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Preconditioned Matrix (preconditioned + matrix)
Selected AbstractsOptimal parameters in the HSS-like methods for saddle-point problemsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 6 2009Zhong-Zhi BaiArticle first published online: 17 NOV 200 Abstract For the Hermitian and skew-Hermitian splitting iteration method and its accelerated variant for solving the large sparse saddle-point problems, we compute their quasi-optimal iteration parameters and the corresponding quasi-optimal convergence factors for the more practical but more difficult case that the (1, 1)-block of the saddle-point matrix is not algebraically equivalent to the identity matrix. In addition, the algebraic behaviors and the clustering properties of the eigenvalues of the preconditioned matrices with respect to these two iterations are investigated in detail, and the formulas for computing good iteration parameters are given under certain principle for optimizing the distribution of the eigenvalues. Copyright © 2008 John Wiley & Sons, Ltd. [source] Static reanalysis of structures with added degrees of freedomINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN BIOMEDICAL ENGINEERING, Issue 4 2006Baisheng Wu Abstract This paper deals with static reanalysis of a structure with added degrees of freedom where the nodes of the original structure form a subset of the nodes of the modified structure. A preconditioned conjugate-gradient approach is developed. The preconditioner is constructed, and the implementation of the approach involves only decomposition of the stiffness matrix corresponding to the newly added degrees of freedom. In particular, the approach can adaptively monitor the accuracy of approximate solutions. The approach is applicable to the reanalysis of the structural layout modifications for the case of addition of some nodes, deletion and addition of elements and further changes in the geometry as well as to the local mesh refinements. Numerical examples show that the condition number of the selected preconditioned matrix is largely reduced. Therefore, the fast convergence and accurate results can be achieved by the approach. Copyright © 2005 John Wiley & Sons, Ltd. [source] A preconditioned conjugate gradient approach to structural reanalysis for general layout modificationsINTERNATIONAL JOURNAL FOR NUMERICAL METHODS IN ENGINEERING, Issue 5 2007Zhengguang Li Abstract This paper presents a preconditioned conjugate gradient approach to structural static reanalysis for general layout modifications. It is suitable for all types of layout modifications, including the general case in which some original members and nodes are deleted and other new members and nodes are added concurrently. The approach is based on the preconditioned conjugate gradient technique. The preconditioner is constructed, and an efficient implementation for applying the preconditioner is presented, which requires the factorization of the stiffness matrix corresponding to the newly added degrees of freedom only. In particular, the approach can adaptively monitor the accuracy of approximate solutions. Numerical examples show that the condition number of the preconditioned matrix is remarkably reduced. Therefore, the fast convergence and accurate results can be achieved by the approach. Copyright © 2006 John Wiley & Sons, Ltd. [source] A new multilevel algebraic preconditioner for the diffusion equation in heterogeneous mediaNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 5 2010Yu Kuznetsov Abstract We develop and analyze a new multilevel preconditioner for algebraic systems arising from the finite volume discretization of 3D diffusion,reaction problems in highly heterogeneous media. The system matrices are assumed to be symmetric M -matrices. The preconditioner is based on a special coarsening algorithm and the inner Chebyshev iterative procedure. The condition number of the preconditioned matrix does not depend on the coefficients in the diffusion operator. Numerical experiments confirm theoretical results and reveal the competitiveness of the new preconditioner with respect to a well-known algebraic multigrid preconditioner. Copyright © 2010 John Wiley & Sons, Ltd. [source] Regularization and preconditioning of KKT systems arising in nonnegative least-squares problemsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 1 2009Stefania Bellavia Abstract A regularized Newton-like method for solving nonnegative least-squares problems is proposed and analysed in this paper. A preconditioner for KKT systems arising in the method is introduced and spectral properties of the preconditioned matrix are analysed. A bound on the condition number of the preconditioned matrix is provided. The bound does not depend on the interior-point scaling matrix. Preliminary computational results confirm the effectiveness of the preconditioner and fast convergence of the iterative method established by the analysis performed in this paper. Copyright © 2008 John Wiley & Sons, Ltd. [source] Augmentation block preconditioners for saddle point-type matrices with singular (1, 1) blocksNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 6 2008Zhi-Hao Cao Abstract We consider the use of block preconditioners for the application of the preconditioned Krylov subspace iterative methods to the solution of large saddle point-type systems with singular (1, 1) blocks. Two block triangular preconditioners are introduced and the block diagonal preconditioner in Greif and Schötzau (Electron. Trans. Numer. Anal. 2006; 22:114,121) is extended to nonsymmetric saddle point systems. All these preconditioners are based on augmentation, using nonsingular weight matrices. If the nullity of the (1, 1) block takes its highest possible value, the preconditioned matrix with either block triangular preconditioner has precisely three distinct eigenvalues, and the preconditioned matrix with the block diagonal preconditioner has precisely two distinct eigenvalues, giving rise to immediate convergence of preconditioned GMRES. Finally, numerical experiments that validate the analysis are reported. Copyright © 2008 John Wiley & Sons, Ltd. [source] Inverse Toeplitz preconditioners for Hermitian Toeplitz systemsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2-3 2005Fu-Rong Lin Abstract In this paper we consider solving Hermitian Toeplitz systems Tnx=b by using the preconditioned conjugate gradient (PCG) method. Here the Toeplitz matrices Tn are assumed to be generated by a non-negative continuous 2,-periodic function ,, i.e. Tn=,,n[,]. It was proved in (Linear Algebra Appl. 1993; 190:181) that if , is positive then the spectrum of ,,n[1/,],,n[,] is clustered around 1. We prove that the trigonometric polynomial q (s,2, cf. (2) and (3)) converges to 1/, uniformly as n,, under the condition that 1/, is in Wiener class. It follows that the computational cost of the PCG method can be reduced by replacing 1/, with q, where N BCCB preconditioners for systems of BVM-based numerical integratorsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 1 2004Siu-Long Lei Abstract Boundary value methods (BVMs) for ordinary differential equations require the solution of non-symmetric, large and sparse linear systems. In this paper, these systems are solved by using the generalized minimal residual (GMRES) method. A block-circulant preconditioner with circulant blocks (BCCB preconditioner) is proposed to speed up the convergence rate of the GMRES method. The BCCB preconditioner is shown to be invertible when the BVM is Ak1,k2 -stable. The spectrum of the preconditioned matrix is clustered and therefore, the preconditioned GMRES method converges fast. Moreover, the operation cost in each iteration of the preconditioned GMRES method by using our BCCB preconditioner is less than that required by using block-circulant preconditioners proposed earlier. In numerical experiments, we compare the number of iterations of various preconditioners. Copyright © 2003 John Wiley & Sons, Ltd. [source] Some observations on the l2 convergence of the additive Schwarz preconditioned GMRES methodNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 5 2002Xiao-Chuan Cai Abstract Additive Schwarz preconditioned GMRES is a powerful method for solving large sparse linear systems of equations on parallel computers. The algorithm is often implemented in the Euclidean norm, or the discrete l2 norm, however, the optimal convergence result is available only in the energy norm (or the equivalent Sobolev H1 norm). Very little progress has been made in the theoretical understanding of the l2 behaviour of this very successful algorithm. To add to the difficulty in developing a full l2 theory, in this note, we construct explicit examples and show that the optimal convergence of additive Schwarz preconditioned GMRES in l2 cannot be obtained using the existing GMRES theory. More precisely speaking, we show that the symmetric part of the preconditioned matrix, which plays a role in the Eisenstat,Elman,Schultz theory, has at least one negative eigenvalue, and we show that the condition number of the best possible eigenmatrix that diagonalizes the preconditioned matrix, key to the Saad,Schultz theory, is bounded from both above and below by constants multiplied by h,1/2. Here h is the finite element mesh size. The results presented in this paper are mostly negative, but we believe that the techniques used in our proofs may have wide applications in the further development of the l2 convergence theory and in other areas of domain decomposition methods. Copyright © 2002 John Wiley & Sons, Ltd. [source] A parallel block overlap preconditioning with inexact submatrix inversion for linear elasticity problemsNUMERICAL LINEAR ALGEBRA WITH APPLICATIONS, Issue 2 2002Igor E. Kaporin We present a parallel preconditioned iterative solver for large sparse symmetric positive definite linear systems. The preconditioner is constructed as a proper combination of advanced preconditioning strategies. It can be formally seen as being of domain decomposition type with algebraically constructed overlap. Similar to the classical domain decomposition technique, inexact subdomain solvers are used, based on incomplete Cholesky factorization. The proper preconditioner is shown to be near optimal in minimizing the so-called K -condition number of the preconditioned matrix. The efficiency of both serial and parallel versions of the solution method is illustrated on a set of benchmark problems in linear elasticity. Copyright © 2002 John Wiley & Sons, Ltd. [source]
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