Generalized skew‐Hermitian triangular splitting iteration methods for saddle‐point linear systems

A generalized skew‐Hermitian triangular splitting iteration method is presented for solving non‐Hermitian linear systems with strong skew‐Hermitian parts. We study the convergence of the generalized skew‐Hermitian triangular splitting iteration methods for non‐Hermitian positive definite linear systems, as well as spectrum distribution of the preconditioned matrix with respect to the preconditioner induced from the generalized skew‐Hermitian triangular splitting. Then the generalized skew‐Hermitian triangular splitting iteration method is applied to non‐Hermitian positive semidefinite saddle‐point linear systems, and we prove its convergence under suitable restrictions on the iteration parameters. By specially choosing the values of the iteration parameters, we obtain a few of the existing iteration methods in the literature. Numerical results show that the generalized skew‐Hermitian triangular splitting iteration methods are effective for solving non‐Hermitian saddle‐point linear systems with strong skew‐Hermitian parts. Copyright © 2013 John Wiley & Sons, Ltd.     

BCCB preconditioners for systems of BVM‐based numerical integrators

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 A_k1,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.     

Modified Hermitian and skew‐Hermitian splitting methods for non‐Hermitian positive‐definite linear systems

To further study the Hermitian and non‐Hermitian splitting methods for a non‐Hermitian and positive‐definite matrix, we introduce a so‐called lopsided Hermitian and skew‐Hermitian splitting and then establish a class of lopsided Hermitian/skew‐Hermitian (LHSS) methods to solve the non‐Hermitian and positive‐definite systems of linear equations. These methods include a two‐step LHSS iteration and its inexact version, the inexact Hermitian/skew‐Hermitian (ILHSS) iteration, which employs some Krylov subspace methods as its inner process. We theoretically prove that the LHSS method converges to the unique solution of the linear system for a loose restriction on the parameter α. Moreover, the contraction factor of the LHSS iteration is derived. The presented numerical examples illustrate the effectiveness of both LHSS and ILHSS iterations. Copyright © 2007 John Wiley & Sons, Ltd.     

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.     

On modified block SSOR iteration methods for linear systems from steady incompressible viscous flow problems

In order to solve the large sparse systems of linear equations arising from numerical solutions of two-dimensional steady incompressible viscous flow problems in primitive variable formulation, we present block SSOR and modified block SSOR iteration methods based on the special structures of the coefficient matrices. In each step of the block SSOR iteration, we employ the block LU factorization to solve the sub-systems of linear equations. We show that the block LU factorization is existent and stable when the coefficient matrices are block diagonally dominant of type-II by columns. Under suitable conditions, we establish convergence theorems for both block SSOR and modified block SSOR iteration methods. In addition, the block SSOR iteration and AF-ADI method are considered as preconditioners for the nonsymmetric systems of linear equations. Numerical experiments show that both block SSOR and modified block SSOR iterations are feasible iterative solvers and they are also effective for preconditioning Krylov subspace methods such as GMRES and BiCGSTAB when used to solve this class of systems of linear equations.     

Diagonal and Toeplitz splitting iteration methods for diagonal‐plus‐Toeplitz linear systems from spatial fractional diffusion equations

The finite difference discretization of the spatial fractional diffusion equations gives discretized linear systems whose coefficient matrices have a diagonal‐plus‐Toeplitz structure. For solving these diagonal‐plus‐Toeplitz linear systems, we construct a class of diagonal and Toeplitz splitting iteration methods and establish its unconditional convergence theory. In particular, we derive a sharp upper bound about its asymptotic convergence rate and deduct the optimal value of its iteration parameter. The diagonal and Toeplitz splitting iteration method naturally leads to a diagonal and circulant splitting preconditioner. Analysis shows that the eigenvalues of the corresponding preconditioned matrix are clustered around 1, especially when the discretization step‐size h is small. Numerical results exhibit that the diagonal and circulant splitting preconditioner can significantly improve the convergence properties of GMRES and BiCGSTAB, and these preconditioned Krylov subspace iteration methods outperform the conjugate gradient method preconditioned by the approximate inverse circulant‐plus‐diagonal preconditioner proposed recently by Ng and Pan (M.K. Ng and J.‐Y. Pan, SIAM J. Sci. Comput. 2010;32:1442‐1464). Moreover, unlike this preconditioned conjugate gradient method, the preconditioned GMRES and BiCGSTAB methods show h‐independent convergence behavior even for the spatial fractional diffusion equations of discontinuous or big‐jump coefficients.     

Strang‐type preconditioners applied to ordinary and neutral differential‐algebraic equations

This paper deals with boundary‐value methods (BVMs) for ordinary and neutral differential‐algebraic equations. Different from what has been done in Lei and Jin (Lecture Notes in Computer Science, vol. 1988. Springer: Berlin, 2001; 505–512), here, we directly use BVMs to discretize the equations. The discretization will lead to a nonsymmetric large‐sparse linear system, which can be solved by the GMRES method. In order to accelerate the convergence rate of GMRES method, two Strang‐type block‐circulant preconditioners are suggested: one is for ordinary differential‐algebraic equations (ODAEs), and the other is for neutral differential‐algebraic equations (NDAEs). Under some suitable conditions, it is shown that the preconditioners are invertible, the spectra of the preconditioned systems are clustered, and the solution of iteration converges very rapidly. The numerical experiments further illustrate the effectiveness of the methods. Copyright © 2011 John Wiley & Sons, Ltd.     

Tuned preconditioners for inexact two‐sided inverse and Rayleigh quotient iteration

Convergence results are provided for inexact two‐sided inverse and Rayleigh quotient iteration, which extend the previously established results to the generalized non‐Hermitian eigenproblem and inexact solves with a decreasing solve tolerance. Moreover, the simultaneous solution of the forward and adjoint problem arising in two‐sided methods is considered, and the successful tuning strategy for preconditioners is extended to two‐sided methods, creating a novel way of preconditioning two‐sided algorithms. Furthermore, it is shown that inexact two‐sided Rayleigh quotient iteration and the inexact two‐sided Jacobi‐Davidson method (without subspace expansion) applied to the generalized preconditioned eigenvalue problem are equivalent when a certain number of steps of a Petrov–Galerkin–Krylov method is used and when this specific tuning strategy is applied. Copyright © 2014 John Wiley & Sons, Ltd.     

Steepest descent preconditioning for nonlinear GMRES optimization

Steepest descent preconditioning is considered for the recently proposed nonlinear generalized minimal residual (N‐GMRES) optimization algorithm for unconstrained nonlinear optimization. Two steepest descent preconditioning variants are proposed. The first employs a line search, whereas the second employs a predefined small step. A simple global convergence proof is provided for the N‐GMRES optimization algorithm with the first steepest descent preconditioner (with line search), under mild standard conditions on the objective function and the line search processes. Steepest descent preconditioning for N‐GMRES optimization is also motivated by relating it to standard non‐preconditioned GMRES for linear systems in the case of a standard quadratic optimization problem with symmetric positive definite operator. Numerical tests on a variety of model problems show that the N‐GMRES optimization algorithm is able to very significantly accelerate convergence of stand‐alone steepest descent optimization. Moreover, performance of steepest‐descent preconditioned N‐GMRES is shown to be competitive with standard nonlinear conjugate gradient and limited‐memory Broyden–Fletcher–Goldfarb–Shanno methods for the model problems considered. These results serve to theoretically and numerically establish steepest‐descent preconditioned N‐GMRES as a general optimization method for unconstrained nonlinear optimization, with performance that appears promising compared with established techniques. In addition, it is argued that the real potential of the N‐GMRES optimization framework lies in the fact that it can make use of problem‐dependent nonlinear preconditioners that are more powerful than steepest descent (or, equivalently, N‐GMRES can be used as a simple wrapper around any other iterative optimization process to seek acceleration of that process), and this potential is illustrated with a further application example. Copyright © 2012 John Wiley & Sons, Ltd.     

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.

     

BFGS‐like updates of constraint preconditioners for sequences of KKT linear systems in quadratic programming

We focus on efficient preconditioning techniques for sequences of Karush‐Kuhn‐Tucker (KKT) linear systems arising from the interior point (IP) solution of large convex quadratic programming problems. Constraint preconditioners (CPs), although very effective in accelerating Krylov methods in the solution of KKT systems, have a very high computational cost in some instances, because their factorization may be the most time‐consuming task at each IP iteration. We overcome this problem by computing the CP from scratch only at selected IP iterations and by updating the last computed CP at the remaining iterations, via suitable low‐rank modifications based on a BFGS‐like formula. This work extends the limited‐memory preconditioners (LMPs) for symmetric positive definite matrices proposed by Gratton, Sartenaer and Tshimanga in 2011, by exploiting specific features of KKT systems and CPs. We prove that the updated preconditioners still belong to the class of exact CPs, thus allowing the use of the conjugate gradient method. Furthermore, they have the property of increasing the number of unit eigenvalues of the preconditioned matrix as compared with the generally used CPs. Numerical experiments are reported, which show the effectiveness of our updating technique when the cost for the factorization of the CP is high.     

Additive block diagonal preconditioning for block two-by-two linear systems of skew-Hamiltonian coefficient matrices

For a class of block two-by-two systems of linear equations with certain skew-Hamiltonian coefficient matrices, we construct additive block diagonal preconditioning matrices and discuss the eigen-properties of the corresponding preconditioned matrices. The additive block diagonal preconditioners can be employed to accelerate the convergence rates of Krylov subspace iteration methods such as MINRES and GMRES. Numerical experiments show that MINRES preconditioned by the exact and the inexact additive block diagonal preconditioners are effective, robust and scalable solvers for the block two-by-two linear systems arising from the Galerkin finite-element discretizations of a class of distributed control problems.     

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.     

Convergence properties of preconditioned Hermitian and skew-Hermitian splitting methods for non-Hermitian positive semidefinite matrices

For the non-Hermitian and positive semidefinite systems of linear equations, we derive necessary and sufficient conditions for guaranteeing the unconditional convergence of the preconditioned Hermitian and skew-Hermitian splitting iteration methods. We then apply these results to block tridiagonal linear systems in order to obtain convergence conditions for the corresponding block variants of the preconditioned Hermitian and skew-Hermitian splitting iteration methods.

     

Minimum residual methods for augmented systems

For large systems of linear equations, iterative methods provide attractive solution techniques. We describe the applicability and convergence of iterative methods of Krylov subspace type for an important class of symmetric and indefinite matrix problems, namely augmented (or KKT) systems. Specifically, we consider preconditioned minimum residual methods and discuss indefinite versus positive definite preconditioning. For a natural choice of starting vector we prove that when the definite and indenfinite preconditioners are related in the obvious way, MINRES (which is applicable in the case of positive definite preconditioning) and full GMRES (which is applicable in the case of indefinite preconditioning) give residual vectors with identical Euclidean norm at each iteration. Moreover, we show that the convergence of both methods is related to a system of normal equations for which the LSQR algorithm can be employed. As a side result, we give a rare example of a non-trivial normal(1) matrix where the corresponding inner product is explicitly known: a conjugate gradient method therefore exists and can be employed in this case. This work was supported by British Council/German Academic Exchange Service Research Collaboration Project 465 and NATO Collaborative Research Grant CRG 960782     

A Class of Nested Iteration Schemes for Linear Systems with a Coefficient Matrix with a Dominant Positive Definite Symmetric Part

We present a class of nested iteration schemes for solving large sparse systems of linear equations with a coefficient matrix with a dominant symmetric positive definite part. These new schemes are actually inner/outer iterations, which employ the classical conjugate gradient method as inner iteration to approximate each outer iterate, while each outer iteration is induced by a convergent and symmetric positive definite splitting of the coefficient matrix. Convergence properties of the new schemes are studied in depth, possible choices of the inner iteration steps are discussed in detail, and numerical examples from the finite-difference discretization of a second-order partial differential equation are used to further examine the effectiveness and robustness of the new schemes over GMRES and its preconditioned variant. Also, we show that the new schemes are, at least, comparable to the variable-step generalized conjugate gradient method and its preconditioned variant.     

On a Chebyshev accelerated splitting iteration method with application to two‐by‐two block linear systems

The Chebyshev accelerated preconditioned modified Hermitian and skew‐Hermitian splitting (CAPMHSS) iteration method is presented for solving the linear systems of equations, which have two‐by‐two block coefficient matrices. We derive an iteration error bound to show that the new method is convergent as long as the eigenvalue bounds are not underestimated. Even when the spectral information is lacking, the CAPMHSS iteration method could be considered as an exponentially converging iterative scheme for certain choices of the method parameters. In this case, the convergence rate is independent of the parameters. Besides, the linear subsystems in each iteration can be solved inexactly, which leads to the inexact CAPMHSS iteration method. The iteration error bound of the inexact method is derived also. We discuss in detail the implementation of CAPMHSS for solving two models arising from the Galerkin finite‐element discretizations of distributed control problems and complex symmetric linear systems. The numerical results show the robustness and the efficiency of the new methods.     

Block‐triangular preconditioners for PDE‐constrained optimization

In this paper we investigate the possibility of using a block‐triangular preconditioner for saddle point problems arising in PDE‐constrained optimization. In particular, we focus on a conjugate gradient‐type method introduced by Bramble and Pasciak that uses self‐adjointness of the preconditioned system in a non‐standard inner product. We show when the Chebyshev semi‐iteration is used as a preconditioner for the relevant matrix blocks involving the finite element mass matrix that the main drawback of the Bramble–Pasciak method—the appropriate scaling of the preconditioners—is easily overcome. We present an eigenvalue analysis for the block‐triangular preconditioners that gives convergence bounds in the non‐standard inner product and illustrates their competitiveness on a number of computed examples. Copyright © 2010 John Wiley & Sons, Ltd.     

Multistep matrix splitting iteration preconditioning for singular linear systems

Multistep matrix splitting iterations serve as preconditioning for Krylov subspace methods for solving singular linear systems. The preconditioner is applied to the generalized minimal residual (GMRES) method and the flexible GMRES (FGMRES) method. We present theoretical and practical justifications for using this approach. Numerical experiments show that the multistep generalized shifted splitting (GSS) and Hermitian and skew-Hermitian splitting (HSS) iteration preconditioning are more robust and efficient compared to standard preconditioners for some test problems of large sparse singular linear systems.     

几乎各向同性的高维空间分数阶扩散方程的分块快速正则Hermite分裂预处理方法

一类空间分数阶扩散方程经过有限差分离散后所得到的离散线性方程组的系数矩阵是两个对角矩阵与Toeplitz型矩阵的乘积之和.在本文中,对于几乎各向同性的二维或三维空间分数阶扩散方程的离散线性方程组,采用预处理Krylov子空间迭代方法,我们利用其系数矩阵的特殊结构和具体性质构造了一类分块快速正则Hermite分裂预处理子.通过理论分析,我们证明了所对应的预处理矩阵的特征值大部分都聚集于1的附近.数值实验也表明,这类分块快速正则Hermite分裂预处理子可以明显地加快广义极小残量(GMRES)方法和稳定化的双共轭梯度(BiCGSTAB)方法等Krylov子空间迭代方法的收敛速度.