Modified Block SSOR Preconditioners for Symmetric Positive Definite Linear Systems

A class of modified block SSOR preconditioners is presented for the symmetric positive definite systems of linear equations, whose coefficient matrices come from the hierarchical-basis finite-element discretizations of the second-order self-adjoint elliptic boundary value problems. These preconditioners include a block SSOR iteration preconditioner, and two inexact block SSOR iteration preconditioners whose diagonal matrices except for the (1,1)-block are approximated by either point symmetric Gauss–Seidel iterations or incomplete Cholesky factorizations, respectively. The optimal relaxation factors involved in these preconditioners and the corresponding optimal condition numbers are estimated in details through two different approaches used by Bank, Dupont and Yserentant (Numer. Math. 52 (1988) 427–458) and Axelsson (Iterative Solution Methods (Cambridge University Press, 1994)). Theoretical analyses show that these modified block SSOR preconditioners are very robust, have nearly optimal convergence rates, and especially, are well suited to difficult problems with rough solutions, discretized using highly nonuniform, adaptively refined meshes.     

HYBRID ALGEBRAIC MULTILEVEL PRECONDITIONING METHODS

ＨＹＢＲＩＤＡＬＧＥＢＲＡＩＣＭＵＬＴＩＬＥＶＥＬＰＲＥＣＯＮＤＩＴＩＯＮＩＮＧＭＥＴＨＯＤＳ￥ＢａｉＺｈｏｎｇｚｈｉ（白中治）（ＦｕｄａｎＵｎｉｖｅｒｓｉｔｙ，复旦大学，邮编：２００４３３）Ａｂｓｔｒａｃｔ：Ａｃｌａｓｓｏｆｈｙｂｒｉｄａｌｇｅｂｒ...     

A multilevel block incomplete cholesky preconditioner for solving normal equations in linear least squares problems

An incomplete factorization method for preconditioning symmetric positive definite matrices is introduced to solve normal equations. The normal equations are form to solve linear least squares problems. The procedure is based on a block incomplete Cholesky factorization and a multilevel recursive strategy with an approximate Schur complement matrix formed implicitly. A diagonal perturbation strategy is implemented to enhance factorization robustness. The factors obtained are used as a preconditioner for the conjugate gradient method. Numerical experiments are used to show the robustness and efficiency of this preconditioning technique, and to compare it with two other preconditioners.     

A MODIFIED INVERSE ITERATION FOR A LARGE SPARSE SPD GENERALIZED EIGENPROBLEM

In this paper, an algorithm based on a shifted inverse power iteration for computing generalized eigenvalues with corresponding eigenvectors of a large scale sparse symmetric positive definite matrix pencil is presented. It converges globally with a cubic asymptotic convergence rate, preserves sparsity of the original matrices and is fully parallelizable. The algebraic multilevel itera-tion method (AMLI) is used to improve the efficiency when symmetric positive definite linear equa-tions need to be solved.     

Using Chebyshev polynomials and approximate inverse triangular factorizations for preconditioning the conjugate gradient method

In order to precondition a sparse symmetric positive definite matrix, its approximate inverse is examined, which is represented as the product of two sparse mutually adjoint triangular matrices. In this way, the solution of the corresponding system of linear algebraic equations (SLAE) by applying the preconditioned conjugate gradient method (CGM) is reduced to performing only elementary vector operations and calculating sparse matrix-vector products. A method for constructing the above preconditioner is described and analyzed. The triangular factor has a fixed sparsity pattern and is optimal in the sense that the preconditioned matrix has a minimum K-condition number. The use of polynomial preconditioning based on Chebyshev polynomials makes it possible to considerably reduce the amount of scalar product operations (at the cost of an insignificant increase in the total number of arithmetic operations). The possibility of an efficient massively parallel implementation of the resulting method for solving SLAEs is discussed. For a sequential version of this method, the results obtained by solving 56 test problems from the Florida sparse matrix collection (which are large-scale and ill-conditioned) are presented. These results show that the method is highly reliable and has low computational costs.     

Preconditioning indefinite discretization matrices

Summary The finite element discretization of many elliptic boundary value problems leads to linear systems with positive definite and symmetric coefficient matrices. Many efficient preconditioners are known for these systems. We show that these preconditioning matrices can also be used for the linear systems arising from boundary value problems which are potentially indefinite due to lower order terms in the partial differential equation. Our main tool is a careful algebraic analysis of the condition numbers and the spectra of perturbed matrices which are preconditioned by the same matrices as in the unperturbed case.     

Preconditioning of GMRES by skew-Hermitian iterations

A class of preconditioners for solving non-Hermitian positive definite systems of linear algebraic equations is proposed and investigated. It is based on Hermitian and skew-Hermitian splitting of the initial matrix. A generalization for saddle point systems having semidefinite or singular (1, 1) blocks is given. Our approach is based on an augmented Lagrangian formulation. It is shown that such preconditioners can be efficiently used for the iterative solution of systems of linear algebraic equations by the GMRES method.     

On the sublinear and superlinear rate of convergence of conjugate gradient methods

The convergence of conjugate gradient methods for solving systems of linear equations with a symmetric positive definite matrix takes typically place in three phases, with a sublinear, linear and superlinear convergence rate, respectively. This behaviour can be explained using various generalized condition numbers which depend on all eigenvalues and on the initial error vector and using annihilating polynomials for the extreme eigenvalues. The analysis also indicates that it can be most efficient to use different preconditioners in different phases.     

A UNIFIED FRAMEWORK FOR THE CONSTRUCTION OF VARIOUS ALGEBRAIC MULTILEVEL PRECONDITIONING METHODS

1.IntroductionThediscretizationofmanysecondorderselfadjointellipticboundaryvalueproblemsbythefiniteelementmethodleadstolargesparsesystemsoflinearequationswithsymmetricpositivedefinite(SPD)coefficientmatrices.Fortheselinearsystems,algebraicmultilevelp...     

Structured preconditioners for nonsingular matrices of block two-by-two structures

For the large sparse block two-by-two real nonsingular matrices, we establish a general framework of practical and efficient structured preconditioners through matrix transformation and matrix approximations. For the specific versions such as modified block Jacobi-type, modified block Gauss-Seidel-type, and modified block unsymmetric (symmetric) Gauss-Seidel-type preconditioners, we precisely describe their concrete expressions and deliberately analyze eigenvalue distributions and positive definiteness of the preconditioned matrices. Also, we show that when these structured preconditioners are employed to precondition the Krylov subspace methods such as GMRES and restarted GMRES, fast and effective iteration solvers can be obtained for the large sparse systems of linear equations with block two-by-two coefficient matrices. In particular, these structured preconditioners can lead to efficient and high-quality preconditioning matrices for some typical matrices from the real-world applications.

     

Massively parallel preconditioners for symmetric positive definite linear systems

In this paper, we address the problem of solving sparse symmetric linear systems on parallel computers. With further restrictive assumptions on the matrix (e.g., bidiagonal or tridiagonal structure), several direct methods may be used. These methods give ideas for constructing efficient data parallel preconditioners for general positive definite symmetric matrices. We describe two examples of such preconditioners for which the factorization (i.e., the construction of the preconditioning matrix) turns out to be parallel. This revised version was published online in June 2006 with corrections to the Cover Date.     

A framework of parallel algebraic multilevel preconditioning iterations

1.IntroductionConsiderthesyllUnetricpositivedeflate(SPD)systemsoflinearequationsthatariseinfiniteelementdiscretisstionsofmanysecond-orderself-adjointellipticboundaryvalueproblems.Tosolvethisclassoflinearsystemsiteratively,AxelssonandVassilevski[1--4]preselltedthealgebraicmultileveliteration(AMLI)methodsbyreasonablyutilizingthemultigridtechniqueandthepolynomialaccelerationstrategy.Thesemethodsareamongthemostefficientiterativesolversbecausetheirpreconditioningmatricesarespectrallyequlvalellt…     

特殊五对角与七对角对称正定矩阵的一类反问题

针对线性代数方程组Ax=b,利用矩阵分解的思想,构造一类特殊五对角与七对角对称正定阵的矩阵分解,获得这类矩阵反问题解存在的充要条件和通解表达式.最后,给出了具体算法与数值算例.     

Multilevel approaches for FSAI preconditioning

Factorized sparse approximate inverse (FSAI) preconditioners are robust algorithms for symmetric positive matrices, which are particularly attractive in a parallel computational environment because of their inherent and almost perfect scalability. Their parallel degree is even redundant with respect to the actual capabilities of the current computational architectures. In this work, we present two new approaches for FSAI preconditioners with the aim of improving the algorithm effectiveness by adding some sequentiality to the native formulation. The first one, denoted as block tridiagonal FSAI, is based on a block tridiagonal factorization strategy, whereas the second one, domain decomposition FSAI, is built by reordering the matrix graph according to a multilevel k‐way partitioning method followed by a bandwidth minimization algorithm. We test these preconditioners by solving a set of symmetric positive definite problems arising from different engineering applications. The results are evaluated in terms of performance, scalability, and robustness, showing that both strategies lead to faster convergent schemes regarding the number of iterations and total computational time in comparison with the native FSAI with no significant loss in the algorithmic parallel degree.     

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.     

Matrix orthogonal polynomials whose derivatives are also orthogonal

In this paper we prove some characterizations of the matrix orthogonal polynomials whose derivatives are also orthogonal, which generalize other known ones in the scalar case. In particular, we prove that the corresponding orthogonality matrix functional is characterized by a Pearson-type equation with two matrix polynomials of degree not greater than 2 and 1. The proofs are given for a general sequence of matrix orthogonal polynomials, not necessarily associated with a hermitian functional. We give several examples of non-diagonalizable positive definite weight matrices satisfying a Pearson-type equation, which show that the previous results are non-trivial even in the positive definite case.A detailed analysis is made for the class of matrix functionals which satisfy a Pearson-type equation whose polynomial of degree not greater than 2 is scalar. We characterize the Pearson-type equations of this kind that yield a sequence of matrix orthogonal polynomials, and we prove that these matrix orthogonal polynomials satisfy a second order differential equation even in the non-hermitian case. Finally, we prove and improve a conjecture of Durán and Grünbaum concerning the triviality of this class in the positive definite case, while some examples show the non-triviality for hermitian functionals which are not positive definite.     

Diagonally compensated reduction and related preconditioning methods

When solving linear algebraic equations with large and sparse coefficient matrices, arising, for instance, from the discretization of partial differential equations, it is quite common to use preconditioning to accelerate the convergence of a basic iterative scheme. Incomplete factorizations and sparse approximate inverses can provide efficient preconditioning methods but their existence and convergence theory is based mostly on M-matrices (H-matrices). In some application areas, however, the arising coefficient matrices are not H-matrices. This is the case, for instance, when higher-order finite element approximations are used, which is typical for structural mechanics problems. We show that modification of a symmetric, positive definite matrix by reduction of positive offdiagonal entries and diagonal compensation of them leads to an M-matrix. This diagonally compensated reduction can take place in the whole matrix or only at the current pivot block in a recursive incomplete factorization method. Applications for constructing preconditioning matrices for finite element matrices are described.     

Preconditioners for the pressure-correction method applied to the unsteady stokes problem

In this paper, the unsteady Stokes problem is considered and also the pressure-correction method for the problem is described. At a fixed time level, we reduce the problem to two summetric positive definite problems which depend on a time step parameter. Linear systems that arise from the problems are large, sparse, symmetric, positive definite and ill-conditioned as the time step tends to zero. Preconditioned problems based on an additive Schwarz method for solving the symmetric positive definite problems are derived and preconditioners are defined implicitly. It will be shown that the rate of convergence is independent of the mesh parameters as well as the time step size     

On SSOR‐like preconditioners for non‐Hermitian positive definite matrices

We construct, analyze, and implement SSOR‐like preconditioners for non‐Hermitian positive definite system of linear equations when its coefficient matrix possesses either a dominant Hermitian part or a dominant skew‐Hermitian part. We derive tight bounds for eigenvalues of the preconditioned matrices and obtain convergence rates of the corresponding SSOR‐like iteration methods as well as the corresponding preconditioned GMRES iteration methods. Numerical implementations show that Krylov subspace iteration methods such as GMRES, when accelerated by the SSOR‐like preconditioners, are efficient solvers for these classes of non‐Hermitian positive definite linear systems. Copyright © 2015 John Wiley & Sons, Ltd.