东北大学学报:自然科学版  2017, Vol. 38 Issue (3): 452-456  
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邵新慧, 李晨, 王心怡. 求解特定鞍点问题的改进 SOR-Like方法[J]. 东北大学学报:自然科学版, 2017, 38(3): 452-456.
[复制中文]
SHAO Xin-hui, LI Chen, WANG Xin-yi. Modified SOR-Like Method for Saddle Point Problems[J]. Journal Of Northeastern University Nature Science, 2017, 38(3): 452-456. DOI: 10.3969/j.issn.1005-3026.2017.03.030.
[复制英文]

基金项目

国家自然科学基金资助项目 (11371081)

作者简介

邵新慧(1970-),女,山东青岛人,东北大学副教授。

文章历史

收稿日期: 2015-10-21
求解特定鞍点问题的改进 SOR-Like方法
邵新慧, 李晨, 王心怡    
东北大学 理学院, 辽宁 沈阳 110819
摘要: 鞍点问题广泛出现在众多的工程研究领域,如流体力学、电磁学、最优化问题、最小二乘问题、椭圆偏微分方程问题等.以SOR类方法为基础,结合HS分裂思想,将经典鞍点问题的求解方法推广到特殊鞍点问题的求解上.给出一种具有新型分裂迭代格式的MSOR-Like方法,用以求解一类含有非对称块的鞍点系统,给出了相应的收敛性分析以及最优松弛参数选取方法.数值算例验证了对于不同的预优矩阵,MSOR-Like方法只有收敛速度的分别,没有收敛性能的影响,且在相同计算精度下,该方法解决特殊鞍点问题的迭代效果优于常规方法解决经典鞍点问题.
关键词鞍点问题    迭代法    HS分裂    SOR方法    收敛    
Modified SOR-Like Method for Saddle Point Problems
SHAO Xin-hui, LI Chen, WANG Xin-yi    
School of Sciences, Northeastern University, Shenyang 110819, China
Corresponding author: SHAO Xin-hui,E-mail:xinhui1002@126.com
Abstract: Saddle point problems exist in many engineering research areas such as fluid mechanics, electromagnetism, optimization problems, the least squares problems, elliptic partial differential equations, and etc. Based on SOR-Like methods in combination of the concept of HS splitting, a new iteration splitting improvement method was presented so as to apply the classic saddle point solutions to special saddle point problems. Then, the MSOR-Like method was proposed to handle the above special saddle point system containing asymmetric blocks, and the convergence analysis as well as the selection of optimal relaxation parameters were also given. Finally, a numerical example was given to verify different optimal matrix of the modified SOR method, and it was found that the only difference is in the convergence rate while there is no difference in the convergence effect. Furthermore, under the same calculation accuracy, the modified SOR method for solving the special saddle point problems is better than the conventional methods in solving the classical saddle point problems.
Key Words: saddle point problem    iterative method    Hermitian and Skew-Hermitian (HS) splitting    SOR method    convergence    

在工程应用的诸多研究中,如涉及流体力学问题、电磁学问题、最优化问题等相关计算时,常常需要求解一类具有特殊结构的鞍点问题[1-5]

(1)

其中:ARm×m为非对称的正定矩阵;BRm×n(m≥n)为列满秩矩阵; x, fRm; y, gRn;BΤB的转置矩阵;f, g为已知向量.鞍点系统呈现一定病态,使得一些经典求解方法难以实现,因此如何行之有效地解决这类问题是近年来国内外学者们研究的热点.通常做法是采用定常迭代与非定常迭代双线并行,结合相应的预条件处理技术, 其中包括Uzawa类方法[6-9],SOR类方法[10-18],HSS类方法[19-21],Krylov子空间类方法[21],以及相应的预处理方法.

1 MSOR-Like方法的迭代格式和收敛性分析

基于HS分裂,令A=H+S, 其中.由于A为正定矩阵,则H也是正定的.作分裂M=D-L-U, 其中D=这里QRn×n为对称正定矩阵,若设ω>0为松弛参数,建立迭代格式:

其中:

(2)

整理得如下迭代算法:

(3)

将式(3)称之为求解鞍点问题的MSOR-Like方法.

若令λ为MSOR-Like方法迭代矩阵Mω的特征值,(x y)TRm+n为相应特征向量,则有

(4)

引理1 设ARm×m为非对称正定矩阵,为对称正定矩阵,为反对称矩阵,BRm×m(m≥n)为列满秩矩阵,Mω如式(2),λMω的任一特征值,则λ≠1.

证明  (反证法)若λ=1,由式(4)可得由于系数矩阵是非奇异阵,则有(x y)T=0,这与其为迭代矩阵Mω的特征向量矛盾,故λ≠1.

引理2  在引理1条件下,设λ为迭代矩阵Mω的任一特征值,(x y)TRm+nMω的相应特征向量,必有x≠0.若y=0,则|λ|<1.

证明  ① (反证法)若x=0,由式(4)可得By=0.由于B为列满秩矩阵,因而有y=0, 与(x y)T为迭代矩阵Mω的特征向量矛盾,故x≠0.② 若x≠0,y=0,由式(4)可得(1-ωλ)HxλωSx=0.用左乘等式两边,则有(1-ωλ则有(1-ωλ)aλωb=0.由于H是对称正定矩阵,则a>0S是反对称矩阵,则b=0.因而,有λ=1-ω.由于松弛参数需满足0<ω<2, 则有|λ|=|1-ω|<1.

引理3[1]  实一元二次方程λ2+c=0两根之模均小于1,当且仅当方程系数满足|c|<1,且 |b|<1+c.

定理1  设ARm×m为非对称的正定矩阵,为对称正定矩阵,为反对称矩阵,BRm×m(m≥n)为列满秩矩阵,QRn×n为适当的对称正定矩阵,则MSOR-Like方法迭代格式(3)收敛,当且仅当松弛参数满足 其中

证明 设λ为迭代矩阵Mω的任一特征值,Mω的相应特征向量.由于Q的非奇异性,由引理1知λ≠1,将式(4)等价变为

整理并同时左乘到等式两边,则有(1-=0.设则有(1-ωλ)aλωb.由于HBQ-1BT是对称正定矩阵,因而a>0,c>0.而 S是反对称矩阵,则b=0,于是有λ2+由引理3知,|λ|<1当且仅当由于a>0,则因而,若使MSOR-Like方法迭代格式收敛,松弛参数ω的范围需满足

推论1 在定理1条件下,若MSOR-Like方法迭代格式收敛,则松弛参数范围可进一步化简为 其中a=

证明  由于HBQ-1BΤ是对称正定矩阵,则分别为HBQ-1BΤRayleigh商.因而,

2 数值实验

考虑模型Stokes问题,有对流扩散方程:

其中:Ω为(0, 1)×(0, 1);方程满足Dirichlet边界条件;Δ为Laplace算子;u为表示速度的向量函数;p为表示压力的数值函数.

将上述问题用迎风差分格式离散化处理,可得形式如下的鞍点问题:

其中:

(-1, 1, 0)∈Rp×p, 为Kronecker量积符号,h=为离散网格值,且m=2p2n=p2.将模型中的T作适当修改,得到(-1.5, 2, -0.5)∈Rp×p.于是问题模型重新定义为

A=H+S, 其中(A-AΤ),有 其中

m=2p2n=p2.取零向量为初始向量,其精确解为(xΤ, yΤ)Τ=(1, 1, …,1)Τ.设IT为迭代步数,CPU为上机运算满足收敛要求所需时间,使用MATLAB编程语言实现,当ERR<10-6时计算中止, 其中ERR表示迭代误差,定义如下:

数值实验结果表 1~表 3表明,对于不同的预优矩阵Q,MSOR-Like方法只有收敛速度的区别,没有收敛性的影响.因而可以得出,在解决非对称鞍点问题上,该方法是切实有效的.

表 1 h=1/9(p=8, m=128, n=64)时,ωopt附近区域收敛情况(最优松弛参数ωopt=0.25) Table 1 Converging situation around optimal omega with h=1/9(p=8, m=128, n=64)(ωopt=0.25)
表 2h=1/9(p=8, m=128, n=64)时,ωopt附近区域收敛情况(最优松弛参数ωopt=1.00) Table 2 Converging situation around optimal omega with h=1/9(p=8, m=128, n=64)(ωopt=1.00)
表 3h=1/9(p=8, m=128, n=64)时,ωopt附近区域收敛情况(最优松弛参数ωopt=1.15) Table 3 Converging situation around optimal omega with h=1/9 (p=8, m=128, n=64) (ωopt=1.15)
3 结 语

本文以SOR-Like方法为基础,融合HS分裂思想,将经典鞍点问题的求解方法推广到特殊鞍点问题的求解,给出了一种具有新型分裂迭代格式的MSOR-Like方法,理论分析和数值算例验证了MSOR-Like方法在解决非对称鞍点问题时的有效性.

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