The solution of linear systems of equations with a structural analysis code on the NAS CRAY-2

Two methods for solving linear systems of equations on the NAS Cray-2 are described. One is a direct method; the other is an iterative method. Both methods exploit the architecture of the Cray-2, particularly the vectorization, and are aimed at structural analysis applications. To demonstrate and evaluate the methods, they were installed in a finite element structural analysis code denoted the Computational Structural Mechanics (CSM) Testbed. A description of the techniques used to integrate the two solvers into the Testbed is given. Storage schemes, memory requirements, operation counts, and reformatting procedures are discussed. Finally, results from the new methods are compared with results from the initial Testbed sparse Choleski equation solver for three structural analysis problems. The new direct solvers described achieve the highest computational rates of the methods compared. The new iterative methods are not able to achieve as high computation rates as the vectorized direct solvers but are best for well conditioned problems which require fewer iterations to converge to the solution

Adapting the interior point method for the solution of LPs on serial, coarse grain parallel and massively parallel computers

In this paper we describe a unified scheme for implementing an interior point algorithm (IPM) over a range of computer architectures. In the inner iteration of the IPM a search direction is computed using Newton's method. Computationally this involves solving a sparse symmetric positive definite (SSPD) system of equations. The choice of direct and indirect methods for the solution of this system, and the design of data structures to take advantage of serial, coarse grain parallel and massively parallel computer architectures, are considered in detail. We put forward arguments as to why integration of the system within a sparse simplex solver is important and outline how the system is designed to achieve this integration

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

A high-accuracy optical linear algebra processor for finite element applications

Optical linear processors are computationally efficient computers for solving matrix-matrix and matrix-vector oriented problems. Optical system errors limit their dynamic range to 30-40 dB, which limits their accuray to 9-12 bits. Large problems, such as the finite element problem in structural mechanics (with tens or hundreds of thousands of variables) which can exploit the speed of optical processors, require the 32 bit accuracy obtainable from digital machines. To obtain this required 32 bit accuracy with an optical processor, the data can be digitally encoded, thereby reducing the dynamic range requirements of the optical system (i.e., decreasing the effect of optical errors on the data) while providing increased accuracy. This report describes a new digitally encoded optical linear algebra processor architecture for solving finite element and banded matrix-vector problems. A linear static plate bending case study is described which quantities the processor requirements. Multiplication by digital convolution is explained, and the digitally encoded optical processor architecture is advanced

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

Sparse Equation-Eigen Solvers for Symmetric/Unsymmetric Positive-Negative-Indefinite Matrices with Finite Element and Linear Programming Applications

Vectorized sparse solvers for direct solutions of positive-negative-indefinite symmetric systems of linear equations and eigen-equations are developed. Sparse storage schemes, re-ordering, symbolic factorization and numerical factorization algorithms are discussed. Loop unrolling techniques are also incorporated in the coding to enhance the vector speed. In the indefinite solver, which employs various pivoting strategies, a simple rotation matrix is introduced to simplify the computer implementation. Efficient usage of the incore memory is accomplished by the proposed restart memory management schemes. A sparse version of the Interior Point Method, IPM, has also been implemented that incorporates the developed indefinite sparse solver for linear programming applications. Numerical performance of the developed software is conducted by performing the static analysis and eigen-analysis of several practical finite elements models, such as the EXXON Offshore Structure, the High Speed Civil Transport (HSCT) Aircraft, and the Space Shuttle Solid Rocket Booster (SRB). The results have been compared to benchmark results provided by the Computational Structural Branch at NASA Langley Research Center. Small to medium-scale linear programming examples have also been used to demonstrate the robustness of the proposed sparse IPM

Solving large scale linear programming

The interior point method (IPM) is now well established as a competitive technique for solving very large scale linear programming problems. The leading variant of the interior point method is the primal dual - predictor corrector algorithm due to Mehrotra. The main computational steps of this algorithm are the repeated calculation and solution of a large sparse positive definite system of equations. We describe an implementation of the predictor corrector IPM algorithm on MasPar, a massively parallel SIMD computer. At the heart of the implemen-tation is a parallel Cholesky factorization algorithm for sparse matrices. Our implementation uses a new scheme of mapping the matrix onto the processor grid of the MasPar, that results in a more efficient Cholesky factorization than previously suggested schemes. The IPM implementation uses the parallel unit of MasPar to speed up the factorization and other computationally intensive parts of the IPM. An impor-tant part of this implementation is the judicious division of data and computation between the front-end computer, that runs the main IPM algorithm, and the par-allel unit. Performanc