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

Complexity of dense-linear-system solution on a multiprocessor ring

Different algorithms, based on Gaussian elimination, for the solution of dense linear systems of equations, are discussed for a multiprocessor ring The number of processors is assumed not to exceed the problem size A fairly general model for data transfer is proposed and the algorithms are analyzed with respect to their requirements of arithmetic as well as communication times This paper lays no claims to being either exhaustive or complete Its objective is to compare a variety of algorithms, which are fairly reasonable to program and to analyze, for the solution of a single problem on a certain class of parallel architectures, thereby leading to a more realistic approach to future algorithm development on multiprocessor machines

Complexity of dense linear system solution on a multiprocessor ring. Research report

This paper proposes a method, based on substructuring, to solve nonlinear structural analysis problems on multiprocessor computers. Background information is given on the use of substructuring in large-scale finite element programs and computational time distributions for the major components for an example nonlinear finite element analysis are discussed. Implementation of the substructuring method on a typical multiprocessor computer is described and estimates are made of expected reductions in computation times based on nonlinear substructuring results obtained on a single processor computer.

Nonlinear substructuring method for concurrent processing computers

Methods for efficient computation of numerical algorithms on a wide variety of MIMD machines are proposed. These techniques reorganize the data dependency patterns to improve the processor utilization. The model problem finds the time-accurate solution to a parabolic partial differential equation discretized in space and implicitly marched forward in time. The algorithms are extensions of Jacobi and SOR. The extensions consist of iterating over a window of several timesteps, allowing efficient overlap of computation with communication. The methods increase the degree to which work can be performed while data are communicated between processors. The effect of the window size and of domain partitioning on the system performance is examined both by implementing the algorithm on a simulated multiprocessor system.

/pdf/towards-developing-robust-algorithms-for-solving-partial-xfpmqxowib.pdf

Towards developing robust algorithms for solving partial differential equations on MIMD machines

Solving large, sparse, linear systems of equations is one of the fundamental problems in large scale scientific and engineering computation. A model of a general class of asynchronous, iterative solution methods for linear systems is developed. In the model, the system is solved by creating several cooperating tasks that each compute a portion of the solution vector. This model is then analyzed to determine the expected intertask data transfer and task computational complexity as functions of the number of tasks. Based on the analysis, recommendations for task partitioning are made. These recommendations are a function of the sparseness of the linear system, its structure (i.e., randomly sparse or banded), and dimension.

A model of asynchronous iterative algorithms for solving large, sparse, linear systems

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A model of asynchronous iterative algorithms for solving large, sparse, linear systems