Halton sequences avoid the origin

SIAM Review

Volumn 48, Issue 3, 2006, Pages 487-503

  Owen, Art B ^a  

^a STANFORD UNIVERSITY   (United States)

Author keywords

Discrepancy; Integrable singularities; Quasi Monte Carlo; Randomized quasi Monte Carlo

Indexed keywords

CONVERGENCE OF NUMERICAL METHODS; INTEGRATION; RANDOM PROCESSES;

CONVERGENCE RATES; HALTON SEQUENCES; INTEGRABLE SINGULARITIES; QUASI-MONTE CARLO (QMC) INTEGRATION; RANDOMIZED QUASI-MONTE CARLO;

MONTE CARLO METHODS;

EID: 33748364105     PISSN: 00361445     EISSN: None     Source Type: Journal    
DOI: 10.1137/S0036144504441573     Document Type: Article

References (27)

1. K.-L. Chung (1974), A Course in Probability Theory, 2nd ed., Academic Press, New York.
2. R. Cranley and T. N. L. Patterson (1976), Randomization of number theoretic methods for multiple integration, SIAM J. Numer. Anal., 13, pp. 904-914.
3. P. J. Davis and P. Rabinowitz (1984), Methods of Numerical Integration, 2nd ed., Academic Press, San Diego.
4. E. de Doncker and Y. Guan (2003), Error bounds for the integration of singular functions using equidistributed sequences, J. Complexity, 19, pp. 259-271.
5. A. Genz and J. Monahan (1998), Stochastic integration rules for infinite regions, SIAM J. Sci. Comput., 19, pp. 426-439.
6. P. Glasserman (2004), Monte Carlo Methods in Financial Engineering, Springer-Verlag, New York.
7. J. Halton (1960), On the efficiency of certain quasi-random sequences of points in evaluating multi-dimensional integrals, Numer. Math., 2, pp. 84-90.
8. J. Hartinger, R. F. Kainhofer, and R. F. Tichy (2004), Quasi-Monte Carlo algorithms for unbounded, weighted integration problems, J. Complexity, 20, pp. 654-668.
9. J. Hartinger, R. F. Kainhofer, and V. Ziegler (2004), On the Corner Avoidance Properties of Various Low-Discrepancy Sequences, Tech. report, Graz University of Technology, Department of Mathematics, Graz, Austria.
10. F. J. Hickernell (1996), The mean square discrepancy of randomized nets, ACM Trans. Model. Comput. Simul., 6, pp. 274-296.
11. s, Math. Comp., 74, p. 1.
12. E. Hlawka and R. Mück (1972), Über eine Transformation von gleichverteilten Folgen. II, Computing, 9, pp. 127-138.
13. B. Klinger (1997), Numerical integration of singular integrands using low-discrepancy sequences, Computing, 59, pp. 223-236.
14. P. L'Ecuyer and C. Lemieux (2002), A survey of randomized quasi-Monte Carlo methods, in Modeling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications, M. Dror, P. L'Ecuyer, and F. Szidarovszki, eds., Kluwer Academic, Boston, pp. 419-474.
15. d, Math. Comp., 73, pp. 827-841.
16. J. Matoušek (1998a), Geometric Discrepancy: An Illustrated Guide, Springer-Verlag, Heidelberg.
17. 2-discrepancy for anchored boxes, J. Complexity, 14, pp. 527-556.
18. H. Niederreiter (1992), Random Number Generation and Quasi-Monte Carlo Methods, SIAM, Philadelphia.
19. A. B. Owen (1995), Randomly permuted (t,m,s)-nets and (t,s)-sequences, in Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, H. Niederreiter and P. J.-S. Shiue, eds., Springer-Verlag, New York, pp. 299-317.
20. A. B. Owen (1997), Scrambled net variance for integrals of smooth functions, Ann. Statist., 25, pp. 1541-1562.
21. A. B. Owen (2005), Multidimensional variation for quasi-Monte Carlo, in International Conference on Statistics in honour of Professor Kai-Tai Fang's 65th birthday, J. Fan and G. Li, eds.
22. A. B. Owen (2006), Randomized QMC and point singularities, in Monte Carlo and Quasi-Monte Carlo Methods 2004, H. Niederreiter and D. Talay, eds., Springer-Verlag, New York.
23. A. Papageorgiou (2003), Sufficient conditions for fast quasi-Monte Carlo convergence, J. Complexity, 19, pp. 332-251.
24. J. K. Patel and C. B. Read (1996), Handbook of the Normal Distribution, 2nd ed., Marcel Dekker, New York.
25. I. H. Sloan and S. Joe (1994), Lattice Methods for Multiple Integration, Oxford Science Publications, Oxford.
26. I. M. Sobol' (1973a), Calculation of improper integrals using uniformly distributed sequences, Soviet Math. Dokl., 14, pp. 734-738.
27. I. M. Sobol' (1973b), On the use of uniformly distributed sequences for approximate computation of improper integrals, in Theory of Cubature Formulas and Applications to Certain Problems in Mathematical Physics, Nauka, Novosibirsk, pp. 62-66 (in Russian).