本文關(guān)鍵詞：鞍點(diǎn)問題及復(fù)對(duì)稱線性系統(tǒng)迭代算法的研究　出處：《華東師范大學(xué)》2017年博士論文　論文類型：學(xué)位論文

  更多相關(guān)文章： 稀疏線性方程組 Uzawa算法 HSS算法 SOR算法 半收斂性 鞍點(diǎn)問題 復(fù)線性系統(tǒng) 奇異線性方程組 GMRES算法 譜半徑 擬譜半徑 Stokes方程

【摘要】：大型稀疏線性方程組的數(shù)值求解問題廣泛存在于電磁學(xué)問題,最小二乘問題,約束優(yōu)化問題及工程中數(shù)值模擬問題等,這些問題經(jīng)過有限元或有限差分等數(shù)值離散方法得到一些具有特殊結(jié)構(gòu)的大型稀疏線性方程組,如鞍點(diǎn)問題,復(fù)線性系統(tǒng)等.該論文主要針對(duì)幾類具有特殊結(jié)構(gòu)的大型稀疏線性方程組:奇異鞍點(diǎn)問題,非奇異鞍點(diǎn)問題和奇異復(fù)對(duì)稱線性系統(tǒng),給出幾種有效的迭代算法和預(yù)處理子,并給出相應(yīng)迭代方法的收斂性質(zhì)和數(shù)值實(shí)驗(yàn),具體如下:首先,針對(duì)奇異鞍點(diǎn)問題,提出了兩類含參數(shù)的不精確Uzawa方法:廣義含參數(shù)不精確Uzawa方法(GPIU)和廣義預(yù)處理含參數(shù)不精確Uzawa方法(GPPIU).首先分別介紹了這兩類方法的迭代格式,然后利用半收斂的定義給出其半收斂的充分條件.其次,結(jié)合Uzawa方法和SOR方法各自的優(yōu)點(diǎn),得到一類Uzawa-SOR迭代方法,分析給出了該方法半收斂的條件,然后通過相應(yīng)的數(shù)值算例驗(yàn)證它的有效性.然后針對(duì)非奇異鞍點(diǎn)問題,利用廣義SOR方法(GSOR)的特點(diǎn),給出推導(dǎo)該方法最優(yōu)參數(shù)的一個(gè)簡(jiǎn)單方法.然后,針對(duì)奇異鞍點(diǎn)問題,首先介紹了正則化的Hermitian和Skew-Hermitian分裂迭代方法(RHSS),然后分析得到該方法是無條件半收斂的.同時(shí),在分析的過程中,我們發(fā)現(xiàn)HSS方法求解奇異鞍點(diǎn)問題時(shí),也是無條件半收斂的,弱化了之前文章的結(jié)果.最后通過一系列的數(shù)值實(shí)驗(yàn)驗(yàn)證該方法的有效性和穩(wěn)定性.再次,針對(duì)非奇異鞍點(diǎn)問題,利用矩陣分裂方法,給出兩類預(yù)處理子:然后針對(duì)廣義鞍點(diǎn)問題,也給出了兩類有效的預(yù)處理子.分別對(duì)這四類預(yù)處理子給出了詳細(xì)的譜分析,他們具有較好的特征值聚集性質(zhì),最后通過一系列的數(shù)值實(shí)驗(yàn)驗(yàn)證這些預(yù)處理子的譜分布的情況及實(shí)際的有效性.最后,針對(duì)奇異復(fù)對(duì)稱線性系統(tǒng),我們將廣義修正的HSS算法(GMHSS)推廣到求解奇異線性系統(tǒng),詳細(xì)給出了半收斂分析,并得到了半收斂的條件.最后給出了詳細(xì)的數(shù)值實(shí)驗(yàn)結(jié)果,進(jìn)一步驗(yàn)證該算法的半收斂性和有效性.
[Abstract]:The numerical solutions of large sparse linear equations are widely used in electromagnetic problems, least-squares problems, constrained optimization problems and numerical simulation problems in engineering and so on. Some large sparse linear equations with special structure, such as saddle point problem, are obtained by finite element method or finite difference method. This paper focuses on several kinds of large sparse linear equations with special structure: singular saddle point problem, nonsingular saddle point problem and singular complex symmetric linear system. Several effective iterative algorithms and preconditioners are given, and the convergence properties and numerical experiments of the corresponding iterative methods are given. The main results are as follows: first, for the singular saddle point problem. Two kinds of imprecise Uzawa methods with parameters are proposed: the generalized imprecise Uzawa method with parameters and the generalized preprocessing Uzawa method with imprecise parameters. First, the iterative schemes of these two methods are introduced. Then the sufficient conditions of semi-convergence are given by using the definition of semi-convergence. Secondly, combining the advantages of Uzawa method and SOR method, a class of Uzawa-SOR iterative method is obtained. The condition of semi-convergence of the method is analyzed, and the validity of the method is verified by corresponding numerical examples. Then, the generalized SOR method is used to solve the nonsingular saddle point problem. A simple method to deduce the optimal parameters of this method is given. Then, the singular saddle point problem is discussed. Firstly, the regularized Hermitian and Skew-Hermitian splitting iterative methods are introduced, and the results show that the method is unconditionally semi-convergent. In the process of analysis, we find that the HSS method is also unconditionally semi-convergent in solving singular saddle point problems. Finally, the validity and stability of the method are verified by a series of numerical experiments. Thirdly, the matrix splitting method is used to solve the nonsingular saddle point problem. Two kinds of preconditioners are given: then for the generalized saddle point problem, two kinds of effective preconditioners are also given. The spectral analysis of the four preconditioners is given in detail, and they have better characteristic of eigenvalue aggregation. Finally, a series of numerical experiments are carried out to verify the spectral distribution of these preconditioners and their effectiveness. Finally, for singular complex symmetric linear systems. We generalize the generalized modified HSS algorithm GMHSS to solve singular linear systems, give the semi-convergence analysis in detail, and obtain the conditions of semi-convergence. Finally, the numerical results are given in detail. The semi-convergence and validity of the algorithm are further verified.
【學(xué)位授予單位】：華東師范大學(xué)
【學(xué)位級(jí)別】：博士
【學(xué)位授予年份】：2017
【分類號(hào)】：O241.6

【相似文獻(xiàn)】

相關(guān)期刊論文 前10條

1 萬維明,遲曉恒;廣義齊三次系統(tǒng)鞍點(diǎn)量問題[J];大連鐵道學(xué)院學(xué)報(bào);2001年03期

2 徐子珊;嚴(yán)格鞍點(diǎn)的查找算法[J];重慶工商大學(xué)學(xué)報(bào)(自然科學(xué)版);2004年05期

3 桑波;朱思銘;;焦點(diǎn)量與鞍點(diǎn)量的關(guān)系[J];數(shù)學(xué)年刊A輯(中文版);2007年02期

4 徐天博;李偉;;缺參數(shù)a_(23),b_(32)的齊五次系統(tǒng)的前四階鞍點(diǎn)量公式[J];大連交通大學(xué)學(xué)報(bào);2008年02期

5 趙景余;張國(guó)鳳;常巖磊;;求解鞍點(diǎn)問題的一種新的結(jié)構(gòu)算法[J];數(shù)值計(jì)算與計(jì)算機(jī)應(yīng)用;2009年02期

6 萬維明;周文;;齊四次系統(tǒng)鞍點(diǎn)量公式[J];大連交通大學(xué)學(xué)報(bào);2010年06期

7 葉惟寅;二次系統(tǒng)鞍點(diǎn)量的計(jì)算[J];南京師大學(xué)報(bào)(自然科學(xué)版);1987年02期

8 李文輝;;鞍點(diǎn)的穩(wěn)定性分析[J];沈陽化工學(xué)院學(xué)報(bào);1992年03期

9 遲曉恒;三次系統(tǒng)第一第二鞍點(diǎn)量計(jì)算公式[J];東北師大學(xué)報(bào)(自然科學(xué)版);1995年01期

10 謝佐恒;動(dòng)態(tài)系統(tǒng)中鞍點(diǎn)處的熵與分維[J];系統(tǒng)科學(xué)與數(shù)學(xué);1996年01期

相關(guān)會(huì)議論文 前2條

1 朱懷念;植t熀，

本文編號(hào)：1440975