Polynomial Integration Lattices

  • Pierre L’Ecuyer
Conference paper

Summary

Lattice rules are quasi-Monte Carlo methods for estimating largedimensional integrals over the unit hypercube. In this paper, after briefly reviewing key ideas of quasi-Monte Carlo methods, we give an overview of recent results, generalize some of them, and provide new results, for lattice rules defined in spaces of polynomials and of formal series with coefficients in the finite ring ℤb. Some of the results are proved only for the case where b is a prime (so ℤb, is a finite field). We discuss basic properties, implementations, a randomized version, and quality criteria (i.e., measures of uniformity) for selecting the parameters. Two types of polynomial lattice rules are examined: dimensionwise lattices and resolutionwise lattices. These rules turn out to be special cases of digital net constructions, which we reinterpret as yet another type of lattice in a space of formal series. Our development underlines the connections between integration lattices and digital nets.

Keywords

Formal Series Linear Feedback Shift Register Integration Lattice Dual Lattice Lattice Rule 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.
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References

  1. 1.
    N. S. Bakhvalov. On the rate of convergence of indeterministic integration processes within the functional classes w p(l). Theory of Probability and its Applications, 7:227, 1962.CrossRefGoogle Scholar
  2. 2.
    R. Couture and P. L’Ecuyer. Lattice computations for random numbers. Mathematics of Computation, 69(230):757–765, 2000.MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    R. Couture, P. L’E cuyer, and S. Tezuka. On the distribution of K-dimensional vectors for simple and combined Tausworthe sequences. Mathematics of Computation, 60(202):749–761, S11-S16, 1993.Google Scholar
  4. 4.
    R. Cranley and T. N. L. Patterson. Randomization of number theoretic methods for multiple integration. SIAM Journal on Numerical Analysis, 13(6):904–914, 1976.MathSciNetMATHCrossRefGoogle Scholar
  5. 5.
    B. L. Fox. Strategies for Quasi-Monte Carlo. Kluwer Academic, Boston, MA, 1999.CrossRefGoogle Scholar
  6. 6.
    M. Fushimi. Increasing the orders of equidistribution of the leading bits of the Tausworthe sequence. Information Processing Letters, 16:189–192, 1983.MathSciNetMATHCrossRefGoogle Scholar
  7. 7.
    P. Hellekalek. On the assessment of random and quasi-random point sets. In P. Hellekalek and G. Larcher, editors, Random and Quasi-Random Point Sets, volume 138 of Lecture Notes in Statistics, pages 49-108. Springer, New York, 1998.Google Scholar
  8. 8.
    F. J. Hickernell. A generalized discrepancy and quadrature error bound. Mathematics of Computation, 67:299–322, 1998.MathSciNetMATHCrossRefGoogle Scholar
  9. 9.
    F. J. Hickernell, H. S. Hong, P. L’E cuyer, and C. Lemieux. Extensible lattice sequences for quasi-Monte Carlo quadrature. SIAM Journal on Scientific Computing, 22(3):1117–1138, 2001.MathSciNetCrossRefGoogle Scholar
  10. 10.
    W. Hoeffding. A class of statistics with asymptotically normal distributions. Annals of Mathematical Statistics, 19:293–325, 1948.MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    S. Joe and I. H. Sloan. Embedded lattice rules for multidimensional integration. SIAM Journal on Numerical Analysis, 29:1119–1135, 1992.MathSciNetMATHCrossRefGoogle Scholar
  12. 12.
    N. M. Korobov. The approximate computation of multiple integrals. Dokl. Akad. Nauk SSSR, 124:1207–1210, 1959. In Russian.MathSciNetMATHGoogle Scholar
  13. 13.
    G. Larcher. Digital point sets: Analysis and application. In P. Hellekalek and G. Larcher, editors, Random and Quasi-Random Point Sets, volume 138 of Lecture Notes in Statistics, pages 167-222. Springer, New York, 1998.Google Scholar
  14. 14.
    G. Larcher, H. Niederreiter, and W. Ch. Schmid. Digital nets and sequences constructed over finite rings and their application to quasi-Monte Carlo integration. Monatshefte für Mathematik, 121(3):231–253, 1996.MathSciNetMATHCrossRefGoogle Scholar
  15. 15.
    P. L’Ecuyer. Maximally equidistributed combined Tausworthe generators. Mathematics of Computation, 65(213):203–213, 1996.MathSciNetMATHCrossRefGoogle Scholar
  16. 16.
    P. L’E cuyer. Tables of maximally equidistributed combined LFSR generators. Mathematics of Computation, 68(225):261–269, 1999.MathSciNetMATHCrossRefGoogle Scholar
  17. 17.
    P. L’E cuyer and C. Lemieux. Variance reduction via lattice rules. Management Science, 46(9):1214–1235, 2000.MATHCrossRefGoogle Scholar
  18. 18.
    P. L’E cuyer and C. Lemieux. Recent advances in randomized quasi-Monte Carlo methods. In M. Dror, P. L’Ecuyer, and F. Szidarovszki, editors, Modeling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications, pages 419-474. Kluwer Academic Publishers, Boston, 2002.Google Scholar
  19. 19.
    P. L’E cuyer and F. Panneton. Construction of equidistributed generators based on linear recurrences modulo 2. In K.-T. Fang, F. J. Hickernell, and H. Niederreiter, editors, Monte Carlo and Quasi-Monte Carlo Methods 2000, pages 318-330. Springer-Verlag, Berlin, 2002.Google Scholar
  20. 20.
    C. Lemieux and P. L’Ecuyer. Selection criteria for lattice rules and other lowdiscrepancy point sets. Mathematics and Computers in Simulation, 55(1-3):139–148, 2001.MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    C. Lemieux and P. L’Ecuyer. Randomized polynomial lattice rules for multivariate integration and simulation. SIAM Journal on Scientific Computing, 24(5):1768–1789, 2003.MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    A. K. Lenstra. Factoring multivariate polynomials over finite fields. Journal of Computer and System Sciences, 30:235–248, 1985.MathSciNetMATHCrossRefGoogle Scholar
  23. 23.
    K. Mahler. An analogue to Minkowski’s geometry of numbers in a field of series. Annals of Mathematics, 42(2):488–522, 1941.Google Scholar
  24. 24.
    K. Mahler. On a theorem in the geometry of numbers in a space of Laurent series. Journal of Number Theory, 17:403–416, 1983.MathSciNetMATHCrossRefGoogle Scholar
  25. 25.
    H. Niederreiter. Low-discrepancy point sets. Monatshefte für Mathematik, 102:155–167, 1986.MathSciNetMATHCrossRefGoogle Scholar
  26. 26.
    H. Niederreiter. Low-discrepancy point sets obtained by digital constructions over finite fields. Czechoslovak Math. Journal, 42:143–166, 1992.MathSciNetMATHGoogle Scholar
  27. 27.
    H. Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods, volume 63 of SIAM CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia, 1992.Google Scholar
  28. 28.
    H. Niederreiter. The existence of good extensible polynomial lattice rules. Monatshefte für Mathematik, 139:295–307, 2003.MathSciNetMATHCrossRefGoogle Scholar
  29. 29.
    H. Niederreiter and G. Pirsic. Duality for digital nets and its applications. Acta Arithmetica, 97:173–182, 2001.MathSciNetMATHCrossRefGoogle Scholar
  30. 30.
    A. B. Owen. Latin supercube sampling for very high-dimensional simulations. ACM Transactions of Modeling and Computer Simulation, 8(l):71–102, 1998.MATHCrossRefGoogle Scholar
  31. 31.
    I. H. Sloan and S. Joe. Lattice Methods for Multiple Integration. Clarendon Press, Oxford, 1994.MATHGoogle Scholar
  32. 32.
    S. Tezuka. A new family of low-discrepancy point sets. Technical Report RT0031, IBM Research, Tokyo Research Laboratory, Jan. 1990.Google Scholar
  33. 33.
    S. Tezuka. The k-dimensional distribution of combined GFSR sequences. Mathematics of Computation, 62(206):809–817, 1994.MathSciNetMATHGoogle Scholar
  34. 34.
    S. Tezuka. Uniform Random Numbers: Theory and Practice. Kluwer Academic Publishers, Norwell, Mass., 1995.MATHCrossRefGoogle Scholar
  35. 35.
    S. Tezuka and P. L’Ecuyer. Efficient and portable combined Tausworthe random number generators. ACM Transactions on Modeling and Computer Simulation, 1(2):99–112, 1991.MATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag Berlin Heidelberg 2004

Authors and Affiliations

  • Pierre L’Ecuyer
    • 1
  1. 1.Département d’informatique et de recherche opérationnelleUniversité de MontréalMontréalCanada

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