Parallel Multisplitting Research Articles

Non-stationary Parallel Multisplitting Two-Stage Iterative Methods with Self-AdaptiveWeighting Schemes

Non-stationary Parallel Multisplitting Two-Stage Iterative Methods with Self-AdaptiveWeighting Schemes

Parallel Statistical Computing for Statistical Inference

Parallel statistical computing is an interesting and topical problem, driven by recent growth in the size of statistical data sets and the availability of network computing. This article reviews parallel statistical computing in regression analysis, nonparametric inference, and stochastic processes. In particular, we describe a range of methods including parallel multisplitting and the parallel QR method for least squares estimation in linear regression, parallel computing methods for nonlinear regression, the theoretical framework of the parallel bootstrap in nonparametric inference, preconditioner methods for Markov chains, and parallel Markov-chain Monte Carlo methods. We conclude that there is a need for further research in parallel statistical computing, and describe some of the important unsolved problems.

GRPM-style methods and comparisons of convergent and divergent rates

Relaxed technique is one of techniques for improving convergence rate of splitting iterative methods. Based on local relaxed method and system relaxed method of parallel multisplitting Frommer and Mayer [A. Frommer, G. Mayer, Convergence of relaxed parallel multisplitting methods, Linear Algebra Appl. 119 (1989) 141–152], we give the global relaxed parallel multisplitting (GRPM) method by introducing some relaxed parameters and study the convergence of our methods (GRPM-style) when the coefficient matrices are H -matrices. Numerical experiments show that, when choosing the approximately optimal relaxed parameters, our methods have faster convergent rate than the methods in Chang [D.W. Chang, Convergence analysis of the parallel multisplitting TOR methods, J. Comput. Appl. Math. 72 (1996) 169–177] Frommer and Mayer [A. Frommer, G. Mayer, Convergence of relaxed parallel multisplitting methods, Linear Algebra Appl. 119 (1989) 141–152]. Furthermore, the convergent and divergent rates of local relaxed parallel multisplitting (LRPM-style) methods about multislitting TOR, AOR, SOR, G-S, extraolated Jacobi methods as well as Jacobi iterative method are compared in detail.

New comparison results for parallel multisplitting iterative methods

The parallel multisplitting nonstationary iterative Model A and Models 1–2 were introduced by Bru et al. [Bru, Elsner, Neumann, Models of parallel chaotic iteration methods, Linear Algebra Appl. 103 (1988) 175–192], and Climent et al. [Climent, Perea, Tortosa, Zamora, Convergence theorems for parallel alternating iterative methods, Appl. Math. Comput. 148 (2004) 497–517], respectively. Using nonnegative splitting theory, we present some new comparison theorems between two general parallel multisplittings of Model A and Models 1–2 when the coefficient matrix is a monotone matrix. Particularly, we establish some sufficient conditions for strict inequalities in comparisons of spectral radii of different iteration matrices.

Comparison results for parallel multisplitting methods with applications to AOR methods

In this paper we present some comparison theorems between two general parallel multisplittings. These comparison theorems can be applied to many classical splittings, such as Jacobi, Gauss–Seidel and AOR splittings. Some significant improvements and generalizations of the existing comparison results are obtained.

Asynchronous multisplitting relaxation methods for linear complementarity problems

To solve the linear complementarity problems efficiently on the high-speed multiprocessor systems, we set up a class of asynchronous parallel matrix multisplitting accelerated over-relaxation (AOR) method by technical combination of the matrix multisplitting and the accelerated overrelaxation techniques. The convergence theory of this new method is thoroughly established under the condition that the system matrix of the linear complementarity problem is an H-matrix with positive diagonal elements. At last, we also make multi-parameter extension for this new asynchronous multisplitting AOR method, and investigate the convergence property of the resulted asynchronous multisplitting unsymmetric AOR method. Thereby, an extensive sequence of asynchronous parallel relaxed iteration methods in the sense of multisplitting is presented for solving the large scale linear complementarity problems in the asynchronous parallel computing environments. This not only affords various choices, but also presents systematic convergence theories about the asynchronous parallel relaxation methods for solving the linear complementarity problems.

Asynchronous multisplitting AOR methods for a class of systems of weakly nonlinear equations

A class of asynchronous parallel multisplitting accelerated overrelaxation (AOR) methods for solving the system of weakly nonlinear equations Au+ϕ( u)= G(u), in which A ε L((ℝ n), ϕ( u)=(ϕ 1( u 1), ϕ 2(u 2), …,φ n( u n ) T and G(u)=,(g 1, g 2( u),…, g n ( u)) T, is proposed and its global convergence theory is set up under suitable conditions.

A parallel multisplitting solution of the least squares problem

The linear least squares problem, minx∥Ax − b∥2, is solved by applying a multisplitting (MS) strategy in which the system matrix is decomposed by columns into p blocks. The b and x vectors are partitioned consistently with the matrix decomposition. The global least squares problem is then replaced by a sequence of local least squares problems which can be solved in parallel by MS. In MS the solutions to the local problems are recombined using weighting matrices to pick out the appropriate components of each subproblem solution. A new two-stage algorithm which optimizes the global update each iteration is also given. For this algorithm the updates are obtained by finding the optimal update with respect to the weights of the recombination. For the least squares problem presented, the global update optimization can also be formulated as a least squares problem of dimension p. Theoretical results are presented which prove the convergence of the iterations. Numerical results which detail the iteration behavior relative to subproblem size, convergence criteria and recombination techniques are given. The two-stage MS strategy is shown to be effective for near-separable problems. © 1998 John Wiley & Sons, Ltd.

A parallel multisplitting solution of the least squares problem

The linear least squares problem, minx∥Ax − b∥2, is solved by applying a multisplitting (MS) strategy in which the system matrix is decomposed by columns into p blocks. The b and x vectors are partitioned consistently with the matrix decomposition. The global least squares problem is then replaced by a sequence of local least squares problems which can be solved in parallel by MS. In MS the solutions to the local problems are recombined using weighting matrices to pick out the appropriate components of each subproblem solution. A new two-stage algorithm which optimizes the global update each iteration is also given. For this algorithm the updates are obtained by finding the optimal update with respect to the weights of the recombination. For the least squares problem presented, the global update optimization can also be formulated as a least squares problem of dimension p. Theoretical results are presented which prove the convergence of the iterations. Numerical results which detail the iteration behavior relative to subproblem size, convergence criteria and recombination techniques are given. The two-stage MS strategy is shown to be effective for near-separable problems. © 1998 John Wiley & Sons, Ltd.

Parallel Multisplitting AOR method for solving a class of system of nonlinear algebraic equations

A class of parallel, multisplitting accelerated overrelaxation (AOR) method is set up for solving large-scale system of nonlinear algebraic equations Aϕ(x)+Bψ(x)=b. Under certain conditions, we prove the existence and uniqueness of the solution of this system of nonlinear equations and set up the global convergence theory of the new method.

Asynchronous multisplitting aor method for a system of nonlinear algebraic equations

An asynchronous parallel multisplitting accelerated overrelaxation (AOR) method is set up for the particularly structured system of nonlinear algebraic equations Aφ(x) + Bψ(x) = b, and the global convergence of the new method is proved.

Asynchronous multisplitting relaxed iterations for weakly nonlinear systems

In this paper, we propose a class of asynchronous parallel multisplitting accelerated overrelaxation methods for solving the system of weakly nonlinear equations Aφ(x) + Bψ(X) = G(x) with A, BeL(Rn), φ, ψ:Rn →Rn being diagonal mappings andG:Rn→Rn a general mapping, which is constantly resulted from the discretization of many classical differential equations.Under suitable conditions of both the coefficient matrices and the nonlinear mappings, as well as reasonable constraints of the multiple splittings and the relaxation parameters, the global convergence theories of these new methods are set up thoroughly.

Parallel multisplitting, block Jacobi type solutions of linear systems of equations

AbstractThree parallel iterative schemes to solve banded systems of equations are presented in this study. The techniques are special implementations of the theory of matrix multisplitting. The resulting algorithms are implemented on a Multiple Instruction Multiple Data (MIMD) grid architecture using 16 Transputer processors. Parametric analyses are performed to develop convergence criteria and examine the speed‐up of solution. It is concluded that the algorithms make very efficient use of the parallel computer architecture, especially if this consists of a large number of nodes.