Taiwanese Journal of Mathematics

MIXED VOLUME COMPUTATION IN PARALLEL

Tianran Chen, Tsung-Lin Lee, and Tien-Yien Li

Full-text: Open access

Abstract

Efficient algorithms for computing mixed volumes, via the computation of mixed cells, have been implemented in DEMiCs [18] and MixedVol-2.0 [13]. While the approaches in those two packages are somewhat different, they follow the same theme and are both highly serial. To fit the need for the parallel computing, a reformulation of the algorithms is inevitable. This article proposes a reformulation of the algorithm for the mixed volume computation rooted from algorithms in graph theory, making it much more fine-grained and scalable. The resulting parallel algorithm can be readily adapted to both distributed and shared memory computing systems. Illustrated by the numerical results on several different architectures, the speedups of our parallel algorithms for the mixed volume computation are remarkable.

Article information

Source
Taiwanese J. Math., Volume 18, Number 1 (2014), 93-114.

Dates
First available in Project Euclid: 10 July 2017

Permanent link to this document
https://projecteuclid.org/euclid.twjm/1499706333

Digital Object Identifier
doi:10.11650/tjm.18.2014.3276

Mathematical Reviews number (MathSciNet)
MR3162115

Zentralblatt MATH identifier
1357.52009

Subjects
Primary: 52A39: Mixed volumes and related topics 65H10: Systems of equations 65H20: Global methods, including homotopy approaches [See also 58C30, 90C30] 90C05: Linear programming

Keywords
mixed volume polyhedral homotopy parallel computing multi-core cluster

Citation

Chen, Tianran; Lee, Tsung-Lin; Li, Tien-Yien. MIXED VOLUME COMPUTATION IN PARALLEL. Taiwanese J. Math. 18 (2014), no. 1, 93--114. doi:10.11650/tjm.18.2014.3276. https://projecteuclid.org/euclid.twjm/1499706333


Export citation

References

  • A. Albouy and A. Chenciner, Le probléme des n corps et les distances mutuelles, Inv. Math., 131 (1998), 151-184.
  • D. P. Anderson, BOINC: A System for Public-Resource Computing and Storage, Proceedings of the 5th IEEE/ACM International Workshop on Grid Computing, 2004, pp. 4-10.
  • G. Björk and R. Fröberg, A faster way to count the solutions of inhomogeneous systems of algebraic equations, J. Symbolic Comput., 12(3) (1991), 329-336.
  • W. Boege, R. Gebauer and H. Kredel, Some examples for solving systems of algebraic equations by calculating Groebner bases, J. Symbolic Comput., 2 (1986), 83-98.
  • W. Cary Huffman and Vera Pless, Fundamentals of Error Correcting Codes, Cambridge University Press, New York, NY, 2003.
  • T. Cormen, C. Leiserson, R. Rivest and C. Stein, Introduction to Algorithms, The MIT Press, Cambridge, MA, 2001.
  • I. Z. Emiris and J. F. Canny, Efficient incremental algorithms for the sparse resultant and the mixed volume, J. Symbolic Comput., 20 (1995), 117-149.
  • J. Frey, T. Tannenbaum, M. Livny, I. Foster and S. Tuecke, Condor-G: A Computation Management Agent for Multi-Institutional Grids, Cluster Computing, 5(3) (2002), 237-246.
  • T. Gao and T. Y. Li, Mixed volume computation for semi-mixed systems, Discrete Comput. Geom., 29(2) (2003), 257-277.
  • T. Gao, T. Y. Li and M. Wu, MixedVol: a software package for mixed volume computation, ACM Trans. on Math. Soft., 31(4) (2005), 555-560.
  • M. Hampton and R. Moeckel, Finiteness of stationary configurations of the four-vortex problem, Trans. of Amer. Math. Soc., 361(3) (2008), 1317-1332.
  • B. Huber and B. Sturmfels, A polyhedral method for solving sparse polynomial systems, Math. Comp., 64 (1995), 1541-1555.
  • T. L. Lee and T. Y. Li, Mixed volume computation in solving polynomial systems, Contemp. Math., 556 (2011), 97-112.
  • T. Y. Li, Numerical solution of multivariate polynomial systems by homotopy continuation methods, ACTA Numerica, (1997), 399-436.
  • T. Y. Li, Solving polynomial systems by polyhedral homotopies, Taiwanese J. Math., 3 (1999), 251-279.
  • T. Y. Li, Solving Polynomial Systems by the Homotopy Continuation Method, Handbook of numerical analysis, Vol. XI, Edited by P. G. Ciarlet, North-Holland, Amsterdam, 2003.
  • T. Y. Li and X. Li, Finding mixed cells in the mixed volume computation, Found. Comput. Math., 1 (2001), 161-181.
  • T. Mizutani, A. Takeda and M. Kojima, Dynamic enumeration of all mixed cells, Discrete Comput. Geom., 37 (2007), 351-367.
  • A. Morgan, Solving Polynomial Systems Using Continuation for Engineering and Scientific Problems, Prentice-Hall, Englewood Cliffs, New Jersey, 1987.
  • V. W. Noonburg, A neural network modeled by an adaptive Lotka-Volterra system, SIAM J. Appl. Math., 49 (1989), 1779-1792.
  • P. Pacheco, Parallel Programming with MPI, Morgan Kaufmann, Burlington, MA, 1996.
  • M. J. Quinn, Parallel Programming in C with MPI and OpenMP, McGraw Hill, New York, NY, 2003.
  • J. Reinders, Intel Threading Building Blocks: Outfitting C++ For Multi-core Processor Parallelism, O'Reilly Media, Sebastopol, CA, 2007.
  • S. Skiena, The Algorithm Design Manual, Springer-Verlag, New York, 1998.
  • M. Snir, S. Otto, S. Hass-Lederman, D. Walker and J. Dongara, ph MPI: The Complete Reference, The MIT Press, Cambridge, MA, 1998.
  • W. R. Stevens, ph TCP/IP Illustrated, Vol. 1, The Protocols, Addison-Wesley, Boston, MA, 1994.
  • D. Thain, T. Tannenbaum and M. Livny, Distributed computing in practice: the condor experience, Concurrency and Computation: Practice and Experience, 17(2-4) (2005), 323-356.