Optimal polynomials for (t,m,s)-nets and numerical integration of multivariate Walsh series

SIAM Journal on Numerical Analysis

Volumn 33, Issue 6, 1996, Pages 2239-2253

  Larcher, G ^a   Lauss, A ^a   Niederreiter, H ^b   Schmid, W Ch ^a  

^a UNIVERSITY OF SALZBURG   (Austria)

^b ERICH SCHMID INSTITUTE OF MATERIALS SCIENCE   (Austria)

Author keywords

Low discrepancy point sets; Quasi Monte Carlo integration; Walsh series

Indexed keywords

EID: 0001714473     PISSN: 00361429     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036142994264705     Document Type: Article

References (23)

1. P. BRATLEY, B. L. FOX, AND H. NIEDERREITER, Implementation and tests of low-discrepancy sequences, ACM Trans. Mod. Comp. Sim., 2 (1992), pp. 195-213.
2. H. FAURE, Discrépance de suites associées à un système de numération (en dimension s), Acta Arith., 41 (1982), pp. 337-351.
3. T. HANSEN, G. L. MULLEN, AND H. NIEDERREITER, Good parameters for a class of node sets in quasi-Monte Carlo integration, Math. Comp., 61 (1993), pp. 225-234.
4. L. K. HUA AND Y. WANG, Applications of Number Theory to Numerical Analysis, Springer-Verlag, Berlin, 1981.
5. N. M. KOROBOV, Properties and calculation of optimal coefficients, Dokl. Akad. Nauk SSSR, 132 (1960), pp. 1009-1012. (In Russian.)
6. G. LARCHER, Optimale Koeffizienten bezüglich zusammengesetzter Zahlen, Monatsh. Math., 100 (1985), pp. 127-135.
7. _, Nets obtained from rational functions over finite fields, Acta Arith., 63 (1993), pp. 1-13.
8. G. LARCHER AND H. NIEDERREITER, Optimal coefficients modulo prime powers in the three-dimensional case, Ann. Mat. Pura Appl., 155 (1989), pp. 299-315.
9. G. LARCHER, W. CH. SCHMID, AND R. WOLF, Representation of functions as Walsh series to different bases and an application to the numerical integration of high-dimensional Walsh series, Math. Comp., 63 (1994), pp. 701-716.
10. G. LARCHER AND C. TRAUNFELLNER, On the numerical integration of Walsh series by number-theoretic methods, Math. Comp., 63 (1994), pp. 277-291.
11. G. L. MULLEN AND G. WHITTLE, Point sets with uniformity properties and orthogonal hypercubes, Monatsh. Math., 113 (1992), pp. 265-273.
12. K. NIEDERDRENK, Die endliche Fourier- und Walsh-Transformation mit einer Einführung in die Bildverarbeitung, Vieweg, Braunschweig, 1982.
13. H. NIEDERREITER, Quasi-Monte Carlo methods and pseudo-random numbers, Bull. Amer. Math. Soc., 84 (1978), pp. 957-1041.
14. _, Point sets and sequences with small discrepancy, Monatsh. Math., 104 (1987), pp. 273-337.
15. _, Low-discrepancy and low-dispersion sequences, J. Number Theory, 30 (1988), pp. 51-70.
16. _, Constructions of low-discrepancy point sets and sequences, in Sets, Graphs and Numbers (Budapest, 1991), Colloquia Math. Soc. János Bolyai, Vol. 60, North-Holland, Amsterdam, 1992, pp. 529-559.
17. _, Low-discrepancy point sets obtained by digital constructions over finite fields, Czechoslovak Math. J., 42 (1992), pp. 143-166.
18. _, Orthogonal arrays and other combinatorial aspects in the theory of uniform point distributions in unit cubes, Discrete Math., 106/107 (1992), pp. 361-367.
19. _, Random Number Generation and Quasi-Monte Carlo Methods, Number 63 in CBMS-NSF Series in Applied Mathematics, Society for Industrial and Applied Mathematics, Philadelphia, PA, 1992.
20. _, Finite fields, pseudorandom numbers, and quasirandom points, in Proc. Internat. Conf. on Finite Fields, Coding Theory, and Advances in Comm. and Comp. (Las Vegas, NV, 1991), Lecture Notes in Pure and Appl. Math., Vol. 141, Dekker, New York, 1993, pp. 375-394.
21. W. CH. SCHMID, Zur numerischen Integration von Walshreihen, Diplomarbeit, Institut für Mathematik, Universität Salzburg, Salzburg, Austria, 1993.
22. I. M. SOBOL', The distribution of points in a cube and the approximate evaluation of integrals, Ž. Vyčisl. Mat. i Mat Fiz., 7 (1967), pp. 784-802. (In Russian.)
23. P. ZINTERHOF AND P. ZINTERHOF JR, Hyperbolic Filtering of Walsh Series; A New Method for Data Compression, Technical report 03, Research Institute for Software Technology, University of Salzburg, Salzburg, Austria, 1994.