Grid Computing for the Bi-CGSTAB Applied to the Solution of the Modified Helmholtz Equation

Authors: E. N. Mathioudakis, E. P. Papadopoulou

Abstract:

The problem addressed herein is the efficient management of the Grid/Cluster intense computation involved, when the preconditioned Bi-CGSTAB Krylov method is employed for the iterative solution of the large and sparse linear system arising from the discretization of the Modified Helmholtz-Dirichlet problem by the Hermite Collocation method. Taking advantage of the Collocation ma-trix's red-black ordered structure we organize efficiently the whole computation and map it on a pipeline architecture with master-slave communication. Implementation, through MPI programming tools, is realized on a SUN V240 cluster, inter-connected through a 100Mbps and 1Gbps ethernet network,and its performance is presented by speedup measurements included.

Keywords: Collocation, Preconditioned Bi-CGSTAB, MPI, Grid and DSM Systems.

Digital Object Identifier (DOI): doi.org/10.5281/zenodo.1331933

References:

[1] S.H.Brill and G.F.Pinder, "Parallel implementation of the Bi- CGSTAB method with Block Red-Black Gauss-Seidel preconditioner applied to the Hermite Collocation discretization of partial differetial equations",Parallel Computing, vol. 28, pp. 399-414, 2002.
[2] J. Dongarra, I. Foster, G. Fox, W. Gropp, K. Kennedy, L. Toczon and A. White, SourceBook of Parallel Computing, San Francisco,CA: Morgan Kaufmann Publishers, 2003.
[3] C.C.Christara, "Parallel solvers for spline collocation equations", Advances in Eng. Software,vol. 27, pp.71-89,1996.
[4] J.F. Guarnaccia and G.F. Pinder, "A Collocation based parallel algorithm to solve immiscible two phase flow in porows media", Procs of the 5th SIAM conf. on Parallel processing for scientific computing, pp. 205-210, 1992.
[5] C. E. Houstis, E. N. Houstis and J.R. Rice, "Partitioning PDE Computations: Methods and Performance Evaluation", Parallel Computing,vol. 5, pp. 141-163, 1997.
[6] E. N. Mathioudakis, E. P. Papadopoulou and Y. G. Saridakis, "Mapping Parallel Iterative Algorithms for PDE Computations on a Distributed Memory Computers", Parallel Alg. and Appl. , vol. 8, pp. 141-154, 1996.
[7] E. N. Mathioudakis, E. P. Papadopoulou and Y. G. Saridakis, "Non-Stationary Iterative Schemes for the Solution of Elliptic Collocation Systems", Procs of the 4nd Hellenic - European conf. on computer math. and its applic., pp. 1044-1051, 1998.
[8] E. N. Mathioudakis, E. P. Papadopoulou and Y. G. Saridakis, "Bi-CGSTAB for collocation equations on distributed memory parallel computers" , Numerical Mathematics and advanced applications - ENUMATH 2001, pp. 957-966, 2003.
[9] E. N. Mathioudakis, E. P. Papadopoulou and Y. G. Saridakis, "Iterative Solution of Elliptic Collocation Systems on a Cognitive Parallel Computer", Computers and Mathematics with applications,vol. 48, pp. 951-970, 2004.
[10] E. N. Mathioudakis, E. P. Papadopoulou and Y. G. Saridakis, "Preconditioning for solving Hermite Collocation by the Bi- CGSTAB", WSEAS Trans. on Mathematics, vol 5, no. 7, pp. 811-816, July 2006.
[11] E. N. Mathioudakis and E. P. Papadopoulou, "MPI Managment of Hermite Collocation computation on a Distributed- Shared memory system", WSEAS Trans. on Mathematics, vol. 5, no. 5, pp. 520-525, May 2006.
[12] Message Passing Interface (MPI) web page, http://www.mpiforum. org
[13] E. P. Papadopoulou, Y. G. Saridakis and T.S. Papatheodorou, "Orderings and Partitions of PDE computations for a fixed size VLSI architecture", Procs of IEEE-ACM Fall Joint Computer Science Conference, pp. 366-374, 1997.
[14] T.S.Papatheodorou and Y.G.Saridakis, "Parallel Algorithms and Architectures for Multisplitting Iterative methods", Parallel Computing, vol. 12, pp. 171-182, 1989.
[15] SGI Origin 350 server web page, http://www.sgi.com/products /remarketed/origin350.
[16] SUN V240z server web page, http://www.sun.com/servers/ entry/v240.
[17] H. A. van der Vorst, "Bi-CGSTAB : A fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems", SIAM J. Sci. Statist. Comput., vol. 13, pp. 631-644, 1992.