2

第二章 问题简述和现有的迭代法 1

2.1 问题概述 1

2.2 线性方程组的常用迭代法 3

第三章 组合算法概述以及收敛分析 7

3.1 Uzawa-SOR方法 7

3.2 Uzawa-Jacobi方法 12

3.3 Uzawa-AOR方法 .............................................................................................13

第四章 数值实验 14

4.1 实验约束条件和方程 14

4.2 MATLAB数据分析 15

参考文献 16

致谢 17

第一章 引言

1.1 研究目的及意义

随着社会的进步和计算机的发展，鞍点问题大量出现在了例如流体力学，经济学，电磁学，图像重构和图像定位等计算机领域与工程学领域中，除此之外，鞍点问题还大量存在于带有约束条件的最优化问题和最小二乘问题中。这些都使得求解鞍点问题成为一个越来越重要的问题，受到了很多学者的重视，所以研究鞍点问题就具有重要的理论价值和实际意义。

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