A new matrix splitting preconditioner for generalized saddle point problems

For generalized saddle point problems, we establish a new matrix splitting preconditioner and give the implementing process in detail. The new preconditioner is much easier to be implemented than the modified dimensional split (MDS) preconditioner. The convergence properties of the new splitting iteration method are analyzed. The eigenvalue distribution of the new preconditioned matrix is discussed and an upper bound for the degree of its minimal polynomial is derived. Finally, some numerical examples are carried out to verify the effectiveness and robustness of our preconditioner on generalized saddle point problems discretizing the incompressible Navier–Stokes equations.     

On parameterized matrix splitting preconditioner for the saddle point problems

In this paper, we present a parameterized matrix splitting (PMS) preconditioner for the large sparse saddle point problems. The preconditioner is based on a parameterized splitting of the saddle point matrix, resulting in a fixed-point iteration. The convergence theorem of the new iteration method for solving large sparse saddle point problems is proposed by giving the restrictions imposed on the parameter. Based on the idea of the parameterized splitting, we further propose a modified PMS preconditioner. Some useful properties of the preconditioned matrix are established. Numerical implementations show that the resulting preconditioner leads to fast convergence when it is used to precondition Krylov subspace iteration methods such as generalized minimal residual method.     

Improving the preconditioning of linear systems from interior point methods

This paper deals with preconditioners for solving linear systems arising from interior point methods, using iterative methods. The main focus is the development of a set of results that allows a more efficient computation of the splitting preconditioner. During the interior point methods iterations, the linear system matrix becomes ill conditioned, leading to numerical difficulties to find a solution, even with iterative methods. Therefore, the choice of an effective preconditioner is essential for the success of the approach. The paper proposes a new ordering for a splitting preconditioner, taking advantage of the sparse structure of the original matrix. A formal demonstration shows that performing this new ordering the preconditioned matrix condition number is limited; numerical experiments reinforce the theoretical results. Case studies show that the proposed idea has better sparsity features than the original version of the splitting preconditioner and that it is competitive regarding the computational time.     

A variant of the HSS preconditioner for complex symmetric indefinite linear systems

Using the equivalent block two-by-two real linear systems, we establish a new variant of the Hermitian and skew-Hermitian splitting (HSS) preconditioner for a class of complex symmetric indefinite linear systems. The new preconditioner is not only a better approximation to the block two-by-two real coefficient matrix than the well-known HSS preconditioner, but also resulting in an unconditional convergent fixed-point iteration. The quasi-optimal parameter, which minimizes an upper bound of the spectral radius of the iteration matrix, is analyzed. Eigen-properties and an upper bound of the degree of the minimal polynomial of the preconditioned matrix are discussed. Finally, two numerical examples are provided to show the efficiency of the new preconditioner.     

Convergence analysis of modified PGSS methods for singular saddle-point problems

Recently, variants of shift-splitting iteration method have been proposed for solving singular saddle-point problems. However, these methods can only be proved to converge to one of the solutions of the consistent singular linear system, not knowing any further information about this solution. In this work, we consider a modified preconditioned generalized shift-splitting (MPGSS) iteration method for solving both consistent and inconsistent singular saddle-point problems. This method is proved to converge to the best least squares solution. Moreover, based on the iteration form, a preconditioner is obtained to accelerate Krylov subspace methods. Theoretical analysis shows that the preconditioned GMRES method also converges to the best least squares solution of the consistent singular saddle-point problem. In addition, numerical results are presented to show the effectiveness and robustness of the proposed iteration method and preconditioner.     

A general class of shift-splitting preconditioners for non-Hermitian saddle point problems with applications to time-harmonic eddy current models

Based on a general splitting of the (1,1) leading block matrix, we first construct a general class of shift-splitting (GCSS) preconditioners for non-Hermitian saddle point problems. Convergence conditions of the corresponding matrix splitting iteration methods and preconditioning properties of the GCSS preconditioned saddle point matrices are analyzed. Then the GCSS preconditioner is specifically applied to the non-Hermitian saddle point problems arising from the finite element discretizations of the hybrid formulations of the time-harmonic eddy current models. With suitable choices of the splittings, the new GCSS preconditioners are easier to implement and have faster convergence rates than the existing shift-splitting preconditioner and its modified variant. Two numerical examples are presented to verify the theoretical results and show effectiveness of the new proposed preconditioners.     

A symmetric positive definite preconditioner for saddle-point problems

For the classical saddle-point problem, we present precisely two intervals containing the positive and the negative eigenvalues of the preconditioned matrix, respectively, when the inexact version of the symmetric positive definite preconditioner introduced in Section 2.1 of Gill et al. [Preconditioners for indefinite systems arising in optimization, SIAM J. Matrix Anal. Appl. 13 (1992), pp. 292–311] is employed. The model of Stokes problem is used to test the effectiveness of the presented bounds as well as the quality of the symmetric positive definite preconditioner.     

求解Maxwell线性棱元鞍点系统的并行Uzawa算法

本文针对一类Maxwell方程组鞍点问题的第一类N啨d啨lec线性棱元离散系统,设计了一种基于节点辅助空间预条件子的并行Uzawa算法(HX-Uzawa-p)。数值实验结果表明,不论是对光滑系数还是对有无浮动子区域及有无内交叉点的跳系数情形,我们所设计的并行算法HX-Uzawa-p的迭代次数都基本不依赖于网格规模及系数跳幅,且具有很好的并行可扩展性。     

求解Maxwell线性元鞍点系统的基于HX预条件子的Uzawa算法

首先对含跳系数的H~1型和H(curl)型椭圆问题的线性有限元方程,分别设计了基于AMG预条件子和基于节点辅助空间预条件子(HX预条件子)的PCG法.数值实验表明,算法的迭代次数基本不依赖于系数跳幅和离散网格"尺寸".然后以此为基础,对Maxwell方程组鞍点问题的第一类N(e)d(e)lec线性棱元离散系统设计并分析了一种基于HX预条件子的Uzawa算法.当系数光滑时,理论上证明了算法的收敛率与网格规模无关.数值实验表明,新算法对跳系数情形也是高效和稳定的.     

Multi space reduced basis preconditioners for parametrized Stokes equations

We introduce a two-level preconditioner for the efficient solution of large scale saddle-point linear systems arising from the finite element (FE) discretization of parametrized Stokes equations. This preconditioner extends the Multi Space Reduced Basis (MSRB) preconditioning method proposed in Dal Santo et al. (2018); it combines an approximated block (fine grid) preconditioner with a reduced basis (RB) solver which plays the role of coarse component. A sequence of RB spaces, constructed either with an enriched velocity formulation or a Petrov–Galerkin projection, is built. Each RB coarse component is defined to perform a single iteration of the iterative method at hand. The flexible GMRES (FGMRES) algorithm is employed to solve the resulting preconditioned system and targets small tolerances with a very small iteration count and in a very short time. Numerical test cases for Stokes flows in three dimensional parameter-dependent geometries are considered to assess the numerical properties of the proposed technique in different large scale computational settings.     

An approach for structural static reanalysis with unchanged number of degrees of freedom

This paper presents an approach for structural static reanalysis with unchanged number of degrees of freedom. Preconditioned conjugate gradient method is employed, and a new preconditioner is constructed by updating the Cholesky factorization of the initial stiffness matrix with little cost. The proposed method preserves the ease of implementation and significantly improves the quality of the results. In particular, the accuracy of the approximate solutions can adaptively be monitored. Numerical examples show that the condition number of preconditioned system using the new preconditioner is much smaller than that using the initial stiffness matrix as the preconditioner. Therefore, the fast convergence and accurate results can be obtained by the proposed approach.     

Preconditioned accelerated generalized successive overrelaxation method for solving complex symmetric linear systems

In this paper, by adopting the preconditioned technique for the accelerated generalized successive overrelaxation method (AGSOR) proposed by Edalatpour et al. (2015), we establish the preconditioned AGSOR (PAGSOR) iteration method for solving a class of complex symmetric linear systems. The convergence conditions, optimal iteration parameters and corresponding optimal convergence factor of the PAGSOR iteration method are determined. Besides, we prove that the spectral radius of the PAGSOR iteration method is smaller than that of the AGSOR one under proper restrictions, and its optimal convergence factor is smaller than that of the preconditioned symmetric block triangular splitting (PSBTS) one put forward by Zhang et al. (2018) recently. The spectral properties of the preconditioned PAGSOR matrix are also proposed. Numerical experiments illustrate the correctness of the theories and the effectiveness of the proposed iteration method and the preconditioner for the generalized minimal residual (GMRES) method.     

基于PCGM的周期非均匀采样信号重构

本文利用Teoeplitz矩阵和正弦变换基预条件矩阵的性质，结合预条件共轭梯度法(PCGM)，对非均匀采样信号提出了一种新的重构方法，该方法针对文献[1]所构造的模型，在不增加运算量的前提下，扩展了原算法的适用空间，提高了信号重构的效率。     

大规模有限元刚度矩阵存储及其并行求解算法

本文提出一种将有限元单元刚度矩阵直接集成压缩格式的总体刚度矩阵的方法,并针对其线性系统设计了预处理的重启动GMRES(m)并行求解器.集成方法使用了一个“关联结点”的数据结构,它用来记录网格中节点的关联信息,作为集成过程的中间媒介.这种方法能减少大量的存储空间,简单且高效.求解器分别使用Jacobi和稀疏近似逆(SPAI)预条件子.二维和三维弹性力学问题的数值试验表明,在二维情形下,SPAI预条件子具有很好的加速收敛效果和并行效率;在三维情形下,Jacobi预条件子更能减少迭代收敛时间.     

On SSOR-like preconditioner for saddle point problems with dominant skew-Hermitian part

ABSTRACT

Based on the SSOR-like iteration method proposed by Bai [Numer. Linear Algebra Appl. 23 (2016), pp. 37–60], we present an SSOR-like preconditioner for the saddle point problems whose coefficient matrix has strongly dominant skew-Hermitian part. The spectral properties, including the bounds on the eigenvalues of the preconditioned matrix, are discussed in this work. Numerical experiments are presented to illustrate the effectiveness of the new preconditioner for saddle point problems.     

Combination of augmented Lagrangian technique and ST preconditioner for saddle point problems

For symmetric indefinite linear systems, we introduce a new triangular preconditioner based on symmetric and triangular (ST) decomposition. A new (1, 1) block is obtained by augmented Lagrangian technique. The new ST preconditioner is introduced by the combination of the new (1, 1) block and symmetric and triangular (ST) decomposition. Then a preconditioned system can be obtained by preconditioning technique, which is superior to the original system in terms of condition number. We study the spectral properties of preconditioned system, such as eigenvalues, the estimation of condition number and then give the quasi-optimal parameter. Numerical examples are given to indicate that the new preconditioner has obvious efficiency advantages. Finally, we conclude that the new ST preconditioner is a better option to deal with large and sparse problems.     

Preconditioning Schur complement matrices based on an aggregation multigrid method for shell structures

An aggregation multigrid method is utilized in constructing a preconditioner for a Schur complement system of automatically partitioned, non-overlapping subdomains. Preserving the relationship of the partitioned subdomains, we apply a rigid body based aggregation method, which employ geometric data, as a coarsening procedure. And then, we derive a new Schur complement coarse grid matrix by an approach of a condensation after the coarsening procedure. Therefore, we generate a multi-level preconditioner of a Krylov subspace method for Schur complement matrices using the coarse grid matrix. Through numerical experiments, the proposed preconditioner shows efficient performance and robust convergences irrespect of the size of elements and subdomains. It also shows better performance than the preconditioned conjugate gradient method (PCG) for the partitioned system and the aggregation multigrid method for the original domain in shell problems of structural mechanics.     

Domain decomposition PCG methods for serial and parallel processing

In this paper two domain decomposition formulations are presented in conjunction with the preconditioned conjugate gradient method (PCG) for the solution of large-scale problems in solid and structural mechanics. In the first approach, the PCG method is applied to the global coefficient matrix, while in the second approach it is applied to the interface problem after eliminating the internal degrees of freedom. For both implementations, a subdomain-by-subdomain (SBS) polynomial preconditioner is employed, based on local information of each subdomain. The approximate inverse of the global coefficient matrix or the Schur complement matrix, which acts as the preconditioner, is expressed by a truncated Neumann series resulting in an additive type local preconditioner. Block type preconditioning, where full elimination is performed inside each block, is also studied and compared with the proposed polynomial preconditioning.     

AILU preconditioning for the finite element formulation of the incompressible Navier–Stokes equations

In this paper, the effect of a variable reordering method on the performance of “adapted incomplete LU (AILU)” preconditioners applied to the P2P1 mixed finite element discretization of the three-dimensional unsteady incompressible Navier–Stokes equations has been studied through numerical experiments, where eigenvalue distribution and convergence histories are examined. It has been revealed that the performance of an AILU preconditioner is improved by adopting a variable reordering method which minimizes the bandwidth of a globally assembled saddle-point type matrix. Furthermore, variants of the existing AILU(1) preconditioner have been suggested and tested for some three-dimensional flow problems. It is observed that the AILU(2) outperforms the existing AILU(1) with a little extra computing time and memory.     

求解鞍点问题的广义正定和反Hermitian分裂方法

探讨了如何求解大型稀疏鞍点问题,给出了一种基于正定分裂的广义正定和反Hermitian分裂(GPSS)方法。该方法首先利用矩阵的正定分裂,构造出鞍点矩阵的2种分裂格式;然后利用这2种分裂格式构造出GPSS迭代;接着给出了迭代收敛的充要条件。最后进行了数值对比实验,实验结果表明,GPSS比正定和反Hermitian分裂(PSS)和Hermitian和反Hermitian分裂(HSS)方法更有效。