广义鞍点问题的松弛维数分解预条件子

本文将Benzi等提出的松弛维数分解(Relaxed dimensionalfactorization, RDF)预条件子进一步推广到广义鞍点问题上,并称为GRDF(Generalized RDF)预条件子.该预条件子可看做是用维数分裂迭代法求解广义鞍点问题而导出的改进维数分裂(Modified dimensional split, MDS)预条件子的松弛形式, 它相比MDS预条件子更接近于系数矩阵, 因而结合Krylov子空间方法(如GMRES)有更快的收敛速度.文中分析了GRDF预处理矩阵特征值的一些性质,并用数值算例验证了新预条件子的有效性.     

广义鞍点问题基于PSS的约束预条件子

对于(1,1)块为非Hermitian阵的广义鞍点问题,本文给出了一种基于正定和反对称分裂(Positive definite andskew-Hermitian splitting, PSS)的约束预条件子.该预条件子的(1,1)块由求解非Hermitian正定线性方程组时的PSS迭代法所构造得到.文中分析了PSS约束预条件子的一些性质并证明了预处理迭代法的收敛性.最后用数值算例验证了该预条件子的有效性.     

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.     

H-矩阵线性方程组的一类预条件并行多分裂SOR迭代法

基于并行多分裂算法的思想及SOR迭代格式, 本文提出一种求解H-矩阵线性方程组新的并行多分裂SOR迭代法, 新方法某种程度上避免了SOR迭代法中选取最优参数的困难. 同时, 选取Kohno等(1997)提出的预条件子$P=I+S_{\alpha}$对原始线性方程组进行预处理, 进而给出了一种实用的预条件并行多分裂SOR迭代法. 理论分析和数值实验均表明, 新算法是实用而有效的.     

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.     

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.     

稀疏近似逆与多层块ILU预条件技术

设计了一种求解一般稀疏线性方程组的健壮且有效的可并行化预条件子,这种预条件子涉及在多层块ILU预条件子(BILUM)中使用稀疏近似逆(AINV)技术．所得的预条件子保持了BILUM的健壮性,它比标准的BILUM预条件子有两点优势:控制稀疏性的能力和增强了并行性．数值例子显示了新预条件子的有效性和效率．     

一种求解H(curl)型椭圆问题的非重叠DDM预条件子

本文对一类H(curl)型椭圆问题的线性棱有限元方程,构造了一种具有简单粗空间的非重叠区域分解型预条件子.在系数为常数的情形下,严格证明了相应的预条件系统的条件数为O(log2d/h),即是渐近最优的.数值实验验证了理论结果的正确性,并说明该预条件子对大跳系数的情形也是有效的.     

实对称正定Toeplitz矩阵的带位移的Sine预处理子

本文研究了求解实对称正定Toeplitz线性方程组的预处理共轭梯度法.基于实对称Toeplitz矩阵都有一个三角变换分裂(TTS)的事实,我们提出了带位移的Sine预处理子TS,分析了预处理矩阵的谱性质,并讨论了每步迭代的计算复杂度.数值实验表明该预处理子比T.Chan预处理子~([2])更有效.     

一类预条件AOR迭代法的比较定理(英文)

本文研究了当线性方程组的系数矩阵是严格对角占优L-矩阵时带有预条件子P1→kα的预条件AOR迭代方法.利用矩阵分裂的相关理论,获得了预条件AOR迭代法的收敛性结论以及参数α和k对收敛速度影响的比较定理.结果表明当α和k取值较大时这类预条件方法更加有效.文中的结论推广了Li等人关于预条件Gauss-Seidel迭代法的相关结论.最后,用数值例子进一步验证了这些结果.     

A SHIFT-SPLITTING PRECONDITIONER FOR NON-HERMITIAN POSITIVE DEFINITE MATRICES

A shift splitting concept is introduced and, correspondingly, a shift-splitting iteration scheme and a shift-splitting preconditioner are presented, for solving the large sparse system of linear equations of which the coefficient matrix is an ill-conditioned non-Hermitian positive definite matrix. The convergence property of the shift-splitting iteration method and the eigenvalue distribution of the shift-splitting preconditioned matrix are discussed in depth, and the best possible choice of the shift is investigated in detail. Numerical computations show that the shift-splitting preconditioner can induce accurate, robust and effective preconditioned Krylov subspace iteration methods for solving the large sparse non-Hermitian positive definite systems of linear equations.     

A relaxed positive semi-definite and skew-Hermitian splitting preconditioner for non-Hermitian generalized saddle point problems

For non-Hermitian saddle point linear systems, Pan, Ng and Bai presented a positive semi-definite and skew-Hermitian splitting (PSS) preconditioner (Pan et al. Appl. Math. Comput. 172, 762–771 2006), to accelerate the convergence rate of the Krylov subspace iteration methods like the GMRES method. In this paper, a relaxed positive semi-definite and skew-Hermitian (RPSS) splitting preconditioner based on the PSS preconditioner for the non-Hermitian generalized saddle point problems is considered. The distribution of eigenvalues and the form of the eigenvectors of the preconditioned matrix are analyzed. Moreover, an upper bound on the degree of the minimal polynomial is also studied. Finally, numerical experiments of a model Navier-Stokes equation are presented to illustrate the efficiency of the RPSS preconditioner compared to the PSS preconditioner, the block diagonal preconditioner (BD), and the block triangular preconditioner (BT) in terms of the number of iteration and computational time.     

On regularized Hermitian splitting iteration methods for solving discretized almost‐isotropic spatial fractional diffusion equations

The shifted finite‐difference discretization of the one‐dimensional almost‐isotropic spatial fractional diffusion equation results in a discrete linear system whose coefficient matrix is a sum of two diagonal‐times‐Toeplitz matrices. For this kind of linear systems, we propose a class of regularized Hermitian splitting iteration methods and prove its asymptotic convergence under mild conditions. For appropriate circulant‐based approximation to the corresponding regularized Hermitian splitting preconditioner, we demonstrate that the induced fast regularized Hermitian splitting preconditioner possesses a favorable preconditioning property. Numerical results show that, when used to precondition Krylov subspace iteration methods such as generalized minimal residual and biconjugate gradient stabilized methods, the fast preconditioner significantly outperforms several existing ones.     

A modified positive-definite and skew-Hermitian splitting preconditioner for generalized saddle point problems from the Navier-Stokes equation

In this paper, we extend the relaxed positive-definite and skew-Hermitian splitting preconditioner (RPSS) for generalized saddle-point problems in [J.-L. Zhang, C.-Q. Gu and K. Zhang, Appl. Math. Comput. 249(2014)468-479] by introducing an additional parameter. The spectral properties of the presented new preconditioned matrix for generalized saddle-point problem are investigated, meanwhile, the infinite termination merit of the iterative step is also discussed if the Krylov subspace method preconditioned by the modified positive-definite and skew-Hermitian splitting preconditioner (MPSS) is applied. Some numerical experiments illustrate that the efficiency of the proposed new preconditioner.     

Regularized DPSS Preconditioners for Singular Saddle Point Problems

Recently, Cao proposed a regularized deteriorated positive and skew-Hermitian splitting (RDPSS) preconditioner for the non-Hermitian nonsingular saddle point problem. In this paper, we consider applying RDPSS preconditioner to solve the singular saddle point problem. Moreover, we propose a two-parameter accelerated variant of the RDPSS (ARDPSS) preconditioner to further improve its efficiency. Theoretical analysis proves that the RDPSS and ARDPSS methods are semi-convergent unconditionally. Some spectral properties of the corresponding preconditioned matrices are analyzed. Numerical experiments indicate that better performance can be achieved when applying the ARDPSS preconditioner to accelerate the GMRES method for solving the singular saddle point problem.     

Improved splitting preconditioner for double saddle point problems arising from liquid crystal director modeling

Numerical Algorithms - To improve the performance of alternating positive semidefinite splitting (APSS) preconditioner, we present an improved APSS (IAPSS) preconditioner for the double saddle...     

A new block preconditioner for complex symmetric indefinite linear systems

Using the equivalent block two-by-two real linear systems and relaxing technique, we establish a new block preconditioner for a class of complex symmetric indefinite linear systems. The new preconditioner is much closer to the original block two-by-two coefficient matrix than the Hermitian and skew-Hermitian splitting (HSS) preconditioner. We analyze the spectral properties of the new preconditioned matrix, discuss the eigenvalue distribution and derive an upper bound for the degree of its minimal polynomial. Finally, some numerical examples are provided to show the effectiveness and robustness of our proposed preconditioner.