Matrix Splittings Research Articles

A class of block alternating splitting implicit iteration methods for double saddle point linear systems

AbstractIn the framework of a special block alternating splitting implicit (BASI) iteration scheme for generalized saddle point problems, we establish some new iteration methods for solving double saddle point problems by means of a suitable partitioning strategy. Convergence analysis of the corresponding BASI iteration methods indicates that they are convergent unconditionally under certain weak requirements for the related matrix splittings, which are satisfied directly for our specific application to double saddle point problems. Numerical examples for liquid crystal director and time‐harmonic eddy current models are presented to demonstrate the efficiency of the proposed BASI preconditioners to accelerate the GMRES method.

On Robustly Convergent and Efficient Iterative Methods for Anisotropic Radiative Transfer

This paper considers the iterative solution of linear systems arising from discretization of the anisotropic radiative transfer equation with discontinuous elements on the sphere. In order to achieve robust convergence behavior in the discretization parameters and the physical parameters we develop preconditioned Richardson iterations in Hilbert spaces. We prove convergence of the resulting scheme. The preconditioner is constructed in two steps. The first step borrows ideas from matrix splittings and ensures mesh independence. The second step uses a subspace correction technique to reduce the influence of the optical parameters. The correction spaces are build from low-order spherical harmonics approximations generalizing well-known diffusion approximations. We discuss in detail the efficient implementation and application of the discrete operators. In particular, for the considered discontinuous spherical elements, the scattering operator becomes dense and we show that mathcal {H}- or mathcal {H}^2-matrix compression can be applied in a black-box fashion to obtain almost linear or linear complexity when applying the corresponding approximations. The effectiveness of the proposed method is shown in numerical examples.

Asynchronous iterations of HSS method for non-Hermitian linear systems

A general asynchronous alternating iterative model is designed, for which convergence is theoretically ensured both under classical spectral radius bound and, then, for a classical class of matrix splittings for -matrices. The computational model can be thought of as a two-stage alternating iterative method, which well suits to the well-known Hermitian and skew-Hermitian splitting (HSS) approach, with the particularity here of considering only one inner iteration. Experimental parallel performance comparison is conducted between the generalized minimal residual (GMRES) algorithm, the standard HSS and our asynchronous variant, on both real and complex non-Hermitian linear systems, respectively, arising from convection–diffusion and structural dynamics problems. A significant gain on execution time is observed in both cases.

Regularized iterative method for ill-posed linear systems based on matrix splitting

In this paper, the concept of matrix splitting is introduced to solve a large sparse ill-posed linear system via Tikhonov?s regularization. In the regularization process, we convert the ill-posed system to a well-posed system. The convergence of such a well-posed system is discussed by using different types of matrix splittings. Comparison analysis of both systems are studied by operating certain types of weak splittings. Further, we have extended the double splitting of [Song J. and Song Y, Calcolo 48(3), 245-260, 2011] to double weak splitting of type II for nonsingular symmetric matrices. In addition to that, some more comparison results are presented with the help of such weak double splittings of type I and type II.

A note on double weak splittings of type II

ABSTRACT Iterative methods based on matrix splittings are useful tools in solving real large sparse linear systems. In this aspect, the type I double splitting approaches are straight forward from the formulation of the iteration scheme and its convergence theory is well established in the literature. However, if a double splitting is of type II, then the convergence of the iteration scheme seems not to be straight forward. In this paper, we develop convergence theory for type II double splittings to make the implementation quite simple. In this direction, we first introduce two new subclasses of double splittings and establish their convergence theory. Using this theory, we prove a new characterization of a monotone matrix. Finally, we apply our theoretical findings to the double splitting of an M-matrix in the Gauss–Seidel double SOR method to obtain a comparison result.

Convergence theory of iterative methods based on proper splittings and proper multisplittings for rectangular linear systems

Multisplitting methods are useful to solve differential-algebraic equations. In this connection, we discuss the theory of matrix splittings and multisplittings, which can be used for finding the iterative solution of a large class of rectangular (singular) linear system of equations of the form Ax = b. In this direction, many convergence results are proposed for different subclasses of proper splittings in the literature. But, in some practical cases, the convergence speed of the iterative scheme is very slow. To overcome this issue, several comparison results are obtained for different subclasses of proper splittings. This paper also presents a few such results. However, this idea fails to accelerate the speed of the iterative scheme in finding the iterative solution. In this regard, Climent and Perea [J. Comput. Appl. Math. 158 (2003), 43-48: MR2013603] introduced the notion of proper multisplittings to solve the system Ax = b on parallel and vector machines, and established convergence theory for a subclass of proper multisplittings. With the aim to extend the convergence theory of proper multisplittings, this paper further adds a few results. Some of the results obtained in this paper are even new for the iterative theory of nonsingular linear systems.

Three-step alternating iterations for index 1 and non-singular matrices

Iterative methods based on matrix splittings are useful in solving large sparse linear systems. In this direction, proper splittings and its several extensions are used to deal with singular and rectangular linear systems. In this article, we introduce a new iteration scheme called three-step alternating iterations using proper splittings and group inverses to find an approximate solution of singular linear systems, iteratively. A preconditioned alternating iterative scheme is also proposed to relax some sufficient conditions and to obtain faster convergence as well. We then show that our scheme converges faster than the existing one. The theoretical findings are then validated numerically.

A Note on Different Subclass of Index-Proper Splittings for Singular Matrices

The theory of matrix splittings is one of the most useful tool for finding an iterative solution of a linear system of equations. In this article, we introduce two new types of matrix splittings for singular matrices, called index-proper weak regular splitting of the second type and index-proper nonnegative splitting of the second type. Under suitable conditions, we derive convergence and comparison theorems for these splittings.

Properweak regular splitting and its application to convergence of alternating iterations

Theory of matrix splittings is a useful tool for finding the solution of a rectangular linear system of equations, iteratively. The purpose of this paper is two-fold. Firstly, we revisit the theory of weak regular splittings for rectangular matrices. Secondly, we propose an alternating iterative method for solving rectangular linear systems by using the Moore-Penrose inverse and discuss its convergence theory, by extending the work of Benzi and Szyld [Numererische Mathematik 76 (1997) 309-321; MR1452511]. Furthermore, a comparison result is obtained which ensures the faster convergence rate of the proposed alternating iterative scheme.

Accelerated Gibbs sampling of normal distributions using matrix splittings and polynomials

Standard Gibbs sampling applied to a multivariate normal distribution with a specified precision matrix is equivalent in fundamental ways to the Gauss–Seidel iterative solution of linear equations in the precision matrix. Specifically, the iteration operators, the conditions under which convergence occurs, and geometric convergence factors (and rates) are identical. These results hold for arbitrary matrix splittings from classical iterative methods in numerical linear algebra giving easy access to mature results in that field, including existing convergence results for antithetic-variable Gibbs sampling, REGS sampling, and generalizations. Hence, efficient deterministic stationary relaxation schemes lead to efficient generalizations of Gibbs sampling. The technique of polynomial acceleration that significantly improves the convergence rate of an iterative solver derived from a symmetric matrix splitting may be applied to accelerate the equivalent generalized Gibbs sampler. Identicality of error polynomials guarantees convergence of the inhomogeneous Markov chain, while equality of convergence factors ensures that the optimal solver leads to the optimal sampler. Numerical examples are presented, including a Chebyshev accelerated SSOR Gibbs sampler applied to a stylized demonstration of low-level Bayesian image reconstruction in a large 3-dimensional linear inverse problem.

Further study of alternating iterations for rectangular matrices

Theory of matrix splittings is a useful tool in the analysis of iterative methods for solving systems of linear equations. When two splittings are given, it is of interest to compare the spectral radii of the corresponding iteration matrices. This helps to arrive at the conclusion that which splitting should one choose so that one can reach the desired solution of accuracy or the exact solution in a faster way. In the case of many splittings are provided, the comparison of the spectral radii is time-consuming. Such a situation can be overcome by introducing another iteration scheme which converges to the same solution of interest in a much faster way. In this direction, the theory of alternating iterations for real rectangular matrices is recently proposed. In this note, some more results to the theory of alternating iterations are added. A comparison result of two different alternating iteration schemes is then presented which will help us to choose the iteration scheme that will guarantee the faster convergence of the alternating iteration scheme. In addition to these results, a comparison result for proper weak regular splittings is also obtained.

Additional Results on Convergence of Alternating Iterations Involving Rectangular Matrices

ABSTRACTDifferent classes of matrix splittings of semi-monotone matrices lead to many comparison results, which are useful tools in examining the convergence rate of iterative methods for solving rectangular systems of linear equations in a faster way. In this context, the theory of alternating iterations for rectangular matrices was introduced recently, in order to arrive at the desired solution of accuracy or at the exact solution. In this article, we expand convergence theory of such alternating iterations and obtain comparison results for such iterations.

Two-step modulus-based matrix splitting iteration method for a class of nonlinear complementarity problems

In this paper, we reformulate the nonlinear complementarity problem as an implicit fixed-point equation. We establish a modulus-based matrix splitting iteration method based on the implicit fixed-point equation and prove its convergence theorem under suitable conditions. Furthermore, we propose a two-step modulus-based matrix splitting iteration method, which may achieve higher computing efficiency. We can obtain many matrix splitting iteration methods by suitably choosing the matrix splittings and the parameters. The proposed methods can be regarded as extensions of the methods for linear complementarity problem. Numerical experiments are presented to show the effectiveness of the proposed methods.

On the Numerical Behavior of Matrix Splitting Iteration Methods for Solving Linear Systems

We study the numerical behavior of stationary one-step or two-step matrix splitting iteration methods for solving large sparse systems of linear equations. We show that inexact solutions of inner linear systems associated with the matrix splittings may considerably influence the accuracy of the approximate solutions computed in finite precision arithmetic. For a general stationary matrix splitting iteration method, we analyze two mathematically equivalent implementations and discuss the conditions when they are componentwise or normwise forward or backward stable. We distinguish two different forms of matrix splitting iteration methods and prove that one of them is significantly more accurate than the other when employing inexact inner solves. The theoretical results are illustrated by numerical experiments with an inexact one-step and an inexact two-step splitting iteration method.

A note on index-proper multisplittings of matrices

Matrix splitting and its convergence theorems are useful tools for finding solution of linear system of equations, iteratively. In this article, we introduce a few matrix splittings arising from index-proper splittings. Then their convergence results and their applications to multisplitting theory are studied.

Subspace Accelerated Matrix Splitting Algorithms for Asymmetric and Symmetric Linear Complementarity Problems

This paper studies the solution of both asymmetric and symmetric linear complementarity problems by two-phase methods that consist of an active set prediction phase and an acceleration phase. The prediction phase employs matrix splitting iterations that are tailored to the structure of the linear complementarity problems studied in this paper. In the asymmetric case, the task of pairing an acceleration phase with matrix splitting iterations is achieved by exploiting a contraction property associated with certain matrix splittings. For symmetric problems, a similar task is achieved by utilizing decent properties of specific matrix splitting iterations and projected searches. The superior optimal active set identification property of matrix splitting iterations is illustrated with numerical experiments, which also demonstrate the general efficiency of the proposed methods.

Stability and Convergence of Product Formulas for Operator Matrices

We present easy to verify conditions implying stability estimates for operator matrix splittings which ensure convergence of the associated Trotter, Strang and weighted product formulas. The results are applied to inhomogeneous abstract Cauchy problems and to boundary feedback systems.

Splitting of the rate matrix as a definition of time reversal in master equation systems

Motivated by recent progress in nonequilibrium fluctuation relations, we present a generalized time reversal for stochastic master equation systems with discrete states, which is defined as a splitting of the rate matrix into irreversible and reversible parts. An immediate advantage of this definition is that a variety of fluctuation relations can be attributed to different matrix splittings. Additionally, we find that the accustomed total entropy production formula and conditions of the detailed balance must be modified appropriately to account for the reversible rate part, which was previously ignored.

DCT- and DST-based splitting methods for Toeplitz systems

New splitting iterative methods for Toeplitz systems are proposed by means of recently developed matrix splittings based on discrete sine and cosine transforms due to Kailath and Olshevsky [Displacement structure approach to discrete-trigonometric transform-based preconditioners of G. Strang type and of T. Chan type, SIAM J. Matrix Anal. Appl. 26 (2005), pp. 706–734]. Theoretical analysis shows that new splitting iterative methods converge to the unique solution of a symmetric Toeplitz linear system. Moreover, an upper bound of the contraction factor of our new splitting iterations is derived. Numerical examples are reported to illustrate the effectiveness of new splitting iterative methods.

An alternating preconditioner for saddle point problems

Based on matrix splittings, a new alternating preconditioner with two parameters is proposed for solving saddle point problems. Some theoretical analyses for the eigenvalues of the associated preconditioned matrix are given. The choice of the parameters is considered and the quasi-optimal parameters are obtained. The new preconditioner with these quasi-optimal parameters significantly improves the convergence rate of the generalized minimal residual (GMRES) iteration. Numerical experiments from the linearized Navier–Stokes equations demonstrate the efficiency of the new preconditioner, especially on the larger viscosity parameter ν . Further extensions of the preconditioner to generalized saddle point matrices are also checked.