The impact of eigenvalue locality on the convergence behavior of the PSD method for two-cyclic matrices

AbstractIn this paper, we analyse the convergence of the preconditioned simultaneous displacement (PSD) method applied to linear systems of the form Au=b where A is a two-cyclic matrix. Convergence conditions and optimum values of the parameters of the method are determined in the cases where the eigenvalues of the associated Jacobi iteration matrix are either all real or all imaginary. It is shown that the convergence behavior of the PSD method is greatly affected by the locality of the eigenvalues of the associated Jacobi iteration matrix. In particular, it is shown that when these eigenvalues are real the PSD method degenerates into the extrapolated Gauss–Seidel method whereas when they are imaginary its convergence is increased by an order of magnitude and becomes equivalent to the extrapolated SOR method. Finally, a comparison with the SSOR method reveals that the PSD method possesses a better convergence behavior in all cases

On the Dynamic Acceleration of the Preconditioned Simultaneous Displacement (PSD) Method

The Preconditioned Simultaneous Displacement (PSD) iterative method is considered for the solution of symmetric, sparse matrix problems. The development of a dynamic algorithm for improving the estimates of the involved parameters is presented. These estimates are then used to accelerate the PSD method by employing semi-iterative techniques. The algorithm determines adaptively a sequence of parameters while the iteration is in progress without requiring preliminary eigenvalue estimates (only trivial input parameters are required). The performance of the algorithm is tested on a number of generalised Dirichlet problems. It is seen that the attained rate of convergence is approximately Q(h1/2) and is better than the algorithm using estimated parameters in certain cases. © 1981, Taylor & Francis Group, LLC. All rights reserved

The Average Diffusion method for the load balancing problem

This paper proposes the Average Diffusion (ADF) method for solving the load balancing problem. It is shown that a sufficient and necessary condition for the ADF method to converge to the uniform distribution of loads is the induced network of processors to be d-regular, connected and not bipartite. Next, we proceed and apply Fourier analysis determining the convergence factor γ in terms of the diffusion parameters cij (weighted case) when the network of processors is a ring and 2D-torus. It is shown that cij = 1/2 and cij ∈ (0, 1/2) when the network is a ring and 2D-torus, respectively, thus solving partially the open problem which concerns the determination of the diffusion parameters cij. © 2002 Springer-Verlag Berlin Heidelberg

Parallel solution of the Navier-Stokes equations on distributed memory MIMD machines

In this paper we study iterative schemes for the solution of the 2-D incompressible Navier-Stokes equations on a mesh network of Transputers. In particular, we consider the local Modified Successive Overrelaxation (MSOR) method and apply Fourier analysis to study its convergence. Parallelism is introduced by decoupling the mesh points with the use of red-black ordering for the 5 point stencil. The determination of optimum set of values, for the parameters involved, results in an increase of convergence rate as compared with the classic SOR method. Results from implementing a block form of the local MSOR method on a transputer mesh network are presented. © Springer-Verlag Berlin Heidelberg 1996

The nine node Extrapolated Diffusion method for weighted torus graphs

The convergence analysis of the Extrapolated Diffusion (EDF) method was developed in Karagiorgos and Missirlis (2008) and Markomanolis and Missirlis (2010) for 2D weighted torus and mesh graphs, respectively using the set N1(i) of the nearest neighbors of a source node i in the graph. In the present work we propose a Diffusion scheme which employs the set N1(i)∪N2(i), where N2(i) denotes the set of the nearest neighbors of a source node i with path length two in an attempt to improve the performance of the new method. We develop the convergence analysis of the new EDF method by considering two subsets of N2(i) for 2D weighted torus graphs. In particular, we study five different communication routes for computing the load of each node i in the torus graph and for each route we find closed form formulae for the optimum values of the edge weights, the extrapolation parameters and the convergence factor of the new EDF scheme. A comparison of the convergence factors of all these EDF schemes reveals a 60% improvement in the performance using the cross communication route compared to the conventional EDF method, a fact which is shown theoretically and experimentally. © 2017 Elsevier Inc

Convergence of the diffusion method for weighted torus graphs using Fourier analysis

This paper studies the diffusion method for the load balancing problem in the case of weighted torus graphs. Closed form formulae for the optimum values of the edge weights are determined using local Fourier analysis. It is shown that an extrapolated version of diffusion can become twice as fast for the stretched torus graphs. © 2008 Elsevier B.V. All rights reserved

The Extrapolated Successive Overrelaxation (ESOR) Method for Consistently Ordered Matrices

This paper develops the theory of the Extrapolated Successive Overrelaxatior (ESOR) method as introduced by Sisler in [1],[2],[3] for the numerical solution of large sparse linear systems of the form Au=b, when A is a consistently ordered 2-cyclic matrix with non-vanishing diagonal elements and the Jacobi iteration matrix B possesses only real eigenvalues. The region of convergence for the ESOR method is described and the optimum values of the involved parameters are also determined. It is shown that if the minimum of the moduli of the eigenvalues of B, [formula omitted] does not vanish, then ESOR attains faster rate of convergence than SOR when [formula omitted], where [formula omitted] denotes the spectral radius of B. © 1984, Hindawi Publishing Corporation. All rights reserved

A comparison of the Extrapolated Successive Overrelaxation and the Preconditioned Simultaneous Displacement methods for augmented linear systems

In this paper we study the impact of two types of preconditioning on the numerical solution of large sparse augmented linear systems. The first preconditioning matrix is the lower triangular part whereas the second is the product of the lower triangular part with the upper triangular part of the augmented system’s coefficient matrix. For the first preconditioning matrix we form the Generalized Modified Extrapolated Successive Overrelaxation (GMESOR) method, whereas the second preconditioning matrix yields the Generalized Modified Preconditioned Simultaneous Displacement (GMPSD) method, which is an extrapolated form of the Symmetric Successive Overrelaxation method. We find sufficient conditions for each aforementioned iterative method to converge. In addition, we develop a geometric approach, for determining the optimum values of their parameters and corresponding spectral radii. It is shown that both iterative methods studied (GMESOR and GMPSD) attain the same rate of convergence. Numerical results confirm our theoretical expectations. © 2015, Springer-Verlag Berlin Heidelberg

Parallel matrix factorizations on a shared memory MIMD computer

This paper is concerned with the study of parallel algorithms for matrix factorization on a shared memory multiprocessor MIMD type computer. We consider the implementation of LU and WZ factorizations of general nonsymmetric matrices when the number of processors p is ∼O(n), where n is the order of the matrix. We show how each of these methods can be divided into noninterfering tasks which can then be executed in parallel. By studying the precedence graph of these tasks we are able to find a schedule for each algorithm which is optimum for a certain number of processors. We also consider the use of the resulting factors to solve a linear system of equations and compare the two algorithms in terms of their speedup and efficiency. It is shown that the parallel WZ algorithm attains a better efficiency using only half the processors of Doolittle's method. © 1988, Springer-Verlag

Optimal diffusion for load balancing in heterogeneous networks

In [7] we studied the local Extrapolated Diffusion (EDF) method for the load balancing problem in case of homogeneous torus networks. The present paper develops the convergence theory of the local EDF for heterogeneous torus networks. In particular, we determine its quasi-optimal iteration parameters and the corresponding quasi-optimal convergence factor using local mode analysis. As a result dynamic load balancing becomes an efficient procedure since the parameters of local EDF are computed via a closed form formulae resulting in the maximization of its rate of convergence. Moreover, it is shown how the convergence factor depends upon the communication edge weights and the processor speeds of the network. © 2014 Springer-Verlag