Showing papers on "Sparse approximation published in 1989"

Sparse matrix test problems

Abstract: We describe the Harwell-Boeing sparse matrix collection, a set of standard test matrices for sparse matrix problems. Our test set comprises problems in linear systems, least squares, and eigenvalue calculations from a wide variety of scientific and engineering disciplines. The problems range from small matrices, used as counter-examples to hypotheses in sparse matrix research, to large test cases arising in large-scale computation. We offer the collection to other researchers as a standard benchmark for comparative studies of algorithms. The procedures for obtaining and using the test collection are discussed. We also describe the guidelines for contributing further test problems to the collection.

Application of sparse matrix solvers as effective preconditioners

Abstract: In this paper the use of a new out-of-core sparse matrix package for the numerical solution of partial differential equations involving complex geometries arising from aerospace applications is discussed. The sparse matrix solver accepts contributions to the matrix elements in random order and assembles the matrix using fast sort/merge routines. Fill-in is reduced through the use of a physically based nested dissection ordering. For very large problems a drop tolerance is used during the matrix decomposition phase. The resulting incomplete factorization is an effective preconditioner for Krylov subspace methods, such as GMRES. Problems involving 200,000 unknowns routinely are solved on the Cray X-MP using 64MW of solid-state storage device (SSD).

Numerical solution of sparse singular systems of equations arising from ergodic markov chains

Abstract: The stationary probability distribution vector, x, associated with an ergodic finite Markov chain satisfies a homogeneous singular system of equations , where A is a real and generally unsymmetric square matrix of the form . Here I is the identity matrix and T is the chain's column stochastic matrix. In many applications A is very large and sparse, and in such cases it is desirable to exploit this property in computing x. In this paper we review some of the literature dealing with sparse techniques for solving the above system of equations, and in so doing attempt to present a variety of methods from a unified point of view

Efficient sparse matrix factorization for circuit simulation on vector supercomputers

Abstract: An efficient approach to sparse matrix factorization on vector supercomputers is described. The approach is suitable for application domains like circuit simulation that require the repeated direct solution of sparse linear systems of equations with identical zero-nonzero structures. An overlap-scatter data structure is used to represent the sparse matrix, enabling the use of multiple operand access modes to achieve higher performance than earlier proposed approaches. The superior performance of the new solver is demonstrated using a number of matrices derived from circuit simulation runs. >

A sparse sequential quadratic programming algorithm

Abstract: Described here is the structure and theory for a sequential quadratic programming algorithm for solving sparse nonlinear optimization problems. Also provided are the details of a computer implementation of the algorithm along with test results. The algorithm maintains a sparse approximation to the Cholesky factor of the Hessian of the Lagrangian. The solution to the quadratic program generated at each step is obtained by solving a dual quadratic program using a projected conjugate gradient algorithm. An updating procedure is employed that does not destroy sparsity.

A new storage scheme for an efficient implementation of the sparse matrix-vector product☆

Abstract: The most critical point in the implementation on a vector computer of iterative methods to solve sparse linear algebraic systems is the sparse matrix-vector product. Therefor we have compared its performance under three different sparse matrix storage schemes found in literature. These schemes have been tested on a FPS M64/330 using matrices similar to those arising in finite element or finite difference discretizations. The results are reported and discussed. On the basis of them a new storage scheme is proposed and tested in similar situations. The obtained performance is never worse and often much better than those of the other schemes we have analyzed.

Efficient Sparse Matrix Factorization for Circuit Simulation on Vector Supercomputers

Abstract: This paper describes an efficient approach to sparse matrix factorization on vector supercomputers. The approach is suitable for application domains like circuit simulation that require the repeated direct solution of unsymmetric sparse linear systems of equations with identical zero-nonzero structure. An Overlap-Scatter data structure is used to represent the sparse matrix, enabling the use of multiple operand access modes to achieve higher performance than earlier proposed approaches. The superior performance of the new solver is demonstrated using a number of matrices derived from circuit simulation runs.

A parallel and vectorial implementation of basic linear algebra subroutines in iterative solving of large sparse linear systems of equations

Abstract: Electromagnetic field analysis by finite element methods, which involve the solution of large sparse systems of linear equations, is discussed. Though no discernible structure for the distribution of nonzero elements can be found (e.g. multidiagonal structures), subsets of independent equations can be determined. Equations that are in the same subset are then solved in parallel. A good choice for the storage scheme of sparse matrices is also very important to speed up the resolution by vectorization. The modifications made to data structures are presented, and the possibility of using other schemes is discussed. >

Architectural mechanisms to support sparse vector processing

Abstract: We discuss the algorithmic steps involved in common sparse matrix problems, with particular emphasis on linear programming by the revised simplex method. We then propose new architectural mechanisms which are being built into an experimental machine, the Edinburgh Sparse Processor, and which enable vector instructions to operate efficiently on sparse vectors stored in compressed form. Finally, we review the use of these new mechanisms on the linear programming problem.

A Large-Grain Parallel Sparse System Solver

Abstract: The eeciency of solving sparse linear systems on parallel processors and more complex multicluster architectures such as Cedar is greatly enhanced if relatively large grain computational tasks can be assigned to each cluster or processor. The ordering of a system into a bordered block upper triangular form facilitates a reasonable large-grain partitioning. A new algorithm which produces this form for unsymmetric sparse linear systems is considered and the associated factorization algorithm is presented. Computational results are presented for the Cedar multiprocessor. Several techniques have been proposed to solve large sparse systems of linear equations on parallel processors. A key task which determines the eeectiveness of these techniques is the identiication and exploitation of the computational granularity appropriate for the target multiprocessor architecture. Many algorithms assume special properties such as symmetric positive deeniteness or exploit knowledge of the application from which the system arises e.g., nite element problems. In this paper, we give an overview of the use of a new ordering technique, the hybrid ordering (H*), and an associated factorization algorithm for unsymmetric unstructured sparse linear systems. More detail on the reordering can be found in 7] and on the merger of the reordering and the factorization algorithm for multicluster architectures in 4]. 1. The Hybrid Ordering. The hybrid ordering H* is composed of two diierent types of orderings: unsymmetric and symmetric. The unsymmetric ordering changes the associated graph of the matrix, mostly by row or column interchanges. The symmetric orderings only relabel the nodes of the associated graphs and maintain certain properties of the system, e.g., symmetry, diagonal dominance. The symmetric orderings are used to obtain a bordered block triangular matrix. The unsymmetric ordering is used to enhance the numerical properties of the matrix.

A systolic accelerator for the iterative solution of sparse linear systems

Abstract: The idea of grouping the nonzero elements of a sparse matrix into a few stripes that are almost parallel is applied to the design of a systolic accelerator for sparse matrix operations. This accelerator is then integrated into a complete systolic system for the solution of large sparse linear systems of equations. The design demonstrates that the application of systolic arrays is not limited to regular computations, and that computationally irregular problems can be solved on systolic networks if local storage is provided in each systolic cell for buffering the irregularity in the data movement and for absorbing the irregularity in the computation. >

Communication reduction for distributed sparse matrix factorization on a processor mesh

Abstract: The problem of reducing the amount of interprocessor communication during the distributed factorization of a sparse matrix on a mesh-connected processor network is investigated. Two strategies are evaluated - 1) use of a fragmented distribution of row/columns of the matrix to limit the number of processors to which each row/column segment is transmitted, and 2) use of the elimination tree to permute the matrix so as to internalize as much of the communication as possible. Empirical evaluation of the schemes using matrices derived from circuit simulation shows significant reduction in the amount of communication for a 64 processor mesh.

Solving Linear Systems with Sparse Matrices on Hypercubes

Abstract: We investigate parallel Gauss elimination for sparse matrices, especially those arising from the discretization of PDEs. We propose an approach which combines minimum degree ordering, nested dissection, domain decomposition and multifront techniques. Neither symbolic factorization nor explicit representation of elimination trees are needed. An effective and economic dynamic data structure is presented along with a grid based subtree~subcube assignment strategy which enhances load balancing, high parallelism and low communication cost The algorithm is implemented on the NCUBE/7 .. Work supported in part by National Science FoundaLion granl CCR·8619817• ... Work supponed in part by the Air Force Office of Scientific Research grants 84-0385. 88-0"-43.

Approximate Sufficiency of Sparse Order Statistics

Abstract: This chapter starts with an introduction to “comparison of statistical models” where in addition to Section 9.1 we also make use of Markov kernels.

Architectural Mechanisms To Support Sparse Vector Processing

Abstract: We discuss the algorithmic stepe involved in common sparse matrix problems, with particular emphasis on linear programming by the revised simplex method. We then propoae new architectural mechanisms which are being built into an experimental machine, the Edinburgh Sparse Processor, and which enable vector instructions to operate eficiently on sparse vectors stored in compressed form. Finally, we review the use of these new mechanisms on the linear programming problem.

Incremental Condition Estimator for Parallel Sparse Matrix Factorization

Abstract: There is often a trade-off between preserving sparsity and numerical stability in sparse matrix factorizations. In applications like the direct solution of Equality Constrained Least Squares problem, the accurate detection of the rank of a large and sparse constraint matrix is a key issue. Column pivoting is not suitable for distributed memory machines because it forces the program into a lock-step mode, preventing any overlapping of computations. So factorization algorithms on such machines need to use a reliable, yet inexpensive incremental condition estimator to decide on which columns to be included. We describe an incremental condition estimator that can be used during a sparse QR factorization. We show that it is quite reliable and is well suited for use on parallel machines. We supply experimental results to support its effectiveness as well as suitability for parallel architectures.

Anova model fitting via sparse matrix computations: a fast direct method

Abstract: A fast and storage-efficient direct method for fitting analysis-of-variance models to unbalanced data is presented This method exploits sparsity and rank deficiency of the model matrix and is based on orthogonal Givens factorization of a set of sparse columns of the model matrix A class of matrices generated by index sets is defined and used to obtain results on linear dependencies between columns of a model matrix and fill during factorization These results are used to develop an algorithm for the selection, ordering, and symbolic factorization of a set of sparse columns of the model matrix This facilitates a fast and storage-efficient numerical factorization and solution A comparison to both a standard direct algorithm and a general-purpose sparse least-squares algorithm shows that the new algorithm reduces time and storage by orders of magnitude for large models and is competitive for small models