Neville elimination on multi- and many-core systems: OpenMP, MPI and CUDA

[EN] This paper describes several parallel algorithmic variations of the Neville elimination. This elimination solves a system of linear equations making zeros in a matrix column by adding to each row an adequate multiple of the preceding one. The parallel algorithms are run and compared on different multi- and many-core platforms using parallel programming techniques as MPI, OpenMP and CUDA. © 2009 Springer Science+Business Media, LLC.This work has been supported by project TIN2007-61273 and FEDER, and by INCO2 (Excellence Group founded by PROMETEO 2009/013, High Performance Computing on Current Architectures in Multiple Signal Processing Problems, Generalitat Valenciana).Alonso, P.; Cortina, R.; Martínez Zaldívar, FJ.; Ranilla, J. (2011). Neville elimination on multi- and many-core systems: OpenMP, MPI and CUDA. Journal of Supercomputing. 58(2):215-225. https://doi.org/10.1007/s11227-009-0360-z215225582Intel (2005) Intel multi-core processor architecture developer backgrounder. White paperOwens JD, Houston M, Luebke D, Green S, Stone JE, Phillips JC (2008) GPU computing. Proc IEEE 96(5):879–899Gasca M, Peña JM (1992) Total positivity and Neville elimination. Linear Algebra Appl 165:25–44Gasca M, Peña JM (1994) A matricial description of Neville elimination with applications to total positivity. Linear Algebra Appl 202:33–45Demmel J, Koev P (2005) The accurate and efficient solution of a totally positive generalized Vandermonde linear system. SIAM J Matrix Anal Appl 27:142–152Gemignani L (2008) Neville elimination for rank-structured matrices. Linear Algebra Appl 428(4):978–991Alonso P, Cortina R, Díaz I, Ranilla J (2004) Neville elimination: a study of the efficiency using checkerboard partitioning. Linear Algebra Appl 393:3–14Alonso P, Díaz I, Cortina R, Ranilla J (2008) Scalability of Neville elimination using checkerboard partitioning. Int J Comput Math 85(3–4):309–317Alonso P, Cortina R, Díaz I, Ranilla J (2009) Blocking Neville elimination algorithm for exploiting cache memories. Appl Math Comput 209(1):2–9Cortina R (2008) El método de Neville: un enfoque basado en Computación de Altas Prestaciones. Ph.D. thesis, Univ. of Oviedo, SpainChandra R et al (2001) Parallel programming in OpenMP. Morgan Kaufmann, San Mate

An efficient and scalable block parallel algorithm of Neville elimination as a tool for the CMB maps problem

Abstract This paper analyses the performance of several versions of a block parallel algorithm in order to apply Neville elimination in a distributed memory parallel computer. Neville elimination is a procedure to transform a square matrix A into an upper triangular one. This analysis must take into account the algorithm behaviour as far as execution time, efficiency and scalability are concerned. Special attention has been paid to the study of the scalability of the algorithms trying to establish the relationship existing between the size of the block and the performance obtained in this metric. It is important to emphasize the high efficiency achieved in the studied cases and that the experimental results confirm the theoretical approximation. Therefore, we have obtained a high predicting ability tool of analysis. Finally, we will present the elimination of Neville as an efficient tool in detecting point sources in cosmic microwave background maps.© Springer Science+Business Media, LLC 2010This work has been partially supported by the Spanish Research Grants TIN2007-61273, TIN2008-06570-C04-02 and TIN2010-14971, and by Valencia Regional Government Grant PRO-METEO/2009/013. Also, we would like to give special thanks to Professor Francisco Argueso of University of Oviedo for his help.Alonso, P.; Cortina, R.; Ranilla, J.; Vidal Maciá, AM. (2012). An efficient and scalable block parallel algorithm of Neville elimination as a tool for the CMB maps problem. Journal of Mathematical Chemistry. 50(2):345-358. https://doi.org/10.1007/s10910-010-9769-0S345358502Alonso P., Cortina R., Díaz I., Ranilla J.: Analyzing scalability of Neville elimination. J. Math. Chem. 40(1), 49 (2006)Alonso P., Cortina R., Díaz I., Ranilla J.: Scalability of Neville elimination using checkerboard partitioning. Int. J. Comput. Math. 85(3–4), 309 (2008)Alonso P., Cortina R., Díaz I., Ranilla J.: Blocking Neville elimination algorithm for exploiting cache memories. Appl. Math. Comput. 209, 2 (2009)Ando T.: Totally positive matrices. Linear Algebra Appl. 90, 165 (1987)Gasca M., Michelli C.A.: Total Positivity and its Applications. Kluwer, Dordrecht (1996)Gasca M., Peña J.M.: Total positivity and Neville elimination. Linear Algebra Appl. 165, 25 (1992)Gemignani L.: Neville elimination for rank-structured matrices. Linear Algebra Appl. 428(4), 978 (2008)Grama A., Gupta A., Karypis G., Kumar V.: Introduction to Parallel Computing. Pearson Education Limited, London (2003)Lin H., Bao H., Wang G.: Totally positive bases and progressive iteration approximation. Comput. Math. Appl. 50, 575 (2005)Lopez-Caniego M. et al.: Comparison of filters for the detection of point sources in Planck simulations. Mon. Not. Roy. Astron. Soc. 370, 2047 (2006)Peña J.M.: Shape Preserving Representations in Computer Aided–Geometric Design. Nova Science Publishers, New York (1999)Penzias A.A., Wilson R.W.: A measurement of excess antenna temperature at 4080 Mc/s. Astrophys. J. 142, 419 (1965)Prieto M., Montero R.S., Llorente I.M., Tirado F.: A parallel multigrid solver for viscous flows on anisotropic structured grids. Parallel Comput. 29, 907 (2003)Smoot G. et al.: Structure in the COBE differential microwave radiometer first-year maps. Astrophys. J. 396, L1 (1992)Spergel D.N. et al.: First-year Wilkinson microwave anisotropy probe (WMAP) observations: determination of cosmological parameters. Astrophys. J. Suppl. 148, 175 (2003)J.A. Tauber, The Planck mission, in New Cosmological Data and the Values of the Fundamental Parameters. Proceedings of IAU Symposium vol. 201 (2005), p. 86, eds. by A. Lasenby, A. WilkinsonToffolatti L. et al.: Extragalactic source counts and contributions to the anisotropies of the cosmic microwave background: predictions for the Planck Surveyor mission. Mon. Not. Roy. Astron. Soc. 297, 117 (1998