Donald C. S. Allison

The parallel complexity of embedding algorithms for the solution of systems of nonlinear equations

Embedding algorithms used to solve nonlinear systems of equations do so by constructing a continuous family of systems and solving the given system by tracking the continuous curve of solutions to the family. Solving nonlinear equations by a globally convergent embedding algorithm requires the evaluation and factoring of a Jacobian matrix at many points along the embedding curve. Ways to optimize the Jacobian matrix on a hypercube are described. Several static and dynamical strategies for assigning components of the Jacobian to processors on the hypercube are investigated. It is found that a static rectangular grid mapping is the preferred choice for inclusion in a robust parallel mathematical software package. The static linear mapping is a viable alternative when there are many common subexpressions in the component evaluation, and the dynamic assignment strategy should only be considered when there is large variation in the evaluation times for the components, leading to a load imbalance on the processors. >

Some performance tests of convex hull algorithms

The two-dimensional convex hull algorithms of Graham, Jarvis, Eddy, and Akl and Toussaint are tested on four different planar point distributions. Some modifications are discussed for both the Graham and Jarvis algorithms. Timings taken of FORTRAN implementations indicate that the Eddy and Akl-Toussaint algorithms are superior on uniform distributions of points in the plane. The Graham algorithm outperforms the others on those distributions where most of the points are on or near the boundary of the hull.

SCALABILITY ANALYSIS OF PARALLEL GMRES IMPLEMENTATIONS

Abstract Applications involving large sparse nonsymmetric linear systems encourage parallel implementations of robust iterative solution methods, such as GMRES(k). Two parallel versions of GMRES(k) based on different data distributions and using Householder reflections in the orthogonalization phase are analyzed with respect to scalability (their ability to maintain fixed efficiency with an increase in problem size and number of processors). A theoretical algorithm-machine model for scalability of GMRES(k) with fixed k is derived and validated by experiments on three parallel computers, each with different machine characteristics. The analysis for an adaptive version of GMRES(k), in which the restart value k is adapted to the problem, is also presented and scalability results for this case are briefly discussed.

Sorting in linear expected time

A new sorting algorithm, Double Distributive Partitioning, is introduced and compared against Sedgewicks quicksort. It is shown that the Double Distributive Partitioning algorithm runs, for all practical purposes, inO(n) time for many distributions of keys. Furthermore, the combined number of comparisons, additions, and assignments required to sort by the new method on these distributions is always less than quicksort.

The Granularity of Parallel Homotopy Algorithms for Polynomial Systems of Equations

Polynomial systems consist of n polynomial functions in n variables, with real or complex coefficients. Finding zeros of such systems is challenging because there may be a large number of solutions, and Newton-type methods can rarely be guaranteed to find the complete set of solutions. There are homotopy algorithms for polynomial systems of equations that are globally convergent from an arbitrary starting point with probability one, are guaranteed to find all the solutions, and are robust, accurate, and reasonably efficient. There is inherent parallelism at several levels in these algorithms. Several parallel homotopy algorithms with different granularities are studied on several different parallel machines, using actual industrial problems from chemical engineering and solid modeling.

Usort: An efficient hybrid of Distributive Partitioning Sorting

A new hybrid of Distributive Partitioning Sorting is described and tested against Quicksort on uniformly distributed items. Pointer sort versions of both algorithms are also tested.

Selection by distributive partitioning

E’rior to 1978, almost all methods of selecting the kth smallest element from a set of elements were based on variations of comparison sorting [2-51. The emergence of distributive partitioning sorting (DPS) [I] suggests that this technique, or a variation of it, might be a convenient starting point for an efficient selection algorithm. This note describes a new selection method based on the original DPS idea described by Dobosiewicz [ 11. In common with comparison selection methods, this method !las an expected time complexity of O(n) for uniform distributicns. The simplicity of the method makes it worth reporting.

Scalable parallel implementations of the GMRES algorithm via Householder reflections

Applications involving large sparse nonsymmetric linear systems encourage parallel implementations of robust iterative solution methods, such as GMRES(k). One variation of GMRES(k) is to adapt the restart value k for any given problem and use Householder reflections in the orthogonalization phase to achieve high accuracy. The Householder transformations can be performed without global communications and modified to use an arbitrary row distribution of the coefficient matrix. The effect of this modification on the GMRES(k) performance is discussed here. This paper compares the abilities of various parallel GMRES(k) implementations to maintain fixed efficiency with increase in problem size and number of processors.

Partitioning rectilinear figures into rectangles

This paper discusses the problem of partitioning rectilinear regions, with or without holes, into a minimum number of rectangles. An algorithm which solves this partitioning problem in time O(n5/2), where n is the number of vertices of the rectilinear figure, is presented.

Parallel Adaptive GMRES Implementations for Homotopy Methods

The success of probability-one homotopy methods in solving large-scale optimization problems and nonlinear systems of equations on parallel architectures may be significantly enhanced by the accurate parallel solution of large sparse nonsymmetric linear systems. Iterative solution techniques, such as GMRES(k), favor parallel implementations. However, their straightforward parallelization usually leads to a poor parallel performance because of global communication incurred by processors. One variation of GMRES(k) considered here is to adapt the restart value k for any given problem and use Householder reflections in the orthogonalization phase, coupled with graph-based matrix partitioning, to achieve high accuracy and reduce the communication overhead. This particular GMRES implementation is tailored to the uniquely stringent requirements imposed on a linear system solver by probability-one homotopy algorithms: occasionally unusually high accuracy, ability to adapt to problems of widely varying difficulty, and parallelism.