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.     

Determination of a good value of the time step and preconditioned Krylov subspace methods for the Navier-Stokes equations

The main purpose of this paper is to develop stable versions of some Krylov subspace methods for solving the linear systems of equations Ax = b which arise in the difference solution of 2-D nonstationary Navier-Stokes equations using implicit scheme and to determine a good value of the time step. Our algorithms are based on the conjugate-gradient method with a suitable preconditioner for solving the symmetric positive definite system and preconditioned GMRES, Orthomin(K), QMR methods for solving the nonsymmetric and (in)definite system. The performance of these methods is compared. In addition, we show that by using the condition number of the first nonsymmetric coefficient matrix, it is possible to determine a good value of the time step.     

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.     

Preconditioning technique for symmetric M-matrices

Abstract Linear systems with M-matrices occur in a wide variety of areas including numerical partial differential equations, input-output production and growth models in economics, linear complementarity problems in operations research and Markov chains in stochastic analysis.In this paper, we propose a new preconditioner for solving a system with symmetric positive definite M-matrix by the preconditioned conjugate gradient (PCG) method. We show that our preconditioner increases the convergence rate of the PCG method and reduces the operation cost. Numerical results are given.     

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.     

A new splitting and preconditioner for iteratively solving non-Hermitian positive definite systems

A new iteration method for solving a linear system with coefficient matrix being non-Hermitian positive definite is presented in this note. We study the spectral radius and contraction properties of the iteration matrix and then analyze the best possible choice of the parameter. With the results obtained, we show that the new method is convergent for a non-Hermitian positive definite linear system and propose a preconditioner to improve the condition number of the system. The numerical examples show that the new method is much more efficient than the HSS (or PSS) iteration method.     

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.     

The improvements of the generalized shift-splitting preconditioners for non-singular and singular saddle point problems

ABSTRACT

To solve the saddle point problems with symmetric positive definite (1,1) parts, the improved generalized shift-splitting (IGSS) preconditioner is established in this paper, which yields the IGSS iteration method. Theoretical analysis shows that the IGSS iteration method is convergent and semi-convergent unconditionally. The choices of the iteration parameters are discussed. Moreover, some spectral properties, including the eigenvalue and eigenvector distributions of the preconditioned matrix are also investigated. Finally, numerical results are presented to verify the robustness and the efficiency of the proposed iteration method and the corresponding preconditioner for solving the non-singular and singular saddle point problems.     

A parallel preconditioned conjugate gradient method using domain decomposition and inexact solvers on each subdomain

We describe a preconditioned conjugate gradient solution strategy for a multiprocessor system with message passing architecture. The preconditioner combines two techniques, a Schurcomplement preconditioning over “coupling boundaries” between the subdomains and an arbitrary choice of classic preconditioning for the inner degrees of freedom on each subdomain. All computational work on the single subdomains is carried out in parallel by distributing the subdomain data over the processor network before starting the finite element solution process (including generating the element matrices and assemblying the local subdomain stiffness matrix). The resulting spectral condition number of the entire preconditioner is estimated. For the important example of choosing MIC(0)-*-preconditioning on the subdomains, the condition number obtained is essentially the product of the two condition numbers involved.     

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.     

块三对角矩阵局部块分解及其在预条件中的应用

该文利用块三对阵角阵分解因子的估值分析了其局部依赖性，并用其构了一类不完全分解型预条件子，给出了五点差分矩阵预条件后的条件数估计，并比较了条件数估计值与实际值，表明了估计值的准确性与预备件的有效性，在具体实现时，考虑了预条件的6个串行实现方案并提出了一个有效的并行化方法，该并行算法具有通信量少的特点，最后在由4中微机通过高速以太网连成的机群系统上作了大量数值实验，并将其与其它较效的预条件方法进行了。结果表明该预条件方法效果较好，尤其适用于并行计算。     

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.     

Numerical performance of a parallel solution method for a heterogeneous 2D Helmholtz equation

The parallel performance of a numerical solution method for the scalar 2D Helmholtz equation written for inhomogeneous media is studied. The numerical solution is obtained by an iterative method applied to the preconditioned linear system which has been derived from a finite difference discretization. The preconditioner is approximately inverted using multigrid iterations. Parallel execution is implemented using the MPI library. Only a few iterations are required to solve numerically the so-called full Marmousi problem [Bourgeois, A., et al. in The Marmousi Experience, Proceedings of the 1990 EAEG Workshop on Practical Aspects of Seismic Data Inversion: Eur. Assoc. Expl. Geophys., pp. 5–16 (1991)] for the high frequency range.     

Additive polynomial preconditioners for parallel computers

We present a polynomial preconditioner that can be used with the conjugate gradient method to solve symmetric and positive definite systems of linear equations. Each step of the preconditioning is achieved by simultaneously taking an iteration of the SOR method and an iteration of the reverse SOR method (equations taken in reverse order) and averaging the results. This yields a symmetric preconditioner that can be implemented on parallel computers by performing the forward and reverse SOR iterations simultaneously. We give necessary and sufficient conditions for additive preconditioners to be positive definite.

We find an optimal parameter, ω, for the SOR-Additive linear stationary iterative method applied to 2-cyclic matrices. We show this method is asymptotically twice as fast as SSOR when the optimal ω is used.

We compare our preconditioner to the SSOR polynomial preconditioner for a model problem. With the optimal ω, our preconditioner was found to be as effective as the SSOR polynomial preconditioner in reducing the number of conjugate gradient iterations. Parallel implementations of both methods are discussed for vector and multiple processors. Results show that if the same number of processors are used for both preconditioners, the SSOR preconditioner is more effective. If twice as many processors are used for the SOR-Additive preconditioner, it becomes more efficient than the SSOR preconditioner when the number of equations assigned to a processor is small. These results are confirmed by the Blue Chip emulator at the University of Washington.     

Comparison of Two-Level Preconditioners Derived from Deflation,Domain Decomposition and Multigrid Methods

For various applications, it is well-known that a multi-level, in particular two-level, preconditioned CG (PCG) method is an efficient method for solving large and sparse linear systems with a coefficient matrix that is symmetric positive definite. The corresponding two-level preconditioner combines traditional and projection-type preconditioners to get rid of the effect of both small and large eigenvalues of the coefficient matrix. In the literature, various two-level PCG methods are known, coming from the fields of deflation, domain decomposition and multigrid. Even though these two-level methods differ a lot in their specific components, it can be shown that from an abstract point of view they are closely related to each other. We investigate their equivalences, robustness, spectral and convergence properties, by accounting for their implementation, the effect of roundoff errors and their sensitivity to inexact coarse solves, severe termination criteria and perturbed starting vectors.     

Gain margin and Lyapunov analysis of discrete-time network synchronization via Riccati design

This paper deals with solution analysis and gain margin analysis of a modified algebraic Riccati matrix equation, and the Lyapunov analysis for discrete-time network synchronization with directed graph topologies. First, the structure of the solution to the Riccati equation associated with a single-input controllable system is analyzed. The solution matrix entries are represented using unknown closed-loop pole variables that are solved via a system of scalar quadratic equations. Then, the gain margin is studied for the modified Riccati equation for both multi-input and single-input systems. A disc gain margin in the complex plane is obtained using the solution matrix. Finally, the feasibility of the Riccati design for the discrete-time network synchronization with general directed graphs is solved via the Lyapunov analysis approach and the gain margin approach, respectively. In the design, a network Lyapunov function is constructed using the Kronecker product of two positive definite matrices: one is the graph positive definite matrix solved from a graph Lyapunov matrix inequality involving the graph Laplacian matrix; the other is the dynamical positive definite matrix solved from the modified Riccati equation. The synchronizing conditions are obtained for the two Riccati design approaches, respectively.     

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.     

Strang-type preconditioners for solving linear systems from neutral delay differential equations

We study the solution of neutral delay differential equations (NDDEs) by using boundary value methods (BVMs). The BVMs require the solution of nonsymmetric, large and sparse linear systems. The GMRES method with Strang-type block-circulant preconditioner is proposed to solve these linear systems. We show that, if an -stable BVM is used for solving a system of NDDEs, then our preconditioner is invertible and the spectrum of the preconditioned system is clustered. It follows that, when the GMRES method is applied to the preconditioned systems, the method can converge rapidly. Numerical results are given to show that our method is effective. Received: July 2002 / Accepted: December 2002 RID="*" ID="*"The research of this author is supported by research grant No. RG024/01-02S/JXQ/FST from the University of Macau.     

Algebraic multigrid methods based on element preconditioning

This paper presents a new algebraic multigrid (AMG) solution strategy for large linear systems with a sparse matrix arising from a finite element discretization of some self-adjoint, second order, scalar, elliptic partial differential equation. The AMG solver is based on Ruge/Stübens method. Ruge/Stübens algorithm is robust for M-matrices, but unfortunately the “region of robustness“ between symmetric positive definite M-matrices and general symmetric positive definite matrices is very fuzzy.

For this reason the so-called element preconditioning technique is introduced in this paper. This technique aims at the construction of an M-matrix that is spectrally equivalent to the original stiffness matrix. This is done by solving small restricted optimization problems. AMG applied to the spectrally equivalent M-matrix instead of the original stiffness matrix is then used as a preconditioner in the conjugate gradient method for solving the original problem.

The numerical experiments show the efficiency and the robustness of the new preconditioning method for a wide class of problems including problems with anisotropic elements.     

An overlapping Schwarz method for virtual element discretizations in two dimensions

A new coarse space for domain decomposition methods is presented for nodal elliptic problems in two dimensions. The coarse space is derived from the novel virtual element methods and therefore can accommodate quite irregular polygonal subdomains. It has the advantage with respect to previous studies that no discrete harmonic extensions are required. The virtual element method allows us to handle polygonal meshes and the algorithm can then be used as a preconditioner for linear systems that arise from a discretization with such triangulations. A bound is obtained for the condition number of the preconditioned system by using a two-level overlapping Schwarz algorithm, but the coarse space can also be used for different substructuring methods. This bound is independent of jumps in the coefficient across the interface between the subdomains. Numerical experiments that verify the result are shown, including some with triangular, square, hexagonal and irregular elements and with irregular subdomains obtained by a mesh partitioner.