Error trends in Quasi-Monte Carlo integration

Computer Physics Communications

Volumn 159, Issue 2, 2004, Pages 93-105

  Schlier, Ch ^a  

^a UNIVERSITY OF FREIBURG   (Germany)

Author keywords

Low discrepancy sequences; Numerical integration; Quasi Monte Carlo methods

Indexed keywords

APPROXIMATION THEORY; COMPUTATIONAL METHODS; ERROR ANALYSIS; MONTE CARLO METHODS; PARAMETER ESTIMATION; PROBABILITY; PROBLEM SOLVING; THEOREM PROVING;

ASYMPTOTIC DISCREPANCIES; LOW-DISCREPANCY SEQUENCES; NUMERICAL INTEGRATION; QUASI-MONTE CARLO METHODS;

INTEGRATION;

EID: 2042548005     PISSN: 00104655     EISSN: None     Source Type: Journal    
DOI: 10.1016/j.cpc.2004.02.004     Document Type: Article

References (37)

1. Berblinger M., Schlier Ch. Monte Carlo integration with quasi-random numbers: some experience. Comp. Phys. Comm. 66:1991;157-166.
2. Berblinger M., Schlier Ch. Monte Carlo integration with quasi-random numbers: experience with discontinuous integrands. Comp. Phys. Comm. 99:1997;151-162.
3. Bratley P., Fox B.L., Niederreiter H. Algorithm 748: programs to generate Niederreiter's low-discrepancy sequences. ACM TOMS. 20:1994;494-495.
4. Braaten E., Weller G. An improved low-discrepancy sequence for multidimensional quasi Monte Carlo integration. J. Comp. Phys. 33:1979;249-258.
5. Caflisch R.E. Monte Carlo and Quasi-Monte Carlo methods. Acta Numer. 7:1998;1-49.
6. Caflisch R.E., Morokoff W., Owen A. Valuation of mortgage-backed securities using Brownian bridges to reduce effective dimension. J. Comput. Finance. 1:1997;27-46.
7. Cranley R., Patterson T.N.L. Randomization of number theoretic methods for numeric integration. SIAM J. Numer. Anal. 13:1976;904-914.
8. Faure H., Tezuka S. Another random scrambling of digital (t, s)-sequences. Fang K.-T., Hickernell F.J., Niederreiter H. Monte Carlo and Quasi-Monte Carlo Methods 2000. 2002;Springer, Berlin.
9. Genz A. Testing multidimensional integration routines. Ford B., Rault J.C., Thomasset F. Tools, Methods and Languages for Scientific and Engineering Computation. 1984;81-94 Elsevier, Amsterdam.
10. Halton J.H. On the efficiency of certain quasi-random sequences of points in evaluating multidimensional integrals. Numer. Math. 2:1960;84-90. and 196.
11. Halton J.H., Smith G.B. Radical inverse quasi-random point sequence, Algorithm 247. Comm. ACM. 7:1964;701.
12. Hong H.S., Hickernell F.J. Implementing scrambled digital sequences. ACM Trans. Math. Software. 29:2003;95-109.
13. Keister B.D. Multidimensional integration algorithms. Comput. Phys. 10:1996;119-122.
14. Keller A. Quasi-Monte Carlo Radiosity. Pueyo X., Schröder P. Rendering Techniques '96. 1996;101-110 Springer, Berlin.
15. Kocics L., Whiten W.J. Computational investigations of low-discrepancy sequences. ACM TOMS. 23:1997;266-294.
16. Lemieux C., L'Ecuyer P. On the use of Quasi-Monte Carlo methods in computational finance. Alexandrow V.N., , et al. ICCS 2001. Lecture Notes in Computer Science. vol. 2073:2001;607-616 Springer, Berlin.
17. An excellent review is L'Ecuyer P., Lemieux C. Recent advances in randomized Quasi-Monte Carlo methods. Dror M., L'Ecuyer P., Szidarovszki F. Modeling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications. 2001;607-616 Kluwer, Amsterdam.
18. Morokoff W.J., Caflisch R.E. Quasi-Monte Carlo integration. J. Comp. Phys. 122:1995;218-230.
19. Niederreiter H. Low-discrepancy and low-dispersion sequences. J. Number Theor. 30:1988;51-70.
20. Niederreiter H. Random Number Generation and Quasi-Monte Carlo Methods. 1992;SIAM, Philadelphia.
21. Niederreiter H., Xing C. The algebraic-geometry approach to low-discrepancy sequences. Niederreiter H., Hellekalek P., Larcher G., Zinterhof P. Monte Carlo and Quasi-Monte Carlo Methods 1996. Lecture Notes in Statistics. vol. 127:1998;139-160 Springer-Verlag, New York.
22. Owen A.B. Randomly permuted (t, m, s)-nets and (t, s) sequences. Niederreiter H., Shiue P.J.-S. Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing. Lecture Notes in Statistics. vol. 106:1995;299-315 Springer, Berlin.
23. Owen A.B. Monte Carlo variance of scrambled net quadrature. SIAM J. Numer. Anal. 34:1997;1884-1910.
24. Owen A.B. Scrambling Sobol' and Niederreiter points. J. Complexity. 14:1998;466-489.
25. Owen A.B. The dimension distribution and quadrature test functions. Stat. Sinica. 13:2003;1-17.
26. Papageorgiou A. Fast convergence of Quasi-Monte Carlo for a class of isotropic integrals. Math. Comput. 70:2000;297-306.
27. Papageorgiou A. Sufficient conditions for fast Quasi-Monte Carlo convergence. J. Complexity. 19:2003;332-351.
28. Papageorgiou A., Traub J.F. Faster evaluation of multidimensional integrals. Comput. Phys. 11:1997;574-579.
29. Press W.H., Teukolsky S.A., Vetterling W.T., Flannery B.P. Numerical Recipes in Fortran. second ed. 1992;Cambridge UP, Cambridge. Chapter 7.7.
30. Pirsic G. A software implementation of Niederreiter-Xing sequences. Fang K.-T., Hickernell F.J., Niederreiter H. Monte Carlo and Quasi-Monte Carlo Methods 2000. 2002;124-140 Springer, Berlin.
31. Roos P., Arnold L. Numerische Experimente zur mehrdimensionalen Quadratur. Österreich. Acad. Wiss. Math.-Natur. Kl. S. B. 172:1963;271-286.
32. Radovic I., Sobol' I.M., Tichy R.F. Quasi-Monte Carlo methods for numerical integration: Comparison of different low discrepancy sequences. Monte Carlo Methods Appl. 2:1996;1-14.
33. Schlier Ch. Discrepancy behaviour in the non-asymptotic regime. Appl. Numer. Math. 2004;. in press.
34. Sobol' I.M. On the distribution of points in a cube and the approximate evaluation of integrals. USSR Comput. Maths. Math. Phys. 7:1967;86-112.
35. Tezuka S. Uniform Random Numbers: Theory and Practice. 1995;Kluwer, Boston.
36. Wang X., Hickernell F.J. Randomized Halton sequences. Mathematical and Computer Modelling. 32:2000;887-899.
37. Wang X., Fang K.-T. The effective dimension and quasi-Monte Carlo integration. J. Complexity. 19:2003;101-124.