March 2018, 8(1): 47-62. doi: 10.3934/naco.2018003

Globalizer: A novel supercomputer software system for solving time-consuming global optimization problems

Lobachevsky State University of Nizhni Novgorod, 23, Gagarin Avenue, Nizhni Novgorod, Russia

* Corresponding author: Konstantin Barkalov

Received  December 2016 Revised  September 2017 Published  March 2018

Fund Project: The study was supported by the Russian Science Foundation, project No 16-11-10150

In this paper, we describe the Globalizer software system for solving the global optimization problems. The system is designed to maximize the use of computational potential of the modern high-performance computational systems in order to solve the most time-consuming optimization problems. The highly parallel computations are facilitated using various distinctive computational schemes: processing several optimization iterations simultaneously, reducing multidimensional optimization problems using multiple Peano space-filling curves, and multi-stage computing based on the nested block reduction schemes. These novelties provide for the use of the supercomputer system capabilities with shared and distributed memory and with large numbers of processors to solve the global optimization problems efficiently.

Citation: Victor Gergel, Konstantin Barkalov, Alexander Sysoyev. Globalizer: A novel supercomputer software system for solving time-consuming global optimization problems. Numerical Algebra, Control & Optimization, 2018, 8 (1) : 47-62. doi: 10.3934/naco.2018003
References:
[1]

K. Barkalov and V. Gergel, Multilevel scheme of dimensionality reduction for parallel global search algorithms, in Proceedings of the 1st International Conference on Engineering and Applied Sciences Optimization, (2014), 2111-2124.

[2]

K. Barkalov and V. Gergel, Parallel global optimization on GPU, J. Glob. Optim., 66 (2016), 3-20. doi: 10.1007/s10898-016-0411-y.

[3]

K. BarkalovV. Gergel and I. Lebedev, Use of Xeon Phi coprocessor for solving global optimization problems, LNCS, 9251 (2015), 307-318. doi: 10.1007/978-3-319-21909-7_31.

[4]

K. BarkalovV. GergelI. Lebedev and A. Sysoev, Solving the global optimization problems on heterogeneous cluster systems, CEUR Workshop Proceedings, 1482 (2015), 411-419.

[5]

K. BarkalovA. PolovinkinI. MeyerovS. Sidorov and N. Zolotykh, SVM regression parameters optimization using parallel global search algorithm, LNCS, 7979 (2013), 154-166. doi: 10.1007/978-3-642-39958-9_14.

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M. R. Bussieck and A. Meeraus, General algebraic modeling system (GAMS), in Modeling Languages in Mathematical Optimization (ed. J. Kallrath), Springer, (2004), 137-157. doi: 10.1007/978-1-4613-0215-5_8.

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Y. Censor and S. A. Zenios, Parallel Optimization: Theory, Algorithms, and Applications, Oxford University Press, 1998.

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R. Čiegis, D. Henty, B. Kågström and J. Žilinskas, Parallel Scientific Computing and Optimization: Advances and Applications, Springer, 2009. doi: 10.1007/978-0-387-09707-7.

[9]

I. N. Egorov, G. V. Kretinin, I. A. Leshchenko and S. V. Kuptzov, IOSO optimization toolkit — novel software to create better design, in 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, 2002. Available from http://www.iosotech.com/text/2002_4329.pdf.

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D. FamularoP. Pugliese and Y. D. Sergeyev, A global optimization technique for checking parametric robustness, Automatica, 35 (1999), 1605-1611. doi: 10.1016/S0005-1098(99)00058-8.

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G. Fasano and J. D. Pintér, Modeling and Optimization in Space Engineering, Springer, 2013. doi: 10.1007/978-1-4614-4469-5.

[12]

C. A. Floudas and M. P. Pardalos, State of the Art in Global Optimization: Computational Methods and Applications, Kluwer Academic Publishers, Dordrecht, 1996. doi: 10.1007/978-1-4613-3437-8.

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C. A. Floudas and M. P. Pardalos, Recent Advances in Global Optimization, Princeton University Press, 2016. doi: 10.1007/s10898-008-9332-8.

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J. M. Gablonsky and C. T. Kelley, A locally-biased form of the DIRECT algorithm, J. Glob. Optim., 21 (2001), 27-37. doi: 10.1023/A:1017930332101.

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M. GavianoD. E. KvasovD. Lera and Y. D. Sergeev, Software for generation of classes of test functions with known local and global minima for global optimization, ACM Trans. Math. Software, 29 (2003), 469-480. doi: 10.1145/962437.962444.

[16]

V. Gergel and I. Lebedev, Heterogeneous parallel computations for solving global optimization problems, Procedia Comput. Sci., 66 (2015), 53-62. doi: 10.1016/j.procs.2015.11.008.

[17]

V. Gergel, A software system for multi-extremal optimization, Eur. J. Oper. Res., 65 (1993), 305-313.

[18]

V. Gergel, A method for using derivatives in the minimization of multiextremum functions, Comput. Math. Math. Phys., 36 (1996), 729-742.

[19]

V. Gergel, A global optimization algorithm for multivariate functions with Lipschitzian first derivatives, J. Glob. Optim., 10 (1997), 257-281. doi: 10.1023/A:1008290629896.

[20]

V. Gergel, et al., High performance computing in biomedical applications, Procedia Computer Science, 18 (2013), 10-19. doi: 10.1016/j.procs.2013.05.164.

[21]

V. Gergel, et al., Recognition of surface defects of cold-rolling sheets based on method of localities, International Review of Automatic Control, 8 (2015), 51-55. doi: 10.15866/ireaco.v8i1.4935.

[22]

V. Gergel and S. Sidorov, A two-level parallel global search algorithm for solving computationally intensive multi-extremal optimization problems, LNCS, 9251 (2015), 505-515. doi: 10.1007/978-3-319-21909-7_49.

[23]

V. A. Grishagin and R. G. Strongin, Optimization of multi-extremal functions subject to monotonically unimodal constraints, Engineering Cybernetics, 5 (1984), 117-122.

[24]

K. Holmstrm and M. M. Edvall, The TOMLAB optimization environment, Modeling Languages in Mathematical Optimization, Springer, (2004), 369-376. doi: 10.1007/978-1-4613-0215-5_19.

[25]

R. Horst and H. Tuy, Global Optimization: Deterministic Approaches, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-662-03199-5.

[26]

D. R. JonesC. D. Perttunen and B. E. Stuckman, Lipschitzian optimization without the Lipschitz constant, J. Optim. Theory Appl., 79 (1993), 157-181. doi: 10.1007/BF00941892.

[27]

R. B. Kearfott, GlobSol user guide, Optim. Methods Softw., 24 (2009), 687-708. doi: 10.1080/10556780802614051.

[28]

D. E. Kvasov and Y. D. Sergeyev, Deterministic approaches for solving practical black-box global optimization problems, Adv. Eng. Softw., 80 (2015), 58-66. doi: 10.1016/j.advengsoft.2014.09.014.

[29]

D. E. KvasovD. MennitiA. PinnarelliY. D. Sergeyev and N. Sorrentino, Tuning fuzzy power-system stabilizers in multi-machine systems by global optimization algorithms based on efficient domain partitions, Electric Power Systems Research, 78 (2008), 1217-1229. doi: 10.1016/j.epsr.2007.10.009.

[30]

D. E. KvasovC. Pizzuti and Y. D. Sergeyev, Local tuning and partition strategies for diagonal GO methods, Numerische Mathematik, 94 (2003), 93-106. doi: 10.1007/s00211-002-0419-8.

[31]

L. Liberti, Writing global optimization software, in Nonconvex Optimization and Its Applications, Springer, 84 (2006), 211-262. doi: 10.1007/0-387-30528-9_8.

[32]

Y. Lin and L. Schrage, The global solver in the LINDO API, Optim. Methods Softw., 24 (2009), 657-668. doi: 10.1080/10556780902753221.

[33]

M. Locatelli and F. Schoen, Global Optimization: Theory, Algorithms and Applications, SIAM, 2013. doi: 10.1137/1.9781611972672.

[34]

G. Luque and E. Alba, Parallel Genetic Algorithms. Theory and Real World Applications, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-22084-5.

[35]

M. MongeauH. KarsentyV. Rouzé and J. B. Hiriart-Urruty, Comparison of public-domain software for black box global optimization, Optim. Methods Softw., 13 (2000), 203-226. doi: 10.1080/10556780008805783.

[36]

K. M. Mullen, Continuous global optimization in R, J. Stat. Softw. , 60 (2014). doi: 10.18637/jss.v060.i06.

[37]

M. P. Pardalos, A. A. Zhigljavsky and J. Žilinskas, Advances in Stochastic and Deterministic Global Optimization, Springer, 2016. doi: 10.1007/978-3-319-29975-4.

[38]

J. D. Pintér, Global Optimization in Action (Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications), Kluwer Academic Publishers, Dordrecht, 1996. doi: 10.1007/978-1-4757-2502-5.

[39]

J. D. Pintér, Software development for global optimization, Lectures on Global Optimization. Fields Institute Communications, 55 (2009), 183-204.

[40]

L. M. Rios and N. V. Sahinidis, Derivative-free optimization: a review of algorithms and comparison of software implementations, J. Glob. Optim., 56 (2013), 1247-1293. doi: 10.1007/s10898-012-9951-y.

[41]

N. V. Sahinidis, BARON: A general purpose global optimization software package, J. Glob. Optim., 8 (1996), 201-205. doi: 10.1007/BF00138693.

[42]

Y. D. Sergeyev, An information global optimization algorithm with local tuning, SIAM J. Optim., 5 (1995), 858-870. doi: 10.1137/0805041.

[43]

Y. D. Sergeyev, Multidimensional global optimization using the first derivatives, Comput. Math. Math. Phys., 39 (1999), 743-752.

[44]

Y. D. Sergeyev and D. E. Kvasov, Global search based on efficient diagonal partitions and a set of Lipschitz constants, SIAM Journal on Optimization, 16 (2006), 910-937. doi: 10.1137/040621132.

[45]

Y. D. Sergeyev and V. A. Grishagin, Parallel asynchronous global search and the nested optimization scheme, J. Comput. Anal. Appl., 3 (2001), 123-145. doi: 10.1023/A:1010185125012.

[46]

Y. D. Sergeyev, R. G. Strongin and D. Lera, Introduction to Global Optimization Exploiting Space-filling Curves, Springer, 2013. doi: 10.1007/978-1-4614-8042-6.

[47]

Y. D. SergeyevD. Famularo and P. Pugliese, Index branch-and-bound algorithm for Lipschitz univariate global optimization with multiextremal constraints, J. Glob. Optim., 21 (2001), 317-341. doi: 10.1023/A:1012391611462.

[48]

R. G. Strongin, Numerical Methods in Multi-Extremal Problems (Information-Statistical Algorithms), Moscow: Nauka, In Russian, 1978.

[49]

R. G. Strongin, Algorithms for multi-extremal mathematical programming problems employing a set of joint space-filling curves, J. Glob. Optim., 2 (1992), 357-378. doi: 10.1007/BF00122428.

[50]

R. G. Strongin, V. P. Gergel, V. A. Grishagin and K. A. Barkalov, Parallel Computations for Global Optimization Problems, Moscow State University (In Russian), Moscow, 2013.

[51]

R. G. Strongin and Y. D. Sergeyev, Global Optimization with Non-convex Constraints. Sequential and Parallel Algorithms, Kluwer Academic Publishers, Dordrecht (2000, 2nd ed. 2013, 3rd ed. 2014). doi: 10.1007/978-1-4615-4677-1.

[52]

A. Törn and A. Žilinskas, Global Optimization, Springer, 1989. doi: 10.1007/3-540-50871-6.

[53]

P. Venkataraman, Applied Optimization with MATLAB Programming, John Wiley & Sons, 2009.

[54]

A. A. Zhigljavsky, Theory of Global Random Search, Kluwer Academic Publishers, Dordrecht, 1991. doi: 10.1007/978-94-011-3436-1.

show all references

References:
[1]

K. Barkalov and V. Gergel, Multilevel scheme of dimensionality reduction for parallel global search algorithms, in Proceedings of the 1st International Conference on Engineering and Applied Sciences Optimization, (2014), 2111-2124.

[2]

K. Barkalov and V. Gergel, Parallel global optimization on GPU, J. Glob. Optim., 66 (2016), 3-20. doi: 10.1007/s10898-016-0411-y.

[3]

K. BarkalovV. Gergel and I. Lebedev, Use of Xeon Phi coprocessor for solving global optimization problems, LNCS, 9251 (2015), 307-318. doi: 10.1007/978-3-319-21909-7_31.

[4]

K. BarkalovV. GergelI. Lebedev and A. Sysoev, Solving the global optimization problems on heterogeneous cluster systems, CEUR Workshop Proceedings, 1482 (2015), 411-419.

[5]

K. BarkalovA. PolovinkinI. MeyerovS. Sidorov and N. Zolotykh, SVM regression parameters optimization using parallel global search algorithm, LNCS, 7979 (2013), 154-166. doi: 10.1007/978-3-642-39958-9_14.

[6]

M. R. Bussieck and A. Meeraus, General algebraic modeling system (GAMS), in Modeling Languages in Mathematical Optimization (ed. J. Kallrath), Springer, (2004), 137-157. doi: 10.1007/978-1-4613-0215-5_8.

[7]

Y. Censor and S. A. Zenios, Parallel Optimization: Theory, Algorithms, and Applications, Oxford University Press, 1998.

[8]

R. Čiegis, D. Henty, B. Kågström and J. Žilinskas, Parallel Scientific Computing and Optimization: Advances and Applications, Springer, 2009. doi: 10.1007/978-0-387-09707-7.

[9]

I. N. Egorov, G. V. Kretinin, I. A. Leshchenko and S. V. Kuptzov, IOSO optimization toolkit — novel software to create better design, in 9th AIAA/ISSMO Symposium on Multidisciplinary Analysis and Optimization, 2002. Available from http://www.iosotech.com/text/2002_4329.pdf.

[10]

D. FamularoP. Pugliese and Y. D. Sergeyev, A global optimization technique for checking parametric robustness, Automatica, 35 (1999), 1605-1611. doi: 10.1016/S0005-1098(99)00058-8.

[11]

G. Fasano and J. D. Pintér, Modeling and Optimization in Space Engineering, Springer, 2013. doi: 10.1007/978-1-4614-4469-5.

[12]

C. A. Floudas and M. P. Pardalos, State of the Art in Global Optimization: Computational Methods and Applications, Kluwer Academic Publishers, Dordrecht, 1996. doi: 10.1007/978-1-4613-3437-8.

[13]

C. A. Floudas and M. P. Pardalos, Recent Advances in Global Optimization, Princeton University Press, 2016. doi: 10.1007/s10898-008-9332-8.

[14]

J. M. Gablonsky and C. T. Kelley, A locally-biased form of the DIRECT algorithm, J. Glob. Optim., 21 (2001), 27-37. doi: 10.1023/A:1017930332101.

[15]

M. GavianoD. E. KvasovD. Lera and Y. D. Sergeev, Software for generation of classes of test functions with known local and global minima for global optimization, ACM Trans. Math. Software, 29 (2003), 469-480. doi: 10.1145/962437.962444.

[16]

V. Gergel and I. Lebedev, Heterogeneous parallel computations for solving global optimization problems, Procedia Comput. Sci., 66 (2015), 53-62. doi: 10.1016/j.procs.2015.11.008.

[17]

V. Gergel, A software system for multi-extremal optimization, Eur. J. Oper. Res., 65 (1993), 305-313.

[18]

V. Gergel, A method for using derivatives in the minimization of multiextremum functions, Comput. Math. Math. Phys., 36 (1996), 729-742.

[19]

V. Gergel, A global optimization algorithm for multivariate functions with Lipschitzian first derivatives, J. Glob. Optim., 10 (1997), 257-281. doi: 10.1023/A:1008290629896.

[20]

V. Gergel, et al., High performance computing in biomedical applications, Procedia Computer Science, 18 (2013), 10-19. doi: 10.1016/j.procs.2013.05.164.

[21]

V. Gergel, et al., Recognition of surface defects of cold-rolling sheets based on method of localities, International Review of Automatic Control, 8 (2015), 51-55. doi: 10.15866/ireaco.v8i1.4935.

[22]

V. Gergel and S. Sidorov, A two-level parallel global search algorithm for solving computationally intensive multi-extremal optimization problems, LNCS, 9251 (2015), 505-515. doi: 10.1007/978-3-319-21909-7_49.

[23]

V. A. Grishagin and R. G. Strongin, Optimization of multi-extremal functions subject to monotonically unimodal constraints, Engineering Cybernetics, 5 (1984), 117-122.

[24]

K. Holmstrm and M. M. Edvall, The TOMLAB optimization environment, Modeling Languages in Mathematical Optimization, Springer, (2004), 369-376. doi: 10.1007/978-1-4613-0215-5_19.

[25]

R. Horst and H. Tuy, Global Optimization: Deterministic Approaches, Springer-Verlag, Berlin, 1990. doi: 10.1007/978-3-662-03199-5.

[26]

D. R. JonesC. D. Perttunen and B. E. Stuckman, Lipschitzian optimization without the Lipschitz constant, J. Optim. Theory Appl., 79 (1993), 157-181. doi: 10.1007/BF00941892.

[27]

R. B. Kearfott, GlobSol user guide, Optim. Methods Softw., 24 (2009), 687-708. doi: 10.1080/10556780802614051.

[28]

D. E. Kvasov and Y. D. Sergeyev, Deterministic approaches for solving practical black-box global optimization problems, Adv. Eng. Softw., 80 (2015), 58-66. doi: 10.1016/j.advengsoft.2014.09.014.

[29]

D. E. KvasovD. MennitiA. PinnarelliY. D. Sergeyev and N. Sorrentino, Tuning fuzzy power-system stabilizers in multi-machine systems by global optimization algorithms based on efficient domain partitions, Electric Power Systems Research, 78 (2008), 1217-1229. doi: 10.1016/j.epsr.2007.10.009.

[30]

D. E. KvasovC. Pizzuti and Y. D. Sergeyev, Local tuning and partition strategies for diagonal GO methods, Numerische Mathematik, 94 (2003), 93-106. doi: 10.1007/s00211-002-0419-8.

[31]

L. Liberti, Writing global optimization software, in Nonconvex Optimization and Its Applications, Springer, 84 (2006), 211-262. doi: 10.1007/0-387-30528-9_8.

[32]

Y. Lin and L. Schrage, The global solver in the LINDO API, Optim. Methods Softw., 24 (2009), 657-668. doi: 10.1080/10556780902753221.

[33]

M. Locatelli and F. Schoen, Global Optimization: Theory, Algorithms and Applications, SIAM, 2013. doi: 10.1137/1.9781611972672.

[34]

G. Luque and E. Alba, Parallel Genetic Algorithms. Theory and Real World Applications, Springer-Verlag, Berlin, 2011. doi: 10.1007/978-3-642-22084-5.

[35]

M. MongeauH. KarsentyV. Rouzé and J. B. Hiriart-Urruty, Comparison of public-domain software for black box global optimization, Optim. Methods Softw., 13 (2000), 203-226. doi: 10.1080/10556780008805783.

[36]

K. M. Mullen, Continuous global optimization in R, J. Stat. Softw. , 60 (2014). doi: 10.18637/jss.v060.i06.

[37]

M. P. Pardalos, A. A. Zhigljavsky and J. Žilinskas, Advances in Stochastic and Deterministic Global Optimization, Springer, 2016. doi: 10.1007/978-3-319-29975-4.

[38]

J. D. Pintér, Global Optimization in Action (Continuous and Lipschitz Optimization: Algorithms, Implementations and Applications), Kluwer Academic Publishers, Dordrecht, 1996. doi: 10.1007/978-1-4757-2502-5.

[39]

J. D. Pintér, Software development for global optimization, Lectures on Global Optimization. Fields Institute Communications, 55 (2009), 183-204.

[40]

L. M. Rios and N. V. Sahinidis, Derivative-free optimization: a review of algorithms and comparison of software implementations, J. Glob. Optim., 56 (2013), 1247-1293. doi: 10.1007/s10898-012-9951-y.

[41]

N. V. Sahinidis, BARON: A general purpose global optimization software package, J. Glob. Optim., 8 (1996), 201-205. doi: 10.1007/BF00138693.

[42]

Y. D. Sergeyev, An information global optimization algorithm with local tuning, SIAM J. Optim., 5 (1995), 858-870. doi: 10.1137/0805041.

[43]

Y. D. Sergeyev, Multidimensional global optimization using the first derivatives, Comput. Math. Math. Phys., 39 (1999), 743-752.

[44]

Y. D. Sergeyev and D. E. Kvasov, Global search based on efficient diagonal partitions and a set of Lipschitz constants, SIAM Journal on Optimization, 16 (2006), 910-937. doi: 10.1137/040621132.

[45]

Y. D. Sergeyev and V. A. Grishagin, Parallel asynchronous global search and the nested optimization scheme, J. Comput. Anal. Appl., 3 (2001), 123-145. doi: 10.1023/A:1010185125012.

[46]

Y. D. Sergeyev, R. G. Strongin and D. Lera, Introduction to Global Optimization Exploiting Space-filling Curves, Springer, 2013. doi: 10.1007/978-1-4614-8042-6.

[47]

Y. D. SergeyevD. Famularo and P. Pugliese, Index branch-and-bound algorithm for Lipschitz univariate global optimization with multiextremal constraints, J. Glob. Optim., 21 (2001), 317-341. doi: 10.1023/A:1012391611462.

[48]

R. G. Strongin, Numerical Methods in Multi-Extremal Problems (Information-Statistical Algorithms), Moscow: Nauka, In Russian, 1978.

[49]

R. G. Strongin, Algorithms for multi-extremal mathematical programming problems employing a set of joint space-filling curves, J. Glob. Optim., 2 (1992), 357-378. doi: 10.1007/BF00122428.

[50]

R. G. Strongin, V. P. Gergel, V. A. Grishagin and K. A. Barkalov, Parallel Computations for Global Optimization Problems, Moscow State University (In Russian), Moscow, 2013.

[51]

R. G. Strongin and Y. D. Sergeyev, Global Optimization with Non-convex Constraints. Sequential and Parallel Algorithms, Kluwer Academic Publishers, Dordrecht (2000, 2nd ed. 2013, 3rd ed. 2014). doi: 10.1007/978-1-4615-4677-1.

[52]

A. Törn and A. Žilinskas, Global Optimization, Springer, 1989. doi: 10.1007/3-540-50871-6.

[53]

P. Venkataraman, Applied Optimization with MATLAB Programming, John Wiley & Sons, 2009.

[54]

A. A. Zhigljavsky, Theory of Global Random Search, Kluwer Academic Publishers, Dordrecht, 1991. doi: 10.1007/978-94-011-3436-1.

Figure 1.  A Peano curve approximation for the third density level
Figure 2.  The computational scheme for obtaining the value of the reduced one-dimensional function $\varphi(y(x))$
Figure 3.  The architecture of the Globalizer system
Table 1.  Averaged number of executed iterations of the compared global optimization methods
$N$Problem classDIRECTDIRECTlMAGS
4 Simple$>$47282(4)1898311953
Hard $>$95708(7)6875425263
5 Simple $>$16057(1)1675815920
Hard $>$217215(16) $>$269064(4) $>$148342(4)
$N$Problem classDIRECTDIRECTlMAGS
4 Simple$>$47282(4)1898311953
Hard $>$95708(7)6875425263
5 Simple $>$16057(1)1675815920
Hard $>$217215(16) $>$269064(4) $>$148342(4)
Table 2.  Averaged numbers of iterations executed by GPMAGS for solving the test optimization problems
p$N=4$$N=5$
Simple Hard Simple Hard
Serial trial computations1119532526315920 $>$148342 (4)
Parallel computations247621117813378109075
on CPU423725972520351868
813932874377351868
Parallel computations601713933823452
on Xeon Phi120851822491306
2404210397381
p$N=4$$N=5$
Simple Hard Simple Hard
Serial trial computations1119532526315920 $>$148342 (4)
Parallel computations247621117813378109075
on CPU423725972520351868
813932874377351868
Parallel computations601713933823452
on Xeon Phi120851822491306
2404210397381
Table 3.  Speedup of parallel computations executed by GPMAGS
p $N=4$ $N=5$
Simple Hard Simple Hard
Serial trial computations. 1 11953 25263 15920 > 148342 (4)
Average number of iterations
Parallel computations 2 2.52 2.32 1.21 1.41
of CPU. 4 5.05 4.24 3.13 2.92
Speedup 8 8.68 8.88 4.24 6.66
Parallel computations 60 8.18 7.37 9.99 6.66
of Xeon Phi. 120 16.316 15.815 15.215 17.317
Speedup 240 33.133 27.827 38.838 59.359
p $N=4$ $N=5$
Simple Hard Simple Hard
Serial trial computations. 1 11953 25263 15920 > 148342 (4)
Average number of iterations
Parallel computations 2 2.52 2.32 1.21 1.41
of CPU. 4 5.05 4.24 3.13 2.92
Speedup 8 8.68 8.88 4.24 6.66
Parallel computations 60 8.18 7.37 9.99 6.66
of Xeon Phi. 120 16.316 15.815 15.215 17.317
Speedup 240 33.133 27.827 38.838 59.359
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