A practical formula for computing optimal parameters in the HSS iteration methods

Some convergence conditions on successive over-relaxed (SOR) iterative method and symmetric SOR (SSOR) iterative method are proposed for non-Hermitian positive definite linear systems. Some examples are given to demonstrate the results obtained.

A generalized preconditioned HSS method for non-Hermitian positive definite linear systems

SUMMARY 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.

We study the role of preconditioning strategies recently developed for coercive problems in connection with a two-step iterative method, based on the Hermitian skew-Hermitian splitting (HSS) of the coefficient matrix, proposed by Bai, Golub and Ng for the solution of nonsymmetric linear systems whose real part is coercive. As a model problem we consider Finite Differences (FD) matrix sequences {A n (a,p)} n discretizing the elliptic (convection-diffusion) problem with Ω being a plurirectangle of R d with a(x) being a uniformly positive function and p(x) denoting the Reynolds function: here for plurirectangle we mean a connected union of rectangles in d dimensions with edges parallel to the axes. More precisely, in connection with preconditioned HSS/GMRES like methods, we consider the preconditioning sequence {P n (a)} n , P n (a):= D n 1/2(a)A n (1,0) D n 1/2(a) where D n (a) is the suitably scaled main diagonal of A n (a,0). If a(x) is positive and regular enough, then the preconditioned sequence shows a strong clustering at unity so that the sequence {P n (a)} n turns out to be a superlinear preconditioning sequence for {A n (a,0)} n where A n (a,0) represents a good approximation of Re(A n (a,p)) namely the real part of A n (a,p). The computational interest is due to the fact that the preconditioned HSS method has a convergence behavior depending on the spectral properties of {P n -1(a)Re(A n (a,p))} n ≈ {P n -1(a)A n (a,0)} n : therefore the solution of a linear system with coefficient matrix A n (a,p) is reduced to computations involving diagonals and to the use of fast Poisson solvers for {A n (1,0)} n . Some numerical experimentations confirm the optimality of the discussed proposal and its superiority with respect to existing techniques.

By further generalizing the concept of Hermitian (or normal) and skew-Hermitian splitting for a non-Hermitian and positive-definite matrix, we introduce a new splitting, called positive-definite and skew-Hermitian splitting (PSS), and then establish a class of PSS methods similar to the Hermitian (or normal) and skew-Hermitian splitting (HSS or NSS) method for iteratively solving the positive-definite systems of linear equations. Theoretical analysis shows that the PSS method converges unconditionally to the exact solution of the linear system, with the upper bound of its convergence factor dependent only on the spectrum of the positive-definite splitting matrix and independent of the spectrum of the skew-Hermitian splitting matrix as well as the eigenvectors of all matrices involved. When we specialize the PSS to block triangular (or triangular) and skew-Hermitian splitting (BTSS or TSS), the PSS method naturally leads to a BTSS or TSS iteration method, which may be more practical and efficient than the HSS and NSS iteration methods. Applications of the BTSS method to the linear systems of block two-by-two structures are discussed in detail. Numerical experiments further show the effectiveness of our new methods.

This paper discusses a class of two-stage iterations whose outer iterative methods are the block SOR methods for parallel solution of linear systems. Convergence is showed for symmetric positive definite linear systems, and an approximate optimal relaxation factor is defined for block tridiagonal matrices. The numerical results for Poisson model problem are presented.

Some results on convergence of two-stage iterative methods for symmetric positive definite linear systems

A new method for computation of positive realizations of given transfer matrices of linear discrete-time linear systems is proposed. Sufficient conditions for the existence of positive realizations of transfer matrices are given. A procedure for computation of the positive realizations is proposed and illustrated by an example.

This paper derives conditions under which co-diagonalization of the controllability and observability Gramians can be performed in a physically consistent manner for the class of linear quantum stochastic systems that are common in the fields of quantum optics and related areas like quantum nano- and optomechanical systems. Such a class is essentially a quantum analogue of classical (non-quantum) linear stochastic systems. However, unlike their classical counterparts, the laws of quantum mechanics impose nonlinear equality constraints on the system matrices of linear quantum stochastic systems, the so-called physical realizability constraints. Therefore, a meaningful model reduction procedure must yield a physically realizable reduced model satisfying these constraints. Previous work has shown that model reduction by subsystem truncation preserves physical realizability, but did not propose a method for truncating a subsystem. One candidate approach for this truncation is a quantum adaptation of the well-known balanced truncation method for classical linear time-invariant systems. It is shown in this paper that balanced realization for linear quantum stochastic systems is only possible under a strongly restrictive necessary and sufficient condition. However, the paper also derives less restrictive necessary and sufficient conditions for other realizations with simultaneously diagonal controllability and observability Gramians, and introduces the notion of a quasi-balanced realization. An example of the application of the results is provided to demonstrate quasi-balanced truncation in the linear quantum stochastic setting.

Direct optimization of BPX preconditioners

The aim of this paper is to introduce a technique for the computation of the principal matrix of the linear system associated to an ordinary differential equation with constant cofficients, through symbolic computation, via an algebraic processor which works with matrices of quasi—polynomials.

Modified HSS iteration methods for a class of non-Hermitian positive-definite linear systems

This issue contains two Survey and Review papers. The first, by Qinmeng Zou and Frédéric Magoulès, is “Delayed Gradient Methods for Symmetric and Positive Definite Linear Systems.” Gradient methods are the oldest and simplest algorithms to minimize a real objective function (f(x)). The ((n+1))st approximation to the minimizer is defined as (x_n+1 = x_n -\alpha_n g_n), where \(g_n\) is the gradient (\nabla f(x_n)) and (\alpha_n>0) is a steplength that depends on the specific method being used. Old and simple as the idea may be, gradient algorithms are of much current interest in the literature; for instance, they played a major role in the influential survey devoted to optimization in machine learning published in Volume 60, Issue 2 of this journal. The paper by Zou and Magoulès focuses on the quadratic case $f(x) = (1/2)x^TAx-b^Tx$ ($A$ symmetric and positive definite), where finding the minimizer is of course equivalent to solving the linear system $Ax=b$ and the gradient $g_n$ coincides with the residual $Ax_n-b$. Well-known strategies to determine the steplength include steepest descent, where $\alpha_n$ is chosen so as to minimize $f(x_{n+1})$, and minimal gradient (or minimal residual), where one rather minimizes the length of $g_{n+1}$. In both of these strategies, the value of $\alpha_n$ depends only on $g_n$. The term “delayed” in the title of the article refers to methods where the recipe to determine $\alpha_n$ includes information from past gradients $g_{n-1}, g_{n-2}$, \dots, and/or past stepsizes $\alpha_{n-1}$, $\alpha_{n-2}$, \dots. The numerical experiments reported clearly indicate that such delayed strategies may give rise to algorithms that are competitive with conjugate gradient methods in large ill-conditioned problems. The paper presents a neat summary of the recent results in this area and of the techniques used to derive them. Mark Van der Boor, Sem C. Borst, Johan S. H. Van Leeuwaarden, and Debankur Mukherjee are the authors of the second paper, “Scalable Load Balancing in Networked Systems: A Survey of Recent Advances.” The problem under consideration is as follows. A dispatcher receives clients that arrive randomly, and her job is to direct them to one of $N\gg 1$ servers. The time required by each client to be served is also random, so that a queue (of random length) of waiting clients will be formed at each server. How should the dispatcher proceed to expedite the service? As one would expect in these days of cloud networks and data systems with massive number of individual centers, the problem is currently receiving much attention in the literature. A strategy that suggests itself is the so-called “join the shortest queue” (JSQ), where on their arrival clients are directed to the server having the shortest queue. While JSQ has been proved to possess several favorable properties, it may not be the best option, due to communication overheads: each time a client arrives, the dispatcher has to communicate with all servers to find the lengths of their queues. The paper analyzes, in the limit $N\rightarrow \infty$, many alternative strategies. Nonspecialists will have little difficulty in reading the easily accessible first few sections and may be interested in discovering how even small tweaks in the algorithms may result in substantial improvements of their performance.

A general class of iterative methods is introduced for solving symmetric, positive definite linear systems. These methods use two different approximations to the inverse of the matrix of the problem, one of which involves the inverse of a smaller matrix. It is shown that the methods of this class reduce the error by a constant factor at each step and that under “ideal” circumstances this constant is equal to ${{(\kappa ' - 1)} / {(\kappa ' + 1)}}$, where $\kappa '$ is the ratio of the largest eigenvalue to the $(J + 1)$ eigenvalue of the matrix, J being the dimension of the smaller matrix involved. A multigrid method is presented as an example of a method of this class, and it is shown that while the multigrid method does not quite achieve this optimal rate of convergence, it does reduce the error at each step by a constant factor independent of the mesh spacing h. The size of this constant factor and properties of the differential equation and the discretization that affect it are also discussed.

Parallel multisplitting two-stage iterative methods with general weighting matrices for non-symmetric positive definite systems