A bibliography on parallel and vector numerical algorithms

This is a bibliography of numerical methods. It also includes a number of other references on machine architecture, programming language, and other topics of interest to scientific computing. Certain conference proceedings and anthologies which have been published in book form are listed also

Solving large scale linear programming problems

The interior point method (IPM) is now well established as a computationaly com-petitive scheme for solving very large scale linear programming problems. The leading variant of the IPM is the primal dual predictor corrector algorithm due to Mehrotra. The main computational efforts in this algorithm are the repeated calculation and solution of a large sparse positive definite system of equations. We describe an implementation of this algorithm for vector processors. At the heart of the implementation is a vectorized matrix multiplication and Cholesky factorization for sparse matrices. We identify the parts where vectorization can be beneficial and discuss in details the merits of alternative vectorization techniques. We show that the best way to utilize a vector processor is by exploiting dense computation within the sparse framework and by unrolling loop operations. We further present an extended definition of supernodes, and describe an implementation based on this new approach. We show that although this approach requires more memory it can increase the scope of dense computation substantially with out adding extra operations. Performance results on standard industrial test problems and comparison between an algorithm that utilizes the extended supernodes and one that utilizes standard supernodes are presented and discussed

Semiannual final report, 1 October 1991 - 31 March 1992

A summary of research conducted at the Institute for Computer Applications in Science and Engineering in applied mathematics, numerical analysis, and computer science during the period 1 Oct. 1991 through 31 Mar. 1992 is presented

Highly parallel sparse Cholesky factorization

Several fine grained parallel algorithms were developed and compared to compute the Cholesky factorization of a sparse matrix. The experimental implementations are on the Connection Machine, a distributed memory SIMD machine whose programming model conceptually supplies one processor per data element. In contrast to special purpose algorithms in which the matrix structure conforms to the connection structure of the machine, the focus is on matrices with arbitrary sparsity structure. The most promising algorithm is one whose inner loop performs several dense factorizations simultaneously on a 2-D grid of processors. Virtually any massively parallel dense factorization algorithm can be used as the key subroutine. The sparse code attains execution rates comparable to those of the dense subroutine. Although at present architectural limitations prevent the dense factorization from realizing its potential efficiency, it is concluded that a regular data parallel architecture can be used efficiently to solve arbitrarily structured sparse problems. A performance model is also presented and it is used to analyze the algorithms

The Czech Republic, 27. 11. -9

Abstract: Our goal is to show on several examples the great progress made in numerical analysis in the past decades together with the principal problems and relations to other disciplines. We restrict ourselves to numerical linear algebra, or, more specifically, to solving Ax = b where A is a real nonsingular n by n matrix and b a real n−dimensional vector, and to computing eigenvalues of a sparse matrix A. We discuss recent developments in both sparse direct and iterative solvers, as well as fundamental problems in computing eigenvalues. The effects of parallel architectures to the choice of the method and to the implementation of codes are stressed throughout the contribution

Group implicit concurrent algorithms in nonlinear structural dynamics

During the 70's and 80's, considerable effort was devoted to developing efficient and reliable time stepping procedures for transient structural analysis. Mathematically, the equations governing this type of problems are generally stiff, i.e., they exhibit a wide spectrum in the linear range. The algorithms best suited to this type of applications are those which accurately integrate the low frequency content of the response without necessitating the resolution of the high frequency modes. This means that the algorithms must be unconditionally stable, which in turn rules out explicit integration. The most exciting possibility in the algorithms development area in recent years has been the advent of parallel computers with multiprocessing capabilities. So, this work is mainly concerned with the development of parallel algorithms in the area of structural dynamics. A primary objective is to devise unconditionally stable and accurate time stepping procedures which lend themselves to an efficient implementation in concurrent machines. Some features of the new computer architecture are summarized. A brief survey of current efforts in the area is presented. A new class of concurrent procedures, or Group Implicit algorithms is introduced and analyzed. The numerical simulation shows that GI algorithms hold considerable promise for application in coarse grain as well as medium grain parallel computers

Solution of partial differential equations on vector and parallel computers

The present status of numerical methods for partial differential equations on vector and parallel computers was reviewed. The relevant aspects of these computers are discussed and a brief review of their development is included, with particular attention paid to those characteristics that influence algorithm selection. Both direct and iterative methods are given for elliptic equations as well as explicit and implicit methods for initial boundary value problems. The intent is to point out attractive methods as well as areas where this class of computer architecture cannot be fully utilized because of either hardware restrictions or the lack of adequate algorithms. Application areas utilizing these computers are briefly discussed

Accelerating data-intensive scientific visualization and computing through parallelization

Many extreme-scale scientific applications generate colossal amounts of data that require an increasing number of processors for parallel processing. The research in this dissertation is focused on optimizing the performance of data-intensive parallel scientific visualization and computing. In parallel scientific visualization, there exist three well-known parallel architectures, i.e., sort-first/middle/last. The research in this dissertation studies the composition stage of the sort-last architecture for scientific visualization and proposes a generalized method, namely, Grouping More and Pairing Less (GMPL), for order-independent image composition workflow scheduling in sort-last parallel rendering. The technical merits of GMPL are two-fold: i) it takes a prime factorization-based approach for processor grouping, which not only obviates the common restriction in existing methods on the total number of processors to fully utilize computing resources, but also breaks down processors to the lowest level with a minimum number of peers in each group to achieve high concurrency and save communication cost; ii) within each group, it employs an improved direct send method to narrow down each processor’s pairing scope to further reduce communication overhead and increase composition efficiency. The performance superiority of GMPL over existing methods is evaluated through rigorous theoretical analysis and further verified by extensive experimental results on a high-performance visualization cluster. The research in this dissertation also parallelizes the over operator, which is commonly used for α-blending in various visualization techniques. Compared with its predecessor, the fully generalized over operator is n-operator compatible. To demonstrate the advantages of the proposed operator, the proposed operator is applied to the asynchronous and order-dependent image composition problem in parallel visualization. In addition, the dissertation research also proposes a very-high-speed pipeline-based architecture for parallel sort-last visualization of big data by developing and integrating three component techniques: i) a fully parallelized per-ray integration method that significantly reduces the number of iterations required for image rendering; ii) a real-time over operator that not only eliminates the restriction of pre-sorting and order-dependency, but also facilitates a high degree of parallelization for image composition. In parallel scientific computing, the research goal is to optimize QR decomposition, which is one primary algebraic decomposition procedure and plays an important role in scientific computing. QR decomposition produces orthogonal bases, i.e.,“core” bases for a given matrix, and oftentimes can be leveraged to build a complete solution to many fundamental scientific computing problems including Least Squares Problem, Linear Equations Problem, Eigenvalue Problem. A new matrix decomposition method is proposed to improve time efficiency of parallel computing and provide a rigorous proof of its numerical stability. The proposed solutions demonstrate significant performance improvement over existing methods for data-intensive parallel scientific visualization and computing. Considering the ever-increasing data volume in various science domains, the research in this dissertation have a great impact on the success of next-generation large-scale scientific applications

Computational methods and software systems for dynamics and control of large space structures

Two key areas of crucial importance to the computer-based simulation of large space structures are discussed. The first area involves multibody dynamics (MBD) of flexible space structures, with applications directed to deployment, construction, and maneuvering. The second area deals with advanced software systems, with emphasis on parallel processing. The latest research thrust in the second area involves massively parallel computers