Solution of Partial Differential Equations on Vector and Parallel Computers

Ortega, James M.; Voigt, Robert G.

doi:10.1137/1.9781611971774

Cited by 147 publications

(29 citation statements)

References 214 publications

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“…Writting (6), (7) in terms of the error vector e(") = u ; -uii , we have e("+Ii R = (1 -wij)eg) + wij Jij e,") for red points (i + j even), (13) …”

Section: The Local Modified Sor Methodsmentioning

confidence: 99%

“…In fact its convergence complexity is O(N) and since the time per iteration is constant, the execution time is also O ( N ) . In order to use the Successive Overrelaxation (SOR) method [14], [15] we have to color the grid points red-black [I], [13] so that sets of points of the same color can be computed in parallel. However, the parameter w which accelerates the rate of convergence of SOR is computed adaptively in terms of u("+ ')and u(") when the method is applied to (2) Other accelerated techniques which improve the rate of convergence of a basic iterative scheme by an order of magnitude are the Semi-Iterative (SI) and the Conjugate Gradient (CG) methods.…”

Section: Introductionmentioning

confidence: 99%

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The parallel local modified sor for nonsymmetric linear systems^∗

Boukas¹,

Missirlis²

1998

International Journal of Computer Mathematics

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In this paper we introduce the local Modified Successive Overrelaxation (MSOR) method and apply Fourier analysis to study its convergence. Parallelism is introduced by decoupling the mesh points with the use of red-black ordering for the 5-point stencil. The optimum set of values for the parameters involved, when the Jacobi iteration operator possesses imaginary eigenvalues, is determined. The performance of the local MSOR method is illustrated by its application to the numerical solution of the convection diffusion equation. It is found that the proposed method is significantly more efficient than local SOR when the absolute value of the smallest eigenvalue of the Jacobi operator is larger than unity. Finally, the parallel implementation of the local MSOR method is discussed and results are presented for distributed memory processors with a mesh topology.

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“…Writting (6), (7) in terms of the error vector e(") = u ; -uii , we have e("+Ii R = (1 -wij)eg) + wij Jij e,") for red points (i + j even), (13) …”

Section: The Local Modified Sor Methodsmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

The parallel local modified sor for nonsymmetric linear systems^∗

Boukas¹,

Missirlis²

1998

International Journal of Computer Mathematics

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“…When we deal with the degenerate case of the BVP described by (38), where the function a(x, y) may vanish in some isolated points, we can use Taylor's expansion in order to evaluate the coefficientsx andŷ. The result of this analysis is similar to that one obtained in the positive case, so that a weak clustering property holds even for the related coefficient matrix A * .…”

Section: The Analysis Of Convergence In the 2d Symmetric Casementioning

confidence: 99%

“…Therefore these methods work much finer then those based on incomplete LU factorizations [36,13,27] and on circulant preconditioners [9,30,35] since the latter techniques do not assure a linear rate of convergence; this method is also competitive in comparison with multigrid algorithms [28, 3,41] which guarantee a linear convergence speed, independent of the mesh size, but not a superlinear convergence rate. Notice that also the classical methods based on separable preconditioners guarantee a linear rate of convergence (not superlinear) [18], but the approach is intrinsically different: actually, the separable preconditioning systems are solved by superfast direct methods (cyclic reduction) [38,17,45] in O(n log n) ops (O(log n) parallel steps) and the preconditioners themselves are devised and analysed starting from a "differential" point of view. In our case, the proposed preconditioners, which cannot be looked, in general, as separable ones, are constructed in a matrix theory context and the related systems are again solved by iterative strategies (e.g., ad hoc multigrids requiring O(n) ops and O(log n) parallel steps [19,20]).…”

Section: Introductionmentioning

confidence: 99%

The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems

Serra

1999

Numerische Mathematik

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In previous works [21][22][23] we proposed the use of τ [5] and band Toeplitz based preconditioners for the solution of 1D and 2D boundary value problems (BVP) by means of the preconditioned conjugate gradient (PCG) methods. As τ and band Toeplitz linear systems can be solved [4] by using fast sine transforms [8], these methods become especially attractive in a parallel environment of computation. In this paper we extend this technique to the nonlinear, nonsymmetric case and, in addition, we prove some clustering properties for the spectra of the preconditioned matrices showing why these methods exhibit a convergence speed which results to be more than linear. Therefore these methods work much finer than those based on separable preconditioners [18,45], on incomplete LU factorizations [36,13,27], and on circulant preconditioners [9,30,35] since the latter two techniques do not assure a linear rate of convergence. On the other hand, the proposed technique has a wider range of application since it can be naturally used for nonlinear, nonsymmetric problems and for BVP in which the coefficients of the differential operator are not strictly positive and only piecewise smooth. Finally the several numerical experiments performed here and in [22,23] confirm the effectiveness of the theoretical analysis.

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“…Parallel implementations of SOR have been studied by Hockney and Jesshope [1981], Saltz and Naik [1988], Ortega and Voigt [1985], and Zhang [1988], among many others. The alternating direction implicit (ADI) method is an efficient PDE technique that has only recently been considered for parallelization [Fatoohi and Grosch 1987;Johnson et al 1987;Lambiotte 1978].…”

Section: Introductionmentioning

confidence: 99%

Efficient decomposition and performance of parallel PDE, FFT, Monte Carlo simulations, simplex, and Sparse solvers

Cvetanovic

Freedman

Nofsinger³

1991

J Supercomput

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In this paper, we describe the decomposition of six algorithms: two partial differential equations (PDE) solvers (successive over-relaxation [SOR] and alternating direction implicit [ADI]), fast Fourier transform (FFT), Monte Carlo simulations, Simplex linear programming, and Sparse solvers. The algorithms were selected not only because of their importance in scientific applications, but also because they represent a variety of computational (structured to irregular) and communication (low to high) requirements. We present the performance results TM 1 of these algorithms on two shared-memory VAX/VMS multiprocessor prototypes: the VAX 6300 series with up to 8 processors and the M31 with up to 22 processors. We demonstrate that by efficient decomposition it is possible to achieve high performance for all algorithms on both prototypes. We describe the efficient decomposition techniques applied to optimize the performance of parallel algorithms. Also, we discuss the performance implications due to different cache designs on two multiprocessors.

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Solution of Partial Differential Equations on Vector and Parallel Computers

Cited by 147 publications

References 214 publications

The parallel local modified sor for nonsymmetric linear systems^∗

The parallel local modified sor for nonsymmetric linear systems^∗

The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems

Efficient decomposition and performance of parallel PDE, FFT, Monte Carlo simulations, simplex, and Sparse solvers

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Solution of Partial Differential Equations on Vector and Parallel Computers

Cited by 147 publications

References 214 publications

The parallel local modified sor for nonsymmetric linear systems∗

The parallel local modified sor for nonsymmetric linear systems∗

The rate of convergence of Toeplitz based PCG methods for second order nonlinear boundary value problems

Efficient decomposition and performance of parallel PDE, FFT, Monte Carlo simulations, simplex, and Sparse solvers

Contact Info

Product

Resources

About

The parallel local modified sor for nonsymmetric linear systems^∗

The parallel local modified sor for nonsymmetric linear systems^∗