Block SOR Method Research Articles

An Inverse-Free Block-SOR Method With Randomly Sampling for Temporal Multiplex PageRank Problems

Ranking nodes in a multiplex network is one of the most pressing and challenging tasks in network analysis. Generalizing centrality measures to multiplex networks is an active area of research. The Multiplex PageRank model is an extension of Google's PageRank, which introduces a new centrality measure to extend the usual PageRank to multiplex networks. In this work, we focus on the Multiplex PageRank problem. First, based on the special structure of the Multiplex PageRank problem, we propose an inverse-free block-SOR method. Second, with the help of randomly sampling, we propose a new strategy for estimating the optimal relaxation parameter. Specifically, the multiplex network is frequently updated in real world applications, and we have to deal with temporal Multiplex PageRank problems including the incremental and the decremental Multiplex PageRank problems. To the best of our knowledge, however, there are few efficient algorithms for solving these type of problems. To fill-in this gap, the third contribution of this work is to propose both incremental and decremental algorithms for solving the temporal Multiplex PageRank problems. Comprehensive numerical experiments are performed to illustrate the numerical behavior of the proposed algorithms, and show the effectiveness of our new strategies.

Block SOR methods for the solution of indefinite least squares problems

This paper describes a technique for constructing block SOR methods for the solution of the large and sparse indefinite least squares problem which involves minimizing a certain type of indefinite quadratic form. Two block SOR-based algorithms and convergence results are presented. The optimum parameters for the methods are also given. It has been shown both theoretically and numerically that the optimum block SOR methods have a faster convergence than block Jacobi and Gauss---Seidel methods.

Continuous-time accelerated block successive overrelaxation methods for time-dependent Stokes equations

In order to solve the time-dependent Stokes equation, we follow the “Method of Lines” to obtain structured linear constant-coefficient differential–algebraic equations (DAEs). By taking advantage of the structure, we propose a class of waveform relaxation methods, called continuous-time accelerated block SOR (CABSOR) methods, for solving the obtained DAEs. The new methods are theoretically analyzed. The theory is applied to a two-dimensional time-dependent Stokes equation and verified by numerical experiments.

Analysis of Block-SOR Iteration for the three dimensional Laplacian

The successive over-relaxation (SOR) iteration method for solving linear systems of equations depends upon a relaxation parameter. A well-known theory for determining this parameter was given by Young for consistently ordered matrices. In this paper, for the three-dimensional Laplacian, we introduce several compact difference schemes and analyse the block-SOR method for the resulting linear systems. Their optimum relaxation parameters are given for the first time. Analysis shows that the value of the optimum relaxation parameter of block-SOR iteration is very sensitive for compact stencils when solving the three-dimensional Laplacian. This paper provides a theoretical solution for determining the optimum relaxation parameter in real applications. doi:10.1017/S1446181109000261

ANALYSIS OF BLOCK-SOR ITERATION FOR THE THREE-DIMENSIONAL LAPLACIAN

AbstractThe successive over-relaxation (SOR) iteration method for solving linear systems of equations depends upon a relaxation parameter. A well-known theory for determining this parameter was given by Young for consistently ordered matrices. In this paper, for the three-dimensional Laplacian, we introduce several compact difference schemes and analyse the block-SOR method for the resulting linear systems. Their optimum relaxation parameters are given for the first time. Analysis shows that the value of the optimum relaxation parameter of block-SOR iteration is very sensitive for compact stencils when solving the three-dimensional Laplacian. This paper provides a theoretical solution for determining the optimum relaxation parameter in real applications.

Block SOR methods for fuzzy linear systems

In this paper, the block SOR iterative methods are studied for n×n fuzzy linear systems and the corresponding convergence theorems are also given out. We know that the coefficient matrix S of the augmented system SX=Y is consistently ordered when S1 is nonsingular, and in this case the optimal parameter ω of the block SOR method is obtained. Numerical examples are presented to illustrate the theory.

Symmetric block-SOR methods for rank-deficient least squares problems

In this article, we develop symmetric block successive overrelaxation (S-block-SOR) methods for finding the solution of the rank-deficient least squares problems. We propose an S2-block-SOR and an S3-block-SOR method for solving such problems and the convergence of these two methods is studied. The comparisons between the S2-block and the S3-block methods are presented with some numerical examples.

A New Block Parallel SOR Method and Its Analysis

As the development of the PSOR method (a point parallel SOR method by mesh domain partitioning proposed in [SIAM J. Sci. Comput., 20 (1999), pp. 2261-2281], this paper introduces a new mesh domain partition and ordering (the multitype partition and ordering), and proposes a new block parallel SOR (BPSOR) method for numerically solving 2-dimensional (2D) or three-dimensional (3D) elliptic boundary problems. A general mathematical analysis shows that the BPSOR method can have the same asymptotic convergence rate as the corresponding sequential block SOR method if the coefficient matrix of the block linear system is consistently ordered. It also shows that the original sequential ordering can be maintained in the parallel implementation of the BPSOR method so that the BPSOR method can be effectively applied to solve complex elliptic boundary problems. Furthermore, three particular multitype orderings are proposed based on strip and block mesh partitions, which lead to three effective BPSOR methods for solving the five-point like linear systems (in two dimensions) and the seven-point like linear systems (in three dimensions). In addition, it is shown that the PSOR method can be generated from BPSOR if each block equation is solved approximately by only one point SOR iteration. Thus, the PSOR method is well defined without involving any interprocessor data communication operations, and extended to solving three-dimensional (3D) problems. Finally, numerical results are presented which confirm the theoretical results and show that BPSOR has a good parallel performance on a parallel MIMD computer.

A domain decomposition method for domains with holes using a complementary decomposition

In this paper, a new non-conforming domain decomposition method based on fictitious domains is considered for linear elasticity problems on domains with holes. Existence and uniqueness of the solution of this alternative approach are showed in the continuous and discrete case. Although the established a priori estimate for the discretization error is not optimal, the scheme is of interest in case of moving geometries and in shape optimization. To solve the arising coupled linear system, we use a block SOR method. The convergence rate does not depend on the mesh size. Numerical results illustrate the flexibility of the discretization scheme.

On parallel multisplitting iterative methods for singular linear systems

In this paper, the semiconvergence of parallel multisplitting iterative methods and extrapolated parallel methods for singular linear systems is discussed. Some sufficient conditions for semiconvergence are presented whenever the multisplittings are P-regular, sub-regular or nonnegative. As special cases the semiconvergence of parallel generalized AOR, block AOR, AOR, parallel generalized SOR, block SOR, SOR, extrapolated parallel (generalized, block) AOR and extrapolated parallel (generalized, block) SOR methods are proved.

Semiconvergence of block SOR method for singular linear systems with p-cyclic matrices

In this paper, we discuss semiconvergence of the block SOR method for solving singular linear systems with p-cyclic matrices. Some sufficient conditions for the semiconvergence of the block SOR method for solving a general p-cyclic singular system are proved.

Comparisons of Block Iterative Methods

Comparison theorems for spectral radii of two block USSOR methods, two block SSOR methods, two block AOR methods, two block SOR methods with respect to a partition π and the corresponding finer partition π* are given, respectively, whenever A is an M-matrix or an H-matrix.

Aor type iterative methods for solving least squares problems

In order to solve least squares problems, the block SOR, block SSOR and block AOR iterative methods are proposed and investigated by some authors. In this paper, we generalize the usual block AOR method. Other five AOR type iterative methods are defined. Then the convergence and divergence regions for the AOR methods are discussed.

Parallel solvers for the transient multigroup neutron diffusion equations

For the safety and the control of a nuclear power plant it is necessary to simulate the constituent processes on a computer system. The three-dimensional multigroup neutron diffusion equations are commonly used to describe the nuclear fission in the reactor core. They form a complicated system of coupled parabolic partial differential equations (PDEs) whose solution can involve very intensive computing. In this paper this system of PDEs is discretized using a special cell-centred mixed finite volume method (NEM-M0) in space, and a method that combines Crank–Nicholson and the BDF(2)-method in time. The linear equation systems which arise are solved with multi-grid as well as with preconditioned BiCGStab. The kernel of both solution methods is an effective Block-SOR method that makes use of the particular structure of the linear equation systems. The parallelization strategy is based on a grid partitioning that distributes the data and the work homogeneously on the processors. Finally, the program was tested for three typical reactor simulation problems on grids with differing coarseness. The speedup achieved by parallelizing multi-grid and preconditioned Bi-CGStab was outstanding for all examples; even superlinear in some cases. Moreover, the parallel execution times were better than the parallel execution times of other established reactor simulation codes. Copyright © 2000 John Wiley & Sons, Ltd.

Preconditioned sor methods for generalized least-squares problems

We consider here iterative methods for the generalized least squares problem defined as min (Ax−b) TW −1, (Ax−b) withW symmetric and positive definite. We develop preconditioned SOR methods specially devised also for the augmented systems of the problem. We establish the convergence region for the relaxation parameter and discuss, for one of the resulting SOR methods, the optimal value of this parameter. The convergence analysis and numerical experiments show that the preconditioned block SOR methods are very good alternatives for solving the problem.

Mathematical modelling of flow towards an oil well

We describe the numerical approximations and applications of a mathematical model that governs the flow of oil towards a well. The flow of a single-phase fluid in a porous medium is governed by a parabolic equation obtained by combining the Darcy's and the continuity equations. In order to account for the spatial variations of porosity and permeability, and for permeability anisotropy, a two-dimensional model is put forward. A mixed initial–boundary value problem is numerically solved by a finite-difference family of numerical schemes, which depends on a parameter θ. The stability—conditional or unconditional, depending on θ—and the convergence of the schemes have been proved. The linear system originated at each time step is solved by the iterative ADI and block-SOR methods, and by a Taylor series of matrix functions (TSMF). These methods are compared and their relative efficiencies are carefully assessed. TSMF is the fastest technique given that adequate values of θ and time step Δt are used—but Δt must remain small. A combination of TSMF and block-SOR with variable Δt seems to be the best policy. Our numerical simulator is tested by reproducing the existing analytical solutions for limiting cases, and then applied in well test analysis. The contributions of this work are: (1) we introduce the TSMF technique to reservoir simulation and (2) vertical permeability and permeability spatial variations are included in a well test simulator for further developments. Copyright © 1999 John Wiley & Sons, Ltd.

Adaptive improved block SOR method with orderings

Recently, a generalized SOR method with multiple relaxation parameters were considered for solving a linear system of equations and it was shown that if a pair of parameter values is computed from the pivots of the Gaussian elimination applied to the system, then the spectral radius of the iterative matrix is reduced to zero. A proper choice of orderings and starting vectors for the iteration were also proposed.

COMPARISONS OF THE PARALLEL PRECONDITIONERS ON THE CRAY-T3E FOR LARGE NONSYMMETRIC LINEAR SYSTEMS

In this paper we consider five types of parallel preconditioners for solving large sparse nonsymmetric linear systems on the CRAY-T3E. They are ILU(0) in the wavefront ordering, ILU(0) in the multi-coloring ordering, SSOR in the wavefront ordering, the SPAI(SParse Approximate Inverse) preconditioner, and finally Multi-color Block SOR preconditioner. The ILU(0) is known to be robust and the wavefront ordering naturally exploits the parallelism but has a limited speedup due to the nonuniform lengths of the wavefronts. Multi-coloring is an efficient way of introducing the parallelism of order(N), where N is the order of the matrix but the convergence rate often deteriorates. The SPAI type preconditioner is inherently parallel and is gaining popularity. Finally, for the 5-point Laplacian matrix SOR method is known to have a nondeteriorating rate of convergence when the multi-coloring order is adopted. Also, Block SOR is expected to incur less communication overheads in a message-passing machine. Hence, Multi-Color Block SOR method is expected to have a good performance. Experiments were conducted for the Finite Difference discretizations of two problems with various meshsizes varying up to 1024×1024. MPI library was used for interprocess communications. The results show that ILU(0) in the multi-coloring ordering gives the best performance.

Preconditioned conjugate gradient method for rank deficient least-squares problems

Rank deficient least squares problems appear in obtaining numerical solution of differential equations, computational genetics and other applications. The usual methods to solve the problem are QR decomposition. It is well-known that for large sparse problems, iterative methods are preferable. Miller and Neumann (1987) proposed the 4-block SOR method, and Santos, Silva and Yuan (1997) proposed the 2-block SOR method and the 3-block SOR method for solving the problem. Here some preconditioned conjugate gradient methods are proposed for solving the problem. The error bound and comparison with block SOR methods are studied. We show the best iterative method is the preconditioned conjugate gradient method for solving rank deficient least squares problems.

Block Stationary Methods for Nonsymmetric Cyclically Reduced Systems Arising from Three-Dimensional Elliptic Equations

We consider a three-dimensional convection-diffusion model problem and examine systems of equations arising from performing one step of cyclic reduction on an equally spaced mesh, discretized using the seven-point operator. We present two ordering strategies and analyze block splittings of the resulting matrices. If the matrices are consistently ordered relative to a given partitioning, Young's analysis for the block Gauss--Seidel and block SOR methods can be applied. We compare partitionings for which this property holds with ones where the matrices do not have Property A yet still give rise to an efficient solution process. Bounds on convergence rates are derived and the work involved in solving the systems is estimated.