Solution of Partial Differential Equations on Vector and Parallel Computers

Ortega, James M.; Voigt, Robert G.

doi:10.1137/1027055

Cited by 227 publications

(38 citation statements)

References 358 publications

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“…The search for parallelism has forced researchers to take a second look at many old techniques sometimes resulting in remarkable success. Ortega and Voigt concluded in [91] that there has been very few "truly parallel algorithms" invented, as opposed to modifications of existing algorithms. This has been even more pronounced for iterative methods, perhaps because there are many known algorithms that do offer a large degree of parallelism or that can be simply modified to improve their parallelism.…”

Section: Resultsmentioning

confidence: 99%

“…As a result, considerations for vector implementations were given early on [49,33,125,61,60]. The survey paper of Ortega and Voigt [91] gives an exhaustive bibliography for research done before 1985 in the general area of solution of partial differential equations on supercomputers. Much still remains to be done as many new techniques for vectorizing and parallelizing standard preconditioners were considered only recently.…”

Section: Introductionmentioning

confidence: 99%

“…For additional reading we recommend other survey papers by Ortega and Voigt [91], by Axelsson [13,14] and the recent book by Ortega [go].…”

Section: Introductionmentioning

confidence: 99%

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Krylov Subspace Methods on Supercomputers

Saad¹

1989

SIAM J. Sci. and Stat. Comput.

207

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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. We describe in detail a few approaches consisting of implementing efficient forward and backward triangular solutions. Then we discuss polynomial preconditioning as an alternative to standard incomplete factorization techniques. Another efficient approach is to reorder the equations so as improve the structure of the matrix to achieve better parallelism or vectorization. We give an overview of these ideas and others and attempt to comment on their effectiveness or potential for different types of architectures.

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Section: Resultsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Krylov Subspace Methods on Supercomputers

Saad¹

1989

SIAM J. Sci. and Stat. Comput.

207

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“…There have been a substantial number of theoretical studies of the performance of algorithms on parallel computers but far fewer actual experimental studies (see Ortega and Voigt, [1985] for a comprehensive, up to date review). Even if one agrees that measured performance on an actual parallel processor is the true measure of computing power, there can be disagreements as to what performance is to be measured.…”

Section: Computationalmentioning

confidence: 99%

Adapting a Navier-Stokes Code to the ICL-DAP

Grosch¹

1987

SIAM J. Sci. and Stat. Comput.

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This is a report of the results of an experiment:to adapt a NavierStokes code, originally developed on a serial computer, to concurrent processing on the ICL Distributed A__rrayP__rocessor (DAP). In this paper the algorithm used in solving the Navier-Stokes equations is briefly described.The architecture of the DAP and DAP Fortran is also described. The modifications of the algorithm so as to fit the DAP are given and discussed.Finally, performance results are given and conclusions are drawn.

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“…A lot of versions of parallel Thomas algorithms for general tridiagonal matrices [11][12][13][14][15][16][17][18], as well as for Toeplitz matrices [19][20][21], have been worked out as of today. Algorithms have been proposed that take into account the presence of diagonal dominance [22].…”

Section: Introductionmentioning

confidence: 99%

A highly scalable parallel algorithm for solving Toeplitz tridiagonal systems of linear equations

Terekhov

2016

Journal of Parallel and Distributed Computing

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Based on a modification of the "Dichotomy Algorithm" (Terekhov, 2010), we propose a parallel procedure for solving tridiagonal systems of equations with Toeplitz matrices. Taking the structure of the Toeplitz matrices, we may substantially reduce the number of the "preliminary calculations" of the Dichotomy Algorithm, which makes it possible to effectively solve a series as well as a single system of equations. On the example of solving of elliptic equations by the Separation Variable Method, we show that the computation accuracy is comparable with the sequential version of the Thomas method, and the dependence of the speedup on the number of processors is almost linear. The proposed modification is aimed at parallel realization of a broad class of numerical methods including the inversion of Toeplitz and quasi-Toeplitz tridiagonal matrices.

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Solution of Partial Differential Equations on Vector and Parallel Computers

Cited by 227 publications

References 358 publications

Krylov Subspace Methods on Supercomputers

Krylov Subspace Methods on Supercomputers

Adapting a Navier-Stokes Code to the ICL-DAP

A highly scalable parallel algorithm for solving Toeplitz tridiagonal systems of linear equations

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