Implementation and tests of low-discrepancy sequences

Abstract

Low-discrepancy sequences are used for numerical integration, in simulation, and in related applications. Techniques for producing such sequences have been proposed by, among others, Halton, Sobol´, Faure, and Niederreiter. Niederreiter's sequences have the best theoretical asymptotic properties. The paper describes two ways to implement the latter sequences on a computer and discusses the results obtained in various practical tests on particular integrals.

References

1. 1 AMERICAN NATIONAL STANDARDS INSTITUTE, INC. American National Standard Programming Language FORTRAN. Standard X3.9-1978, New York, 1978.
2. 2 ANTONOV, I. A., AND SALEEV, V.M. An economic method of computing LP~-sequences. USSR Comput. Math. Math. Phys. 19, I (1979), 252-256.
3. 3 BRATLEY, P., AND FOX, B.L. Algorithm 659: Implementing Sobol's quasi-random sequence generator. ACM Trans. Math. Softw. 14 (1988), 88-100.
4. 4 DAVENPORT, T. R., AND CHENC, R. C.H. Implementation of quasi-random generators and their use in discrete event simulation. In Proceedings of the 1989 Winter Simulation Conference (Washington, DC, Dec. 4 6, 1989), E. A. MacNair, K. J. Musselman, and P. Heidelberger, Eds., pp. 419-427.
5. 5 FAURE, H. Discr~pance de suites associ~es ~ un syst~me de num6ration (en dimension s). Acta Arith. 41 (1982), 337-351.
6. 6 Fox, B.L. Algorithm 647: Implementation and relative efficiency of quasirandom sequence generators. ACM Trans. Math. Softw. 12 (1986), 362-376.
7. 7 HALTON, J. $. On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals. Numer. Math. 2 (1960), 84 90; and Correction ibid 2 (1960), 196.
8. 8 HANSEN, T., MULLEN, G. L., AND NIEDERREITER, H. Good parameters for a class of node sets in quasi-Monte Carlo integration. To appear, Math. Comput.
9. 9 HUA, L. K., ANn WANG, Y. Applications of Number Theory to Numerical Analysis. Springer, Berlin, 1981.
10. 10 LIDL, R., AND NIEDERREITER, H. Fintte Fields. Addison-Wesley, Reading, Mass., 1983. (Now distributed by Cambridge University Press, Cambridge, 1984.)
11. 11 NIEDERREITER, H Quasi-Monte Carlo methods and pseudo-random numbers. Bull. Am. Math. Soc. 84 (1978), 957-1041.
12. 12 NIEDERREITER, $. Point sets and sequences with small discrepancy. Monatsh. Math. 104 (1987), 273-337.
13. 13 NIEDERREITER, H,, Low-discrepancy and low-dispersion sequences. J. Number Theor. 30 (1988), 51 70.
14. 14 RIESEL, H. Prime Numbers and Computer Methods for Factorization. Birkh~iuser, Boston, 1985.
15. 15 SOBOL', I. M. The distribution of points in a cube and the approximate evaluation of integrals. USSR Comput. Math. Math. Phys. 7, 4 (1967), 86-112.

Index Terms

• Computing methodologies

  • Modeling and simulation

    • Simulation types and techniques

• Mathematics of computing

  • Mathematical analysis

    • Numerical analysis

      • Numerical differentiation

    • Quadrature

  • Probability and statistics

    • Probabilistic reasoning algorithms

      • Markov-chain Monte Carlo methods

      • Sequential Monte Carlo methods