Improvements on low discrepancy one-dimensional sequences and two-dimensional point sets

Monte Carlo and Quasi-Monte Carlo Methods 2006

Volumn , Issue , 2008, Pages 327-341

  Faure, Henri ^a  

^a AIX MARSEILLE UNIVERSITÉ   (France)

Author keywords

[No Author keywords available]

Indexed keywords

GEOMETRY;

LOWER BOUNDS; POINT SET; SCRAMBLED SEQUENCE; STAR-DISCREPANCY;

MONTE CARLO METHODS;

EID: 70349938038     PISSN: None     EISSN: None     Source Type: Conference Proceeding    
DOI: 10.1007/978-3-540-74496-2_19     Document Type: Conference Paper

References (26)

1. [Bej78] R. Béjian. Sur certaines suites présentant une faible discrépance à l'origine. C. R. Acad. Sc.,Paris, 286A, 135-138 (1978).
2. [Bej82] R. Béjian. Minoration de la discrépance d'une suite quelconque sur T. Acta Arith., 41, 185-202 (1982).
3. [CF93] H. Chaix and H. Faure. Discrépance et diaphonie en dimension un. Acta Arith., 63, 103-141 (1993).
4. [DeC86] L. De Clerck. A method for the exact calculation of the star-discrepancy of plane sets applied to the sequences of Hammersley. Mh. Math., 101, 261-278 (1986).
5. [DLP05] M. Drmota, G. Larcher, and F. Pillichshammer. Precise distribution properties of the van der Corput sequence and related sequences. Manuscripta Math. 118, 11-41 (2005).
6. [Fau81] H. Faure. Discrépance de suites associées à un système de numération (en dimension un). Bull. Soc. Math. France, 109, 143-182 (1981).
7. [Fau86] H. Faure. On the star-discrepancy of generalized Hammersley sequences in two dimensions. Mh. Math., 101, 291-300 (1986).
8. [Fau92] H. Faure. Good permutations for extreme discrepancy. J. Number Theory, 41, 47-56 (1992).
9. [Fau05a] H. Faure. Discrepancy and diaphony of digital (0,1)-sequences in prime base. Acta Arith. 117.2, 125-148 (2005).
10. [Fau05b] H. Faure. Irregularities of distribution of digital (0, 1)-sequences in prime base. Integers, Electronic Journal of Combinatorial Number Theory 5(3) A07, 1-12 (2005).
11. [Fau06] H. Faure. Selection criteria for (random) generation of digital (0, s)-sequences. In: Niederreiter, H., Talay, D. (Eds.) Monte Carlo and Quasi-Monte Carlo Methods 2004. Springer, Berlin Heidelberg New York, 113-126 (2006).
12. [Fau07] H. Faure. Van der Corput sequences towards general (0, 1)-sequences in base b. J. Théor. Nombres Bordeaux, to appear (2007).
13. [HZ69] J.H. Halton and S.K. Zaremba. The extreme and L2-discrepancies of some plane sets. Mh. Math., 73, 316-328 (1969).
14. [Kri05] P. Kritzer. A new upper bound on the star discrepancy of (0, 1)-sequences. Integers, Electronic Journal of Combinatorial Number Theory 5(3) A11, 1-9 (2005).
15. [Kri06] P. Kritzer. On some remarkable properties of the two-dimensional Hammersley point set in base 2. J. Théor. Nombres Bordeaux 18, 203-221 (2006).
16. [KLP06] P. Kritzer, G. Larcher, and F. Pillichshammer. A thorough analysis of the discrepancy of shifted Hammersley and van der Corput point sets. Ann. Math. Pura Appl., to appear (2006).
17. [LP01] G. Larcher and F. Pillichshammer. Walsh series analysis of the L2-discrepancy of symmetrisized points sets. Mh. Math., 132, 1-18 (2001).
18. [LP03] G. Larcher and F. Pillichshammer. Sums of distances to the nearest integer and the discrepancy of digital nets. Acta Arith., 106.4, 379-408 (2003).
19. [Mat98] J. Matoušek. On the L2-discrepancy for anchored boxes. J. Complexity, 14, 527-556 (1998).
20. [Nie92] H. Niederreiter. Random Number Generation and Quasi-Monte Carlo Methods. CBMS-NSF Regional Conference Series in Applied Mathematics. SIAM, Philadelphia (1992).
21. [NX96] H. Niederreiter and C.P. Xing. Quasirandom points and global functions fields. In: Cohen, S. and Niederreiter, H. (eds) Finite Fields and Applications. London Math. Soc. Lectures Notes Series, 233, 269-296 (1996).
22. [Owe95] A.B. Owen. Randomly permuted (t, m, s)-nets and (t, s)-sequences. In: Niederreiter, H., Shiue, P. (eds.) Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing. Springer, Lectures Notes in Statistics, 126, 299-317 (1995).
23. [Pil04] F. Pillichshammer. On the discrepancy of (0,1)-sequences. J. Number Theory, 104, 301-314 (2004).
24. [Tez94] S. Tezuka. A generalization of Faure sequences and its efficient implementation. Research Report IBM RT0105 (1994).
25. [Tez95] S. Tezuka. Uniform Random Numbers: Theory and Practice. Kluwer Academic Publishers, Boston (1995).
26. [Whi75] B.E. White. Mean-square discrepancies of the Hammersley and Zaremba sequences for arbitrary radix. Mh. Math., 80, 219-229 (1975).