Monte Carlo and Quasi-Monte Carlo Methods 2008 pp 561-572 | Cite as
Recent Progress in Improvement of Extreme Discrepancy and Star Discrepancy of One-Dimensional Sequences
Abstract
In this communication, we report on recent progress in improvement of extreme discrepancy and star discrepancy of one-dimensional sequences. Namely, we present a permutation of “Babylonian” sequences in base 60, which improves the best known results for star discrepancy obtained by Henri Faure in 1981 [Bull. Soc. Math. France, 109, 143–182 (1981)], and a permutation of sequences in base 84, which improves the best known results for extreme discrepancy obtained by Henri Faure in 1992 [J. Numb. Theory, 42, 47–56 (1992)]. Our best result for star discrepancy in base 60 is 32209/(35400log 60)≈0.222223 (Faure’s best result in base 12 is 1919/(3454log 12)≈0.223585); our best result for extreme discrepancy in base 84 is 130/(83log 84)≈0.353494 (Faure’s best result in base 36 is 23/(35log 6)≈0.366758).
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References
- 1.Béjian, R.: Minoration de la discrépance d’une suite quelconque sur T. Acta Arith. 41, 185–202 (1982) MATHMathSciNetGoogle Scholar
- 2.Béjian, R., Faure, H.: Discrépance de la suite de van der Corput. C. R. Acad. Sci. Paris, Série A 285, 313–316 (1977) MATHGoogle Scholar
- 3.Borel, J.P.: Self-similar measures and sequences. J. of Numb. Theory 31(2), 208–241 (1989) MATHCrossRefMathSciNetGoogle Scholar
- 4.Braaten, E., Weller, W.: An improved low-discrepancy sequence for multidimensional quasi-Monte Carlo integration. J. Comput. Phys. 33, 249–258 (1979) MATHCrossRefGoogle Scholar
- 5.van der Corput, J.G.: Verteilungsfunktionen I. Akademie van Wetenschappen 38, 813–821 (1935) MATHGoogle Scholar
- 6.Faure, H.: Discrépance des suites associées à un système de numération (en dimension un). Bull. Soc. Math. France 109, 143–182 (1981) MATHMathSciNetGoogle Scholar
- 7.Faure, H.: Good permutations for extreme discrepancy. J. Numb. Theory 42, 47–56 (1992) MATHCrossRefMathSciNetGoogle Scholar
- 8.Lapeyre, B., Pagès, G.: Familles de suites discrépance faible obtenues par itération de transformations de [0,1]. Note aux C.R.A.S., Série I 17, 507–509 (1989) Google Scholar
- 9.Niederreiter, H.: Random Number Generation and Quasi-Monte Carlo Methods, SIAM CBMS-NSF Regional Conference Series in Applied Mathematics, vol. 63. SIAM, Philadelphia, PA (1992) MATHGoogle Scholar
- 10.Schmidt, W.M.: Irregularities of distribution. VII. Acta Arith. 21, 45–50 (1972) MATHMathSciNetGoogle Scholar
- 11.Thomas, A.: Discrépance en dimension un. Ann. Fac. Sci. Toulouse, Série Math. 10(3), 369–399 (1989) MATHGoogle Scholar