In this work we review the present status of numerical methods for partial differential equations on vector and parallel computers. A discussion of the relevant aspects of these computers 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. A brief discussion of application areas utilizing these computers is included.

/pdf/solution-of-partial-differential-equations-on-vector-and-3mkk75xyo6.pdf

Solution of Partial Differential Equations on Vector and Parallel Computers

This paper presents a short survey of recent research on Krylov subspace methods with emphasis on implementation on vector and parallel computers. Conjugate gradient methods have proven very useful on traditional scalar computers, and their popularity is likely to increase as three-dimensional models gain importance. A conservative approach to derive effective iterative techniques for supercomputers has been to find efficient parallel/vector implementations of the standard algorithms. The main source of difficulty in the incomplete factorization preconditionings is in the solution of the triangular systems at each step. A few approaches consisting of implementing efficient forward and backward triangular solutions are described in detail. Then polynomial preconditioning as an alternative to standard incomplete factorization techniques is discussed. Another efficient approach is to reorder the equations so as to improve the structure of the matrix to achieve better parallelism or vectorization. An overview of these ideas and others is given in this article, as well as an attempt to comment on their effectiveness or potential for different types of architectures.

/pdf/krylov-subspace-methods-on-supercomputers-5bbepuzurh.pdf

Krylov Subspace Methods on Supercomputers

This paper is divided into three parts. The first part surveys the more common techniques for generating and testing uniformly distributed random numbers. The second part considers the efficient ge...

Random number generations generators on Vector supercomputers and other advanced architectures

We present vectorizable versions of the block preconditioners introduced by Concus, Golub & Meurant (2) for use with the conjugate gradient method. In (2) it was shown that the block preconditioners give less computational work than the classical point ones on conventional serial computers. Here we give numerical results for various vector computers, CDC Cyber 205, CRAY l-S, CRAY X- MP and for several problems, which show that for most cases the block method with slight modifications, gives better results also on vector computers.

The block preconditioned conjugate gradient method on vector computers

We present vectorizable versions of the block preconditioners introduced by Concus. Golub & Meurant [2] for use with the conjugate gradient method. In [2] it was shown that the block preconditioners give less computational work than the classical point ones on conventional serial computers. Here we give numerical results for various vector computers, CDC Cyber 205, CRAY 1-S, CRAY X-MP and for several problems, which show that for most cases the block method with slight modifications, gives better results also on vector computers.

We present the concept of a pseudo-random tree, and generalize the Lehmer pseudo-random number generator as an efficient implementation of the concept. Pseudo-random trees can be used to give reproducibility, as well as speed, in Monte Carlo computations on parallel computers with either the SIMD architecture of the current generation of supercomputer or the MIMD architecture characteristic of the next generation. Monte Carlo simulations based on pseudo-random trees are free of certain pitfalls, even for sequential computers, which can make them considerably more useful.

Pseudo-random trees in Monte Carlo

Publisher Summary This chapter discusses conjugate gradient preconditioners for vector and parallel processors. Vector and parallel processors have caused program designers to select or develop algorithms that allow efficient use of these computers. The relatively simple tasks of solving linear systems of equations or performing matrix multiplication require, as a minimum, a reformulation of existing methods. At another extreme, radically new approaches, such as chaotic relaxation, are invented to accommodate large numbers of asynchronous processors, commonly referred to as multiple instructions, multiple data (MIMD) computers. Preconditioners considered in the chapter include Jacobi, incomplete Cholesky, polynomial approximation, block Jacobi, and block cyclic reduction. Performance data on Cray-1 is provided for finite five- and nine-point difference approximations to elliptic partial differential equations. The variable elements are generated randomly such that the diagonal element of a row is the negative sum of the off-diagonal elements except for the boundary equations.

Conjugate gradient preconditioners for vector and parallel processors

Solving Very Large Elliptic Problems on a Supercomputer with Solid State Disk

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Pseudo-random trees in Monte Carlo

Conjugate gradient preconditioners for vector and parallel processors

Solving Very Large Elliptic Problems on a Supercomputer with Solid State Disk