Parameterized preconditioning for generalized saddle point problems arising from the Stokes equation

A parameterized preconditioning framework is proposed to improve the conditions of the generalized saddle point problems. Based on the eigenvalue estimates for the generalized saddle point matrices, a strategy to minimize the upper bounds of the spectral condition numbers of the matrices is given, and the explicit expression of the quasi-optimal preconditioning parameter is obtained. In numerical experiment, parameterized preconditioning techniques are applied to the generalized saddle point problems derived from the mixed finite element discretization of the stationary Stokes equation. Numerical results demonstrate that the involved preconditioning procedures are efficient.     

On nonsingularity of block two-by-two matrices

We derive necessary and sufficient conditions for guaranteeing the nonsingularity of a block two-by-two matrix by making use of the singular value decompositions and the Moore–Penrose pseudoinverses of the matrix blocks. These conditions are complete, and much weaker and simpler than those given by Decker and Keller [D.W. Decker, H.B. Keller, Multiple limit point bifurcation, J. Math. Anal. Appl. 75 (1980) 417–430], and may be more easily examined than those given by Bai [Z.-Z. Bai, Eigenvalue estimates for saddle point matrices of Hermitian and indefinite leading blocks, J. Comput. Appl. Math. 237 (2013) 295–306] from the computational viewpoint. We also derive general formulas for the rank of the block two-by-two matrix by utilizing either the unitarily compressed or the orthogonally projected sub-matrices.     

Evaluation of ST preconditioners for saddle point problems

The general block ST decomposition of the saddle point problem is used as a preconditioner to transform the saddle point problem into an equivalent symmetric and positive definite system. Such a decomposition is called a block ST preconditioner. Two general block ST preconditioners are proposed for saddle point problems with symmetric and positive definite (1,1)-block. Some estimations of the condition number of the preconditioned system are given. The same study is done for singular (1,1)-block.     

On nonsymmetric saddle point matrices that allow conjugate gradient iterations

Linear systems in saddle point form are usually highly indefinite,which often slows down iterative solvers such as Krylov subspace methods. It has been noted by several authors that negating the second block row of a symmetric indefinite saddle point matrix leads to a nonsymmetric matrix ${{\mathcal A}}Linear systems in saddle point form are usually highly indefinite,which often slows down iterative solvers such as Krylov subspace methods. It has been noted by several authors that negating the second block row of a symmetric indefinite saddle point matrix leads to a nonsymmetric matrix whose spectrum is entirely contained in the right half plane. In this paper we study conditions so that is diagonalizable with a real and positive spectrum. These conditions are based on necessary and sufficient conditions for positive definiteness of a certain bilinear form,with respect to which is symmetric. In case the latter conditions are satisfied, there exists a well defined conjugate gradient (CG) method for solving linear systems with . We give an efficient implementation of this method, discuss practical issues such as error bounds, and present numerical experiments. In memory of Gene Golub (1932–2007), our wonderful friend and colleague, who had a great interest in the conjugate gradient method and the numerical solution of saddle point problems. The work of J?rg Liesen was supported by the Emmy Noether-Program and the Heisenberg-Program of the Deutsche Forschungsgemeinschaft.     

ImplicitQR factorization of a product of three matrices

We present a numerical algorithm for computing the implicit QR factorization of a product of three matrices, and we illustrate the technique by applying it to the generalized total least squares and the restricted total least squares problems. We also demonstrate how to take advantage of the block structures of the underlying matrices in order to reduce the computational work.     

Tensor properties of multilevel Toeplitz and related matrices

A general proposal is presented for fast algorithms for multilevel structured matrices. It is based on investigation of their tensor properties and develops the idea recently introduced by Kamm and Nagy in the block Toeplitz case. We show that tensor properties of multilevel Toeplitz matrices are related to separation of variables in the corresponding symbol, present analytical tools to study the latter, expose truncation algorithms preserving the structure, and report on some numerical results confirming advantages of the proposal.     

Modified block preconditioners for the discretized time-harmonic Maxwell equations in mixed form

In this paper, building on the previous work by Greif and Schötzau [Preconditioners for the discretized time-harmonic Maxwell equations in mixed form, Numer. Linear Algebra Appl. 14 (2007) 281–297] and Benzi and Olshanskii [An augmented lagrangian-based approach to the Oseen problem, SIAM J. Sci. Comput. 28 (2006) 2095–2113], we present the improved preconditioning techniques for the iterative solution of the saddle point linear systems, which arise from the finite element discretization of the mixed formulation of the time-harmonic Maxwell equations. The modified block diagonal and triangular preconditioners considered are based on augmentation with using the symmetric nonsingular weighted matrix. We discuss the spectral properties of the preconditioned matrix in detail and generalize the results of the above-mentioned paper by Greif and Schötzau. Numerical experiments are given to demonstrate the efficiency of the presented preconditioners.     

On algebraic multi-level methods for non-symmetric systems - Comparison results

We establish theoretical comparison results for algebraic multi-level methods applied to non-singular non-symmetric M-matrices. We consider two types of multi-level approximate block factorizations or AMG methods, the AMLI and the MAMLI method. We compare the spectral radii of the iteration matrices of these methods. This comparison shows, that the spectral radius of the MAMLI method is less than or equal to the spectral radius of the AMLI method. Moreover, we establish how the quality of the approximations in the block factorization effects the spectral radii of the iteration matrices. We prove comparisons results for different approximations of the fine grid block as well as for the used Schur complement. We also establish a theoretical comparison between the AMG methods and the classical block Jacobi and block Gauss-Seidel methods.     

Eigenvalue bounds for the Schur complement with a pressure convection–diffusion preconditioner in incompressible flow computations

If the stationary Navier–Stokes system or an implicit time discretization of the evolutionary Navier–Stokes system is linearized by a Picard iteration and discretized in space by a mixed finite element method, there arises a saddle point system which may be solved by a Krylov subspace method or an Uzawa type approach. For each of these resolution methods, it is necessary to precondition the Schur complement associated to the saddle point problem in question. In the work at hand, we give upper and lower bounds of the eigenvalues of this Schur complement under the assumption that it is preconditioned by a pressure convection–diffusion matrix.     

Limiting accuracy of segregated solution methods for nonsymmetric saddle point problems

Nonsymmetric saddle point problems arise in a wide variety of applications in computational science and engineering. The aim of this paper is to discuss the numerical behavior of several nonsymmetric iterative methods applied for solving the saddle point systems via the Schur complement reduction or the null-space projection approach. Krylov subspace methods often produce the iterates which fluctuate rather strongly. Here we address the question whether large intermediate approximate solutions reduce the final accuracy of these two-level (inner–outer) iteration algorithms. We extend our previous analysis obtained for symmetric saddle point problems and distinguish between three mathematically equivalent back-substitution schemes which lead to a different numerical behavior when applied in finite precision arithmetic. Theoretical results are then illustrated on a simple model example.     

Spectral estimates for saddle point matrices arising in weak constraint four‐dimensional variational data assimilation

We consider the large sparse symmetric linear systems of equations that arise in the solution of weak constraint four‐dimensional variational data assimilation, a method of high interest for numerical weather prediction. These systems can be written as saddle point systems with a 3 × 3 block structure but block eliminations can be performed to reduce them to saddle point systems with a 2 × 2 block structure, or further to symmetric positive definite systems. In this article, we analyse how sensitive the spectra of these matrices are to the number of observations of the underlying dynamical system. We also obtain bounds on the eigenvalues of the matrices. Numerical experiments are used to confirm the theoretical analysis and bounds.     

Convergence of block iterative methods for linear systems arising in the numerical solution of Euler equations

Summary We discuss block matrices of the formA=[A _ij ], whereA _ij is ak×k symmetric matrix,A _ij is positive definite andA _ij is negative semidefinite. These matrices are natural block-generalizations of Z-matrices and M-matrices. Matrices of this type arise in the numerical solution of Euler equations in fluid flow computations. We discuss properties of these matrices, in particular we prove convergence of block iterative methods for linear systems with such system matrices.     

Some properties of generalized K-centrosymmetric H-matrices

Every n×n$n\times n$ generalized K-centrosymmetric matrix A   can be reduced into a 2×2$2\times 2$ block diagonal matrix (see [Z. Liu, H. Cao, H. Chen, A note on computing matrix–vector products with generalized centrosymmetric (centrohermitian) matrices, Appl. Math. Comput. 169 (2) (2005) 1332–1345]). This block diagonal matrix is called the reduced form of the matrix A. In this paper we further investigate some properties of the reduced form of these matrices and discuss the square roots of these matrices. Finally exploiting these properties, the development of structure-preserving algorithms for certain computations for generalized K-centrosymmetric H-matrices is discussed.     

On HSS and AHSS iteration methods for nonsymmetric positive definite Toeplitz systems

Two iteration methods are proposed to solve real nonsymmetric positive definite Toeplitz systems of linear equations. These methods are based on Hermitian and skew-Hermitian splitting (HSS) and accelerated Hermitian and skew-Hermitian splitting (AHSS). By constructing an orthogonal matrix and using a similarity transformation, the real Toeplitz linear system is transformed into a generalized saddle point problem. Then the structured HSS and the structured AHSS iteration methods are established by applying the HSS and the AHSS iteration methods to the generalized saddle point problem. We discuss efficient implementations and demonstrate that the structured HSS and the structured AHSS iteration methods have better behavior than the HSS iteration method in terms of both computational complexity and convergence speed. Moreover, the structured AHSS iteration method outperforms the HSS and the structured HSS iteration methods. The structured AHSS iteration method also converges unconditionally to the unique solution of the Toeplitz linear system. In addition, an upper bound for the contraction factor of the structured AHSS iteration method is derived. Numerical experiments are used to illustrate the effectiveness of the structured AHSS iteration method.     

Block triangular preconditioners for nonsymmetric saddle point problems: field-of-values analysis

A preconditioned minimal residual method for nonsymmetric saddle point problems is analyzed. The proposed preconditioner is of block triangular form. The aim of this article is to show that a rigorous convergence analysis can be performed by using the field of values of the preconditioned linear system. As an example, a saddle point problem obtained from a mixed finite element discretization of the Oseen equations is considered. The convergence estimates obtained by using a field–of–values analysis are independent of the discretization parameter h. Several computational experiments supplement the theoretical results and illustrate the performance of the method. Received March 20, 1997 / Revised version received January 14, 1998     

The spectral properties of the preconditioned matrix for nonsymmetric saddle point problems

In this paper, on the basis of matrix splitting, two preconditioners are proposed and analyzed, for nonsymmetric saddle point problems. The spectral property of the preconditioned matrix is studied in detail. When the iteration parameter becomes small enough, the eigenvalues of the preconditioned matrices will gather into two clusters—one is near (0,0) and the other is near (2,0)—for the PPSS preconditioner no matter whether A is Hermitian or non-Hermitian and for the PHSS preconditioner when A is a Hermitian or real normal matrix. Numerical experiments are given, to illustrate the performances of the two preconditioners.     

Towards an optimal condition number of certain augmented Lagrangian‐type saddle‐point matrices

We present an analysis for minimizing the condition number of nonsingular parameter‐dependent 2 × 2 block‐structured saddle‐point matrices with a maximally rank‐deficient (1,1) block. The matrices arise from an augmented Lagrangian approach. Using quasidirect sums, we show that a decomposition akin to simultaneous diagonalization leads to an optimization based on the extremal nonzero eigenvalues and singular values of the associated block matrices. Bounds on the condition number of the parameter‐dependent matrix are obtained, and we demonstrate their tightness on some numerical examples. Copyright © 2016 John Wiley & Sons, Ltd.     

Fast transforms for tridiagonal linear equations

In this paper we study the use of the Fourier, Sine and Cosine Transform for solving or preconditioning linear systems, which arise from the discretization of elliptic problems. Recently, R. Chan and T. Chan considered circulant matrices for solving such systems. Instead of using circulant matrices, which are based on the Fourier Transform, we apply the Fourier and the Sine Transform directly. It is shown that tridiagonal matrices arising from the discretization of an onedimensional elliptic PDE are connected with circulant matrices by congruence transformations with the Fourier or the Sine matrix. Therefore, we can solve such linear systems directly, using only Fast Fourier Transforms and the Sherman-Morrison-Woodbury formula. The Fast Fourier Transform is highly parallelizable, and thus such an algorithm is interesting on a parallel computer. Moreover, similar relations hold between block tridiagonal matrices and Block Toeplitz-plus-Hankel matrices of ordern ²×n ² in the 2D case. This can be used to define in some sense natural approximations to the given matrix which lead to preconditioners for solving such linear systems.     

New choices of preconditioning matrices for generalized inexact parameterized iterative methods

For large sparse saddle point problems, Chen and Jiang recently studied a class of generalized inexact parameterized iterative methods (see [F. Chen, Y.-L. Jiang, A generalization of the inexact parameterized Uzawa methods for saddle point problems, Appl. Math. Comput. 206 (2008) 765-771]). In this paper, the methods are modified and some choices of preconditioning matrices are given. These preconditioning matrices have advantages in solving large sparse linear system. Numerical experiments of a model Stokes problem are presented.     

Spectral Analysis for HSS Preconditioners

In this paper,we are interested in HSS preconditioners for saddle point lin- ear systems with a nonzero(2,2)-th block.We study an approximation of the spectra of HSS preconditioned matrices and use these results to illustrate and explain the spectra obtained from numerical examples,where the previous spectral analysis of HSS precon- ditioned matrices does not cover.