Abstract
New parallel computational techniques are introduced for the parallelization of Generic Approximate Sparse Inverse multigrid methods, based on Portable Operating System Interface for UniX (POSIX) threads, for multicore systems. Parallelization of the Generic Approximate Sparse Inverse Matrix (GenAspI) algorithm is achieved based on a new computational approach, namely “strip,” which utilizes the data independence of the rows assigned in each available processor. Additionally, new parallel computational techniques are proposed for the parallelization of a modified multigrid V-Cycle method, based on POSIX Threads, for multicore systems. The modified V-Cycle utilized a Parallel PGenAspI Preconditioned Bi-Conjugate Gradient STABilized (BiCGSTAB) as a coarse solver to ensure better parallel performance of the multigrid method. For parallelization purposes, a replication of the multigrid method function is executed on each processor with different index bands and with proper synchronization points to ensure less thread-creation overhead and to maximize parallel performance. Theoretical estimates on speedups and efficiency are also presented. Finally, numerical results for the performance of the PGenAspI algorithm and the PGenAspI–MGV method for solving classical two-dimensional boundary value problems on multicore computer systems are presented. The implementation issues of the proposed method are also discussed using POSIX threads on multicore systems.
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References
Bank RE, Douglas CC (1985) Sharp estimates for multigrid rates of convergence with general smoothing and acceleration. SIAM J Numer Anal 22:617–633
Briggs LW, Henson VE, McCormick FS (2000) A multigrid tutorial. SIAM, Philadelphia
Bröker O, Grote MJ, Mayer C, Reusken A (2001) Robust parallel smoothing for multigrid via sparse approximate inverses. SIAM J Sci Comput 23(4):1396–1417
Butenhof DR (1997) Programming with POSIX® threads. Addison-Wesley, Reading
Chow E (2000) A priori sparsity patterns for parallel sparse approximate inverse preconditioners. SIAM J Sci Comput 21:1804–1822
Chow E (2001) Parallel implementation and practical use of sparse approximate inverses with a priori sparsity patterns. Int J High Perform Comput Appl 15:56–74
Filelis-Papadopoulos CK, Gravvanis GA (2013) On the multigrid method based on finite difference approximate inverses. Comput Model Eng Sci 90(3):233–253
Filelis-Papadopoulos CK, Gravvanis GA (2013) Generic approximate sparse inverse matrix techniques. Int J Comput Methods (to appear)
Giannoutakis KM, Gravvanis GA (2008) High performance finite element approximate inverse preconditioning. Appl Math Comput 201:293–304
Gravvanis GA (2009) High performance inverse preconditioning. Arch Comput Methods Eng 16(1):77–108
Gravvanis GA (1996) The rate of convergence of explicit approximate inverse preconditioning. Int J Comput Math 60:77–89
Gravvanis GA, Filelis-Papadopoulos CK, Matskanidis PI (2012) Algebraic multigrid methods based on generic approximate inverse matrix techniques. TR/ECE/ASC-AMA/2012/13
Hackbusch W (1985) Multigrid methods and applications. Springer, Berlin
Hackbusch W (1985) Iterative solution of large sparse systems of equations. Springer, Berlin
Haelterman R, Viederndeels J, van Heule D (2009) Non-stationary two-stage relaxation based on the principle of aggregation multi-grid. In: Computational fluid dynamics 2006, part 3. Springer, Berlin, pp 243–248
Karamouta A, Filelis-Papadopoulos CK, Gravvanis GA, Chryssomallis MT (2012) On the numerical solution of a time harmonic 3D wave equation by explicit approximate inverse preconditioning. In: 16th panhellenic conference on informatics (PCI) 2012, vol 5–7, pp 223–227
Kyziropoulos PE, Filelis-Papadopoulos CK, Gravvanis GA (2013) N-Body simulation based on the particle mesh method using multigrid schemes. Comput Asp Numer Algorithms (CANA 2013), pp 471–478
Lipitakis EA, Evans DJ (1987) Explicit semi-direct methods based on approximate inverse matrix techniques for solving boundary-value problems on parallel processors. Math Comput Simul 29:1–17
Lipitakis EA, Gravvanis GA (1995) Explicit preconditioned iterative methods for solving large unsymmetric finite element systems. Computing 54(2):167–183
Saad Y (1996) Iterative methods for sparse linear systems. PWS, Boston
Saad Y, van der Vorst HA (2000) Iterative solution of linear systems in the 20th century. J Comput Appl Math 123:1–33
Smith IM, Margets L (2006) The convergence variability of parallel iterative solvers. Eng Comput 23(2):154–165
Trottenberg U, Osterlee CW, Schuller A (2000) Multigrid. Academic Press, San Diego
Van der Vorst HA (1992) Bi-CGSTAB: a fast and smoothly converging variant of Bi-CG for the solution of nonsymmetric linear systems. SIAM J Sci Stat Comput 13(2):631–644
Yun JH, Kim SW (1997) Parallel implementation of hybrid iterative methods for nonsymmetric linear systems. Korean J Comp Appl Math 4(1):1–16
Acknowledgements
The authors would like to express their thanks to Professor K.G. Margaritis, Parallel Distributed Processing Laboratory, Department of Applied Informatics, University of Macedonia, for the provision of suitable computational facilities.
The authors would like also to thank the anonymous reviewers for constructive suggestions and criticism for improving the manuscript.
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Appendix
Appendix
In the following the PGenAspI-BiCGSTAB method is presented by the following algorithm:
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Filelis-Papadopoulos, C.K., Gravvanis, G.A. Parallel multigrid algorithms based on generic approximate sparse inverses: an SMP approach. J Supercomput 67, 384–407 (2014). https://doi.org/10.1007/s11227-013-1006-8
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DOI: https://doi.org/10.1007/s11227-013-1006-8