Block SOR Research Articles

In Case You Haven't Heard

States are using their Substance Abuse Prevention and Treatment (SAPT) block grant and STR and SOR grants for naloxone, which used to be considered prevention but is now considered treatment. Prevention advocates said it was not prevention of substance use disorder but prevention of death. In the second year of STR, naloxone fell under treatment. How much money is going directly to naloxone manufacturers as a pass‐through and how much to the organizations that are distributing it? Apparently, most is going to the manufacturers, who have free distribution networks in the recovery programs that are working on this. Good, bad or indifferent, the money is being used to save lives. But there is a limited amount of money. After people are rescued from overdoses, what happens? Who's watching? One thing is clear: the state directors with authority over the SAPT block grant, STR and SOR are in the lead when it comes to knowing what their states need. Stay tuned.

PATH PLANNING SIMULATION USING HARMONIC POTENTIAL FIELDS THROUGH FOUR POINT-EDGSOR METHOD VIA 9-POINT LAPLACIAN

This paper presents our study on a simulation of path planning for indoor robot that relies on the use of Laplace’s equation to constrain the generation of Harmonic Potential Fields (HPF). The computation of HPF requires immense amount of computing resources, particularly when the size of environment is large. In the past, fast iterative methods that apply the use of half-sweep iteration and block technique are suggested. In this study, faster iterative method known as Four Point-Explicit Decoupled Group Successive Over relaxation via 9-Point Laplacian (4-EDGSOR-9L) is introduced. Essentially, the 4-EDGSOR-9L is actually a variant of block SOR iterative method based on four points that employs half-sweep iteration and utilizes 9-Point Laplacian discretization scheme. Once the HPF is obtained, the standard Gradient Descent Search (GDS) technique is performed for path tracing to the goal point.

On parallel multisplitting block iterative methods for linear systems arising in the numerical solution of Euler equations

The paper studies the convergence of some parallel multisplitting block iterative methods for the solution of linear systems arising in the numerical solution of Euler equations. Some sufficient conditions for convergence are proposed. As special cases the convergence of the parallel block generalized AOR (BGAOR), the parallel block AOR (BAOR), the parallel block generalized SOR (BGSOR), the parallel block SOR (BSOR), the extrapolated parallel BAOR and the extrapolated parallel BSOR methods are presented. Furthermore, the convergence of the parallel block iterative methods for linear systems with special block tridiagonal matrices arising in the numerical solution of Euler equations are discussed. Finally, some examples are given to demonstrate the convergence results obtained in this paper.

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.

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.

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.

Block SOR Preconditioned Projection Methods for Kronecker Structured Markovian Representations

Kronecker structured representations are used to cope with the state space explosion problem in Markovian modeling and analysis. Currently, an open research problem is that of devising strong preconditioners to be used with projection methods for the computation of the stationary vector of Markov chains (MCs) underlying such representations. This paper proposes a block successive overrelaxation (BSOR) preconditioner for hierarchical Markovian models (HMMs; throughout the paper, the HMM acronym stands for hierarchical Markovian models and should not be confused with the HMM that is sometimes used for hidden Markov models.) that are composed of multiple low-level models and a high-level model that defines the interaction among low-level models. The Kronecker structure of an HMM yields nested block partitionings in its underlying continuous-time MC which may be used in the BSOR preconditioner. The computation of the BSOR preconditioned residual in each iteration of a preconditioned projection method becomes the problem of solving multiple nonsingular linear systems whose coefficient matrices are the diagonal blocks of the chosen partitioning. The proposed BSOR preconditioner solves these systems using sparse LU or real Schur factors of diagonal blocks. The fill-in of sparse LU factorized diagonal blocks is reduced using the column approximate minimum degree (COLAMD) ordering. A set of numerical experiments is presented to show the merits of the proposed BSOR preconditioner.

Iterative solution of elliptic collocation systems on a cognitive parallel computer

The performance of parallel algorithms implementing the block SOR iterative method, used for the solution of linear systems arising from the discretization of elliptic BVPs by the finite element collocation method, is investigated in this paper. By making use of a recently introduced block tridiagonal partitioning of the collocation matrix, which yields faster rates of convergence, and its associated modified red-black ordering of unknowns and equations, we succeeded to increase the scalability of the problem. The study of the communicational and computational needs of the problem on a virtual parallel machine made possible the mapping of the resulted SOR-algorithm on a fixed size distributed memory parallel system, utilizing a simple pipeline architecture among its processors, in an optimum way. The proposed parallel schemes are being realized on a Parsytec cognitive computer via an SPMD programming model. Speedup measurements are used to reveal the efficiency of our implementation.

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.

COMPARISONS OF THE PARALLEL PRECONDITIONERS FOR LARGE NONSYMMETRIC SPARSE LINEAR SYSTEMS ON A PARALLEL COMPUTER

In this paper we compare various parallel preconditioners for solving large sparse nonsymmetric linear systems. They are Block Jacobi, Point-SSOR, ILU(0) in the wavefront order, ILU(0) in the multi-color order, SPAI(SParse Approximate Inverse), and Multi-Color Block SOR. The Block Jacobi and Point-SSOR are well-known, and ILU(0) is one of the most popular preconditioners, but it is inherently serial. ILU(0) in the wavefront order maximizes the parallelism, and ILU(0) in the multi-color order achieves the parallelism of order (N), where N is the order of the matrix. The SPAI tries to capture the approximate inverse in sparse form, which, then, is expected to be a scalable preconditioner. Finally, we implemented the Multi-Color Block SOR preconditioner combined with direct sparse matrix solver. For the Laplacian matrix the SOR method is known to have a non-deteriorating rate of convergence when used with Multi-Color ordering. Since most of the time is spent on the diagonal inversion, which is done on each processor, we expect it to be a good scalable preconditioner. Finally, due to the blocking effect, it will be effective for ill-conditioned problems. Experiments were conducted for the Finite Difference discretizations of two problems with various meshsizes varying up to 1024×1024, and for an ill-conditioned matrix from the shell problem from the Harwell–Boeing collection. CRAY-T3E with 128 nodes was used. MPI library was used for interprocess communications. The results show that Multi-Color Block SOR and ILU(0) with Multi-Color ordering give the best performances for the finite difference matrices and for the shell problem only the Multi-Color Block SOR and Block Jacobi converges. Based on this we recommend that the Multi-Color Block SOR is the most robust preconditioner out of the preconditioners considered.

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.

Block SOR for Kronecker structured representations

The Kronecker structure of a hierarchical Markovian model (HMM) induces nested block partitionings in the transition matrix of its underlying Markov chain. This paper shows how sparse real Schur factors of certain diagonal blocks of a given partitioning induced by the Kronecker structure can be constructed from smaller component matrices and their real Schur factors. Furthermore, it shows how the column approximate minimum degree (COLAMD) ordering algorithm can be used to reduce fill-in of the remaining diagonal blocks that are sparse LU factorized. Combining these ideas, the paper proposes three-level block successive over-relaxation (BSOR) as a competitive steady state solver for HMMs. Finally, on a set of numerical experiments it demonstrates how these ideas reduce storage required by the factors of the diagonal blocks and improve solution time compared to an all LU factorization implementation of the BSOR solver.

An iterative back substitution algorithm for the solution of tridiagonal matrix systems with fringes

For tridiagonal matrix systems, a simple direct algorithm giving the solution exists, but in the most general case of tridiagonal matrix with fringes, the direct solving algorithms are more complicated. For big systems, direct methods are not well fitted and iterative algorithms are preferable. In this paper a relaxation type iterative algorithm is presented. It is an extension of the backward substitution method used for simple tridiagonal matrix systems. The performances show that this algorithm is a good compromise between a direct method and other iterative methods as block SOR. Its nature suggests its use as inner solver in the solution of problems derived by application of a decomposition domain method. A special emphasis is done on the programming aspect. The solving Fortran subroutines implementing the algorithm have been generated automatically from their specification by using a computer algebra system technique.

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.

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.

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.

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.

Comparison of Partitioning Techniques for Two-Level Iterative Solvers on Large, Sparse Markov Chains

Experimental results for large, sparse Markov chains, especially the ill-conditioned nearly completely decomposable (NCD) ones, are few. We believe there is need for further research in this area, specifically to aid in the understanding of the effects of the degree of coupling of NCD Markov chains and their nonzero structure on the convergence characteristics and space requirements of iterative solvers. The work of several researchers has raised the following questions that led to research in a related direction: How must one go about partitioning the global coefficient matrix into blocks when the system is NCD and a two-level iterative solver (such as block SOR) is to be employed? Are block partitionings dictated by the NCD form of the stochastic one-step transition probability matrix necessarily superior to others? Is it worth investigating alternative partitionings? Better yet, for a fixed labeling and partitioning of the states, how does the performance of block SOR (or even that of point SOR) compare to the performance of the iterative aggregation-disaggregation (IAD) algorithm? Finally, is there any merit in using two-level iterative solvers when preconditioned Krylov subspace methods are available? We seek answers to these questions on a test suite of 13 Markov chains arising in 7 applications.

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.