Block colouring schemes for the SOR method on local memory parallel computers

doi:10.1016/0167-8191(90)90096-R

Parallel Computing

Volume 14, Issue 1, May 1990, Pages 61-75

https://doi.org/10.1016/0167-8191(90)90096-R Get rights and content

Abstract

We introduce the concept of a block colouring which we use to obtain parallel versions of the SOR method. Block colourings are particularly well suited for banded linear systems and finite difference discretizations. We show that on a local memory parallel computer block colour SOR often induces less communicational overhead than the traditional multicolour SOR method. This is confirmed by numerical experiments with two simple examples on a 64-processor binary tree computer.

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Citation Excerpt :
This divides the nodes into two groups, namely the inner nodes and the interface nodes. Next, the block multi-color ordering method [7,21,22] is applied to the inner nodes, while the multi-color ordering method is applied to the interface nodes. In the block multi-color ordering method, convergence can be improved by increasing the block size.
This paper reports on an investigation into large-scale parallel time-harmonic electromagnetic field analysis based on the finite element method. The parallel geometric multigrid preconditioned iterative solver for the resulting linear system was developed on a cluster of shared memory parallel computers. We propose a hybrid parallel ordering method for the parallelization of a multiplicative Schwarz smoother, which is a key component of the multigrid solver for electromagnetic field analysis. The method, using domain decomposition ordering for multi-process parallelism and introducing block multi-color ordering for multi-thread parallel processing, attains a high convergence rate with a small number of message passing interface communications and thread synchronizations. The numerical test confirms that the proposed method attains a solver performance more than twice as good as the conventional method based on multi-color ordering. Furthermore, an approximately 800 million degrees of freedom problem is successfully solved on 256 quad-core processors.
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We present a parallelizable SSOR preconditioning scheme for Krylov subspace iterative solvers which proves to be efficient in lattice QCD applications involving Wilson fermions. Our preconditioner is based on a locally lexicographic ordering of the lattice points. In actual Hybrid Monte Carlo applications with the bi-conjugate gradient stabilized method BiCGstab, we achieve a gain factor of about 2 in the number of iterations compared to conventional odd-even preconditioning. Whether this translates into similar reductions in run time will depend on the parallel computer in use. We discuss implementation issues using the ‘Eisenstat-trick’ and machine specific advantages of the method for the APE100/Quadrics parallel computer. In a full QCD simulation on a 512-processor Quadrics QH4 we find a gain in cpu-time of a factor of 1.7 over odd-even preconditioning for a 24³ × 40 lattice.
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A detailed model is presented for the speed-up of parallel linear iterative solvers. The mapping of a topologically rectangular discrete grid onto a processor array of N × M transputers leads to step-like behaviour in the local communication and calculation time. This results in a typical oscillatory behaviour of ‘speed-up’ when problem sizes are increased monotonously. The theory is compared with experimental results obtained on a Parsytec Super Cluster Model 64 using a black-red SOR algorithm coded in 3L Parallel Fortran.
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1994, Parallel Computing
A detailed model is presented for the speedup of parallel linear iterative solvers. The mapping of a topologically rectangular discrete grid onto a processor array of N×M transputers leads to step-like behaviour in the local communication and calculation time. This results in a typical oscillatory behaviour of ‘speedup’ when problem sizes are increased monotonously. The theory is compared with experimental results obtained on a Parsytec Super Cluster Model 64 using a black-red SOR algorithm coded in 3L Parallel Fortran.

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Block colouring schemes for the SOR method on local memory parallel computers

Abstract

Parallel Comput.

Parallel Comput.

Parallel Comput.

Parallel Comput.

Is SOR color-blind?

SIAM J. Sci. Statist. Comput.

Analysis of the SOR iteration for the 9-point Laplacian

SIAM J. Numer. Anal.

A multicolour SOR method for parallel computation

Parallele Algorithmen für Baumrechner

The Numerical Treatment of Differential Equations