解不可压缩Navier-Stokes方程的非精确块因子分解预处理子

针对相关于不可压缩Navier-Stokes方程数值求解的一类3×3块结构的线性方程组,基于线性方程组的等价形式,构造了一个非精确的块因子分解预处理子,在新的特征值等价矩阵形式的基础上,得到了预处理矩阵特征值实部和虚部的上下界估计.数值实验表明,与已有的预处理子相比,所构造的预处理子可以使得GMRES迭代方法对网格尺寸,网格形式以及粘度系数的依赖性都比较弱,且在迭代步数和CPU时间上都占优.     

关于H-矩阵的H-预处理子（英文）

设A为一实对称正定的严格对角占优矩阵.设A=D-B为A的Jacobi分裂.为了求解线性方程组Ax=b,在新提出的预处理子的基础上,我们采用预处理共轭梯度方法(PCG)来求解该问题.新提出的预处理子Pv=D+νvv~T,其中v=|B|e,e=(1,...,1)~T,ν=v~TBv/||v||_2~4,且ν使||cvv~T-B||_F达到极小.我们得到了预处理矩阵P_v~(-1)A特征值的上下界,它的界比JIN提出的预处理子的界简单紧凑.数值结果表明我们的预处理子的有效性.     

椭圆PDE-约束优化问题的一个预条件子

针对由Galerkin有限元离散椭圆PDE-约束优化问题产生的具有特殊结构的3×3块线性鞍点系统,提出了一个预条件子并给出了预处理矩阵特征值及特征向量的具体表达形式.数值结果表明了该预条件子能够有效地加速Krylov子空间方法的收敛速率,同时也验证了理论结果.     

鞍点问题基于半增广的松弛分裂预条件子

对于(1,1)块为正定的鞍点问题,本文给出了半增广松弛分裂预条件子.文中分析了预条件矩阵特征值分布情况,并用数值实验验证了半增广松弛分裂预条件子的有效性.     

一类Hermitian鞍点矩阵的特征值估计

本文研究了一类大型稀疏Hermitian鞍点线性系统Az=(B E E* 0)(x y)=(f g)=b系数矩阵的特征值,其中B∈C~(p×p)是Hermitian正定阵矩阵,E∈C~(p×q)是列降秩.本文分别给出了该系数矩阵正特征值与负特征值界的一个估计式,同时通过数值算例验证本文所给出的特征值界的估计是合理且有效的.     

速度追踪问题中鞍点系统的新分裂迭代

 有限元离散一类速度追踪问题后得到具有鞍点结构的线性系统,针对该鞍点系统,本文提出了一种新的分裂迭代技术.证明了新的分裂迭代方法的无条件收敛性,详细分析了新的分裂预条件子对应的预处理矩阵的谱性质.数值结果验证了对于大范围的网格参数和正则参数,新的分裂预条件子在求解有限元离散速度追踪问题得到的鞍点系统时的可行性和有效性.     

非对称鞍点问题的块三角预处理方法

本文研究了非对称广义鞍点问题的不完全块上三角预处理方法,特别是对于(1,2)块不等于(2,1)块的转置的情况,利用矩阵扰动技术给出了预处理后矩阵的特征值分布情况.并由数值试验验证了结果的正确性和有效性.     

速度追踪问题产生的鞍点系统的新的分裂迭代技术

有限元离散一类速度追踪问题后得到具有鞍点结构的线性系统,针对该鞍点系统,本文提出了一种新的分裂迭代技术.证明了新的分裂迭代方法的无条件收敛性,详细分析了新的分裂预条件子对应的预处理矩阵的谱性质.数值结果验证了对于大范围的网格参数和正则参数,新的分裂预条件子在求解有限元离散速度追踪问题得到的鞍点系统时的可行性和有效性.     

图的无符号拉普拉斯特征值α次幂总和的界北大核心CSCD

令G为简单图.sα（G）等于图G的无符号拉普拉斯特征值α次幂的总和,其中α为实数且α≠0,1.本文我们得到一些连通图的sα（G）的新的界,并给出了正则图的Mycielskian图、正则图及半正则二部图的Double图这些特殊图类的sα（G）的新的界.由这些结论的特殊情况可得到相应图的关联能量的界.     

一类非对称结构线性方程组的子结构预处理子(英文)

针对一类具结构的非对称线性方程组提出了一类子结构预处理子,该预处理子只保留了约束条件的一半项.研究表明,预处理矩阵只有三个离散的特征值.为了避免计算Schur补的逆,还给出了正则化的子结构预处理子,同样对预处理矩阵进行了谱分析.这些结果将Zhou和Niu(Zhou J T,Niu Q.Substructure preconditioners for a class of structuredlinear systems of equations.Math.Comput.Model.,2010,52:1547-1553)的结果推广到非对称结构线性方程组.数值算例验证了提出的子结构预处理子的有效性.     

On parameterized block triangular preconditioners for generalized saddle point problems

In this paper, we discuss two classes of parameterized block triangular preconditioners for the generalized saddle point problems. These preconditioners generalize the common block diagonal and triangular preconditioners. We will give distributions of the eigenvalues of the preconditioned matrix and provide estimates for the interval containing the real eigenvalues. Numerical experiments of a model Stokes problem are presented.     

Shift-splitting preconditioners for a class of block three-by-three saddle point problems

In this paper, shift-splitting preconditioners are studied for a special class of block three-by-three saddle point problems, which arise from many practical problems and are different from the traditional saddle point problems. It is proved that the block three-by-three saddle point matrix is positive stable and the corresponding shift-splitting stationary iteration method is unconditionally convergent, which leads to a nice clustering property of the eigenvalues of the shift-splitting preconditioned matrix. Numerical results show that the proposed shift-splitting preconditioners outperform much better than some existing block diagonal preconditioners studied recently.     

Spectral properties of primal-based penalty preconditioners for saddle point problems

For large and sparse saddle point linear systems, this paper gives further spectral properties of the primal-based penalty preconditioners introduced in [C.R. Dohrmann, R.B. Lehoucq, A primal-based penalty preconditioner for elliptic saddle point systems, SIAM J. Numer. Anal. 44 (2006) 270-282]. The regions containing the real and non-real eigenvalues of the preconditioned matrix are obtained. The model of the Stokes problem is supplemented to illustrate the theoretical results and to test the quality of the primal-based penalty preconditioner.     

Optimization of the parameterized Uzawa preconditioners for saddle point matrices

The parameterized Uzawa preconditioners for saddle point problems are studied in this paper. The eigenvalues of the preconditioned matrix are located in (0, 2) by choosing the suitable parameters. Furthermore, we give two strategies to optimize the rate of convergence by finding the suitable values of parameters. Numerical computations show that the parameterized Uzawa preconditioners can lead to practical and effective preconditioned GMRES methods for solving the saddle point problems.     

The spectral properties of the preconditioned matrix for nonsymmetric saddle point problems

In this paper, on the basis of matrix splitting, two preconditioners are proposed and analyzed, for nonsymmetric saddle point problems. The spectral property of the preconditioned matrix is studied in detail. When the iteration parameter becomes small enough, the eigenvalues of the preconditioned matrices will gather into two clusters—one is near (0,0) and the other is near (2,0)—for the PPSS preconditioner no matter whether A is Hermitian or non-Hermitian and for the PHSS preconditioner when A is a Hermitian or real normal matrix. Numerical experiments are given, to illustrate the performances of the two preconditioners.     

On HSS and AHSS iteration methods for nonsymmetric positive definite Toeplitz systems

Two iteration methods are proposed to solve real nonsymmetric positive definite Toeplitz systems of linear equations. These methods are based on Hermitian and skew-Hermitian splitting (HSS) and accelerated Hermitian and skew-Hermitian splitting (AHSS). By constructing an orthogonal matrix and using a similarity transformation, the real Toeplitz linear system is transformed into a generalized saddle point problem. Then the structured HSS and the structured AHSS iteration methods are established by applying the HSS and the AHSS iteration methods to the generalized saddle point problem. We discuss efficient implementations and demonstrate that the structured HSS and the structured AHSS iteration methods have better behavior than the HSS iteration method in terms of both computational complexity and convergence speed. Moreover, the structured AHSS iteration method outperforms the HSS and the structured HSS iteration methods. The structured AHSS iteration method also converges unconditionally to the unique solution of the Toeplitz linear system. In addition, an upper bound for the contraction factor of the structured AHSS iteration method is derived. Numerical experiments are used to illustrate the effectiveness of the structured AHSS iteration method.     

Spectral Analysis for HSS Preconditioners

In this paper,we are interested in HSS preconditioners for saddle point lin- ear systems with a nonzero(2,2)-th block.We study an approximation of the spectra of HSS preconditioned matrices and use these results to illustrate and explain the spectra obtained from numerical examples,where the previous spectral analysis of HSS precon- ditioned matrices does not cover.     

On the eigenvalue distribution of preconditioned nonsymmetric saddle point matrices

In this paper, we derive bounds for the complex eigenvalues of a nonsymmetric saddle point matrix with a symmetric positive semidefinite (2,2) block, that extend the corresponding previous bounds obtained by Bergamaschi. For the nonsymmetric saddle point problem, we propose a block diagonal preconditioner for the conjugate gradient method in a nonstandard inner product. Numerical experiments are also included to test the performance of the presented preconditioner. Copyright © 2013 John Wiley & Sons, Ltd.     

Unified analysis of preconditioning methods for saddle point matrices

Short and unified proofs of spectral properties of major preconditioners for saddle point problems are presented. The need to sufficiently accurately construct approximations of the pivot block and Schur complement matrices to obtain real eigenvalues or eigenvalues with positive real parts and non‐dominating imaginary parts are pointed out. The use of augmented Lagrangian methods for more ill‐conditioned problems are discussed. Copyright © 2014 John Wiley & Sons, Ltd.     

Analysis of some projection method based preconditioners for models of incompressible flow

In this paper, several projection method based preconditioners for various incompressible flow models are studied. In the derivations of these projection method based preconditioners, we use three different types of the approximations of the inverse of the Schur complement, i.e., the exact inverse, the Cahouet–Chabard type approximation and the BFBt type approximation. We illuminate the connections and the distinctions between these projection method based preconditioners and those related preconditioners. For the preconditioners using the Cahouet–Chabard type approximation, we show that the eigenvalues of the preconditioned systems have uniform bounds independent of the parameters and most of them are equal to 1. The analysis is based on a detailed discussion of the commutator difference operator. Moreover, these results demonstrate the stability of the staggered grid discretization and reveal the effects of the boundary treatment. To further illustrate the effectiveness of these projection method based preconditioners, numerical experiments are given to compare their performances with those of the related preconditioners. Generalizations of the projection method based preconditioners to other saddle point problems are also discussed.