Comparison analysis for improving preconditioned SOR-type iterative method

On the basis of nonnegative matrices, some (I + S)-type preconditioners based on the SOR method are studied. Moreover, we prove the monotonicity of spectral radii of iterative matrices with respect to the parameters in [12]. Also, some splittings and preconditioners are compared and derived by comparisons. A numerical example is given to illustrate our results.     

Preconditioning Indefinite Systems in Interior Point Methods for Optimization

Every Newton step in an interior-point method for optimization requires a solution of a symmetric indefinite system of linear equations. Most of today's codes apply direct solution methods to perform this task. The use of logarithmic barriers in interior point methods causes unavoidable ill-conditioning of linear systems and, hence, iterative methods fail to provide sufficient accuracy unless appropriately preconditioned. Two types of preconditioners which use some form of incomplete Cholesky factorization for indefinite systems are proposed in this paper. Although they involve significantly sparser factorizations than those used in direct approaches they still capture most of the numerical properties of the preconditioned system. The spectral analysis of the preconditioned matrix is performed: for convex optimization problems all the eigenvalues of this matrix are strictly positive. Numerical results are given for a set of public domain large linearly constrained convex quadratic programming problems with sizes reaching tens of thousands of variables. The analysis of these results reveals that the solution times for such problems on a modern PC are measured in minutes when direct methods are used and drop to seconds when iterative methods with appropriate preconditioners are used.     

Block ILU Preconditioners for a Nonsymmetric Block-Tridiagonal M-Matrix

We propose block ILU (incomplete LU) factorization preconditioners for a nonsymmetric block-tridiagonal M-matrix whose computation can be done in parallel based on matrix blocks. Some theoretical properties for these block ILU factorization preconditioners are studied and then we describe how to construct them effectively for a special type of matrix. We also discuss a parallelization of the preconditioner solver step used in nonstationary iterative methods with the block ILU preconditioners. Numerical results of the right preconditioned BiCGSTAB method using the block ILU preconditioners are compared with those of the right preconditioned BiCGSTAB using a standard ILU factorization preconditioner to see how effective the block ILU preconditioners are.     

Preconditioners for non‐Hermitian Toeplitz systems

In this paper, we construct new ω‐circulant preconditioners for non‐Hermitian Toeplitz systems, where we allow the generating function of the sequence of Toeplitz matrices to have zeros on the unit circle. We prove that the eigenvalues of the preconditioned normal equation are clustered at 1 and that for (N, N)‐Toeplitz matrices with spectral condition number 𝒪(N^α) the corresponding PCG method requires at most 𝒪(N log² N) arithmetical operations. If the generating function of the Toeplitz sequence is a rational function then we show that our preconditioned original equation has only a fixed number of eigenvalues which are not equal to 1 such that preconditioned GMRES needs only a constant number of iteration steps independent of the dimension of the problem. Numerical tests are presented with PCG applied to the normal equation, GMRES, CGS and BICGSTAB. In particular, we apply our preconditioners to compute the stationary probability distribution vector of Markovian queuing models with batch arrival. Copyright © 2001 John Wiley & Sons, Ltd.     

Some extensions of the improved modified Gauss–Seidel iterative method for H‐matrices

In this paper, first we present a convergence theorem of the improved modified Gauss–Seidel iterative method, referred to as the IMGS method, for H‐matrices and compare the range of parameters α_i with that of the parameter ω of the SOR iterative method. Then with a more general splitting, the convergence analysis of this method for an H‐matrix and its comparison matrix is given. The spectral radii of them are also compared. Copyright © 2006 John Wiley & Sons, Ltd.     

Repeated Red-Black ordering: a new approach

Hereafter, we describe and analyze, from both a theoretical and a numerical point of view, an iterative method for efficiently solving symmetric elliptic problems with possibly discontinuous coefficients. In the following, we use the Preconditioned Conjugate Gradient method to solve the symmetric positive definite linear systems which arise from the finite element discretization of the problems. We focus our interest on sparse and efficient preconditioners. In order to define the preconditioners, we perform two steps: first we reorder the unknowns and then we carry out a (modified) incomplete factorization of the original matrix. We study numerically and theoretically two preconditioners, the second preconditioner corresponding to the one investigated by Brand and Heinemann [2]. We prove convergence results about the Poisson equation with either Dirichlet or periodic boundary conditions. For a meshsizeh, Brand proved that the condition number of the preconditioned system is bounded byO(h ^–1/2) for Dirichlet boundary conditions. By slightly modifying the preconditioning process, we prove that the condition number is bounded byO(h ^–1/3).     

A multigrid‐based shifted Laplacian preconditioner for a fourth‐order Helmholtz discretization

In this paper, an iterative solution method for a fourth‐order accurate discretization of the Helmholtz equation is presented. The method is a generalization of that presented in (SIAM J. Sci. Comput. 2006; 27 :1471–1492), where multigrid was employed as a preconditioner for a Krylov subspace iterative method. The multigrid preconditioner is based on the solution of a second Helmholtz operator with a complex‐valued shift. In particular, we compare preconditioners based on a point‐wise Jacobi smoother with those using an ILU(0) smoother, we compare using the prolongation operator developed by de Zeeuw in (J. Comput. Appl. Math. 1990; 33 :1–27) with interpolation operators based on algebraic multigrid principles, and we compare the performance of the Krylov subspace method Bi‐conjugate gradient stabilized with the recently introduced induced dimension reduction method, IDR(s). These three improvements are combined to yield an efficient solver for heterogeneous problems. Copyright © 2009 John Wiley & Sons, Ltd.     

Conservative discretizations and parameter‐robust preconditioners for Biot and multiple‐network flux‐based poroelasticity models

The parameters in the governing system of partial differential equations of multiple‐network poroelasticity models typically vary over several orders of magnitude, making its stable discretization and efficient solution a challenging task. In this paper, we prove the uniform Ladyzhenskaya–Babu?ka–Brezzi (LBB) condition and design uniformly stable discretizations and parameter‐robust preconditioners for flux‐based formulations of multiporosity/multipermeability systems. Novel parameter‐matrix‐dependent norms that provide the key for establishing uniform LBB stability of the continuous problem are introduced. As a result, the stability estimates presented here are uniform not only with respect to the Lamé parameter λ but also to all the other model parameters, such as the permeability coefficients K_i; storage coefficients $c_{p_i}$; network transfer coefficients β_{i j},i,j = 1,…,n; the scale of the networks n; and the time step size τ. Moreover, strongly mass‐conservative discretizations that meet the required conditions for parameter‐robust LBB stability are suggested and corresponding optimal error estimates proved. The transfer of the canonical (norm‐equivalent) operator preconditioners from the continuous to the discrete level lays the foundation for optimal and fully robust iterative solution methods. The theoretical results are confirmed in numerical experiments that are motivated by practical applications.     

Preconditioners Based on Fit Techniques for the Iterative Regularization in the Image Deconvolution Problem

For large-scale image deconvolution problems, the iterative regularization methods can be favorable alternatives to the direct methods. We analyze preconditioners for regularizing gradient-type iterations applied to problems with 2D band Toeplitz coefficient matrix. For problems having separable and positive definite matrices, the fit preconditioner we have introduced in a previous paper has been shown to be effective in conjunction with CG. The cost of this preconditioner is of O(n²) operations per iteration, where n² is the pixels number of the image, whereas the cost of the circulant preconditioners commonly used for this type of problems is of O(n² log n) operations per iteration. In this paper the extension of the fit preconditioner to more general cases is proposed: namely the nonseparable positive definite case and the symmetric indefinite case. The major difficulty encountered in this extension concerns the factorization phase, where a further approximation is required. Three approximate factorizations are proposed. The preconditioners thus obtained have still a cost of O(n²) operations per iteration. A numerical experimentation shows that the fit preconditioners are competitive with the regularizing Chan preconditioner, both in the regularizing efficiency and the computational cost. AMS subject classification (2000) 65F10, 65F22.Received October 2003. Accepted December 2004. Communicated by Lars Eldén.     

Regularizing properties of a class of matrices including the optimal and the superoptimal preconditioners

In this work, given a positive definite matrix A, we introduce a class of matrices related to A, which is obtained by suitably combining projections of its powers onto algebras of matrices simultaneously diagonalized by a unitary transform. After a detailed theoretical study of some spectral properties of the matrices of this class, which suggests their use as regularizing preconditioners, we prove that such matrices can be cheaply computed when the matrix A has a Toeplitz structure. We provide numerical evidence of the advantages coming from the employment of the proposed preconditioners when used in regularizing procedures.     

Solution of large‐scale weighted least‐squares problems

A sequence of least‐squares problems of the form min_y∥G^1/2(A^T y?h)∥₂, where G is an n×n positive‐definite diagonal weight matrix, and A an m×n (m?n) sparse matrix with some dense columns; has many applications in linear programming, electrical networks, elliptic boundary value problems, and structural analysis. We suggest low‐rank correction preconditioners for such problems, and a mixed solver (a combination of a direct solver and an iterative solver). The numerical results show that our technique for selecting the low‐rank correction matrix is very effective. Copyright © 2002 John Wiley & Sons, Ltd.     

Block two-stage preconditioners

Linear systems of the form Ax = b, where the matrix A is symmetric and positive definite, often arise from the discretization of elliptic partial differential equations. A very successful method for solving these linear systems is the preconditioned conjugate gradient method. In this paper, we study parallel preconditioners for the conjugate gradient method based on the block two-stage iterative methods. Sufficient conditions for the validity of these preconditioners are given. Computational results of these preconditioned conjugate gradient methods on two parallel computing systems are presented.     

Limited memory preconditioners for symmetric indefinite problems with application to structural mechanics

This paper presents a class of limited memory preconditioners (LMP) for solving linear systems of equations with symmetric indefinite matrices and multiple right‐hand sides. These preconditioners based on limited memory quasi‐Newton formulas require a small number k of linearly independent vectors and may be used to improve an existing first‐level preconditioner. The contributions of the paper are threefold. First, we derive a formula to characterize the spectrum of the preconditioned operator. A spectral analysis of the preconditioned matrix shows that the eigenvalues are all real and that the LMP class is able to cluster at least k eigenvalues at 1. Secondly, we show that the eigenvalues of the preconditioned matrix enjoy interlacing properties with respect to the eigenvalues of the original matrix provided that the k linearly independent vectors have been prior projected onto the invariant subspaces associated with the eigenvalues of the original matrix in the open right and left half‐plane, respectively. Third, we focus on theoretical properties of the Ritz‐LMP variant, where Ritz information is used to determine the k vectors. Finally, we illustrate the numerical behaviour of the Ritz limited memory preconditioners on realistic applications in structural mechanics that require the solution of sequences of large‐scale symmetric saddle‐point systems. Numerical experiments show the relevance of the proposed preconditioner leading to a significant decrease in terms of computational operations when solving such sequences of linear systems. A saving of up to 43% in terms of computational effort is obtained on one of these applications. Copyright © 2016 John Wiley & Sons, Ltd.     

Extra Precise Preconditioning for Non-Hermitian Eigenvalue Problems

For badly conditioned Non-Hermite eigenvalue problems known iterative refinement procedures often fail. It is proposed to use extra precise iterative refinement in a precondition step. As preconditioners the matrix Q from the QR decomposition or at least the matrix of - even not very accurate - eigenvalues can be used. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Preconditioned HSS method for large multilevel block Toeplitz linear systems via the notion of matrix‐valued symbol

We perform a spectral analysis of the preconditioned Hermitian/skew‐Hermitian splitting (PHSS) method applied to multilevel block Toeplitz linear systems in which the coefficient matrix T_n(f) is associated with a Lebesgue integrable matrix‐valued function f. When the preconditioner is chosen as a Hermitian positive definite multilevel block Toeplitz matrix T_n(g), the resulting sequence of PHSS iteration matrices M_n belongs to the generalized locally Toeplitz class. In this case, we are able to compute the symbol ?(f,g) describing the asymptotic eigenvalue distribution of M_nwhen n→∞ and the matrix size diverges. By minimizing the infinity norm of the spectral radius of the symbol ?(f,g), we are also able to identify effective PHSS preconditioners T_n(g) for the matrix T_n(f). A number of numerical experiments are presented and commented, showing that the theoretical results are confirmed and that the spectral analysis leads to efficient PHSS methods. Copyright © 2015 John Wiley & Sons, Ltd.     

Improvements for some condition number estimates for preconditioned system in p‐FEM

In this paper, we consider domain decomposition preconditioners for a system of linear algebraic equations arising from the p‐version of the FEM. We analyse several multi‐level preconditioners for the Dirichlet problems in the sub‐domains in two and three dimensions. It is proved that the condition number of the preconditioned system is bounded by a constant independent of the polynomial degree. Relations between the p‐version of the FEM and the h‐version are helpful in the interpretations of the results. Copyright © 2006 John Wiley & Sons, Ltd.     

Spectral discretizations of 3-D elliptic problems and fast domain decomposition methods

An important for applications, the class of hp discretizations of second-order elliptic equations consists of discretizations based on spectral finite elements. The development of fast domain decomposition algorithms for them was restrained by the absence of fast solvers for the basic components of the method, i.e., for local interior problems on decomposition subdomains and their faces. Recently, the authors have established that such solvers can be designed using special factorized preconditioners. In turn, factorized preconditioners are constructed using an important analogy between the stiffness matrices of spectral and hierarchical basis hp-elements (coordinate functions of the latter are defined as tensor products of integrated Legendre polynomials). Due to this analogy, for matrices of spectral elements, fast solvers can be developed that are similar to those for matrices of hierarchical elements. Based on these facts and previous results on the preconditioning of other components, fast domain decomposition algorithms for spectral discretizations are obtained.     

on hybrid preconditioning methods for large sparse saddle-point problems

Based on the block-triangular product approximation to a 2-by-2 block matrix, a class of hybrid preconditioning methods is designed for accelerating the MINRES method for solving saddle-point problems. The appropriate values for the parameters involved in the new preconditioners are estimated, so that the numerical conditioning and the spectral property of the saddle-point matrix of the linear system can be substantially improved. Several practical hybrid preconditioners and the corresponding preconditioning iterative methods are constructed and studied, too.     

Efficient linear solvers for incompressible flow simulations using Scott‐Vogelius finite elements

Recent research has shown that in some practically relevant situations like multiphysics flows (Galvin et al., Comput Methods Appl Mech Eng, to appear) divergence‐free mixed finite elements may have a significantly smaller discretization error than standard nondivergence‐free mixed finite elements. To judge the overall performance of divergence‐free mixed finite elements, we investigate linear solvers for the saddle point linear systems arising in ((P_k)^d,P _k‐1^disc) Scott‐Vogelius finite element implementations of the incompressible Navier–Stokes equations. We investigate both direct and iterative solver methods. Due to discontinuous pressure elements in the case of Scott‐Vogelius (SV) elements, considerably more solver strategies seem to deliver promising results than in the case of standard mixed finite elements such as Taylor‐Hood elements. For direct methods, we extend recent preliminary work using sparse banded solvers on the penalty method formulation to finer meshes and discuss extensions. For iterative methods, we test augmented Lagrangian and \begin{align*}\mathcal{H}\end{align*} ‐LU preconditioners with GMRES, on both full and statically condensed systems. Several numerical experiments are provided that show these classes of solvers are well suited for use with SV elements and could deliver an interesting overall performance in several applications.© 2012 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq, 2013