Optimal quadrature for haar wavelet spaces

Mathematics of Computation

Volumn 73, Issue 245, 2004, Pages 259-277

  Heinrich, Stefan ^a   Hickernell, Fred J ^b   Yue, Rong Xian ^c  

^a Rheinland Pfälzische Technische Universität Kaiserslautern Landau   (Germany)

^b HONG KONG BAPTIST UNIVERSITY   (Hong Kong)

^c SHANGHAI NORMAL UNIVERSITY   (China)

Author keywords

High dimensional integration; Lower bounds; Monte Carlo methods; Quasi Monte Carlo methods

Indexed keywords

EID: 0942278836     PISSN: 00255718     EISSN: None     Source Type: Journal    
DOI: 10.1090/S0025-5718-03-01531-X     Document Type: Article

References (29)

1. [Ent97] K. Entacher, Quasi-Monte Carlo methods for numerical integration of multivariate Haar series, BIT 37 (1997), 846-861. MR 98m:65011
2. [Ent98] K. Entacher, Quasi-Monte Carlo methods for numerical integration of multivariate Haar series II, BIT 38 (1998), 283-292. MR 99h:65010
3. [Hei93] S. Heinrich, Random approximation in numerical analysis, Functional analysis. Proceedings of the Essen Conference, held in Essen, Germany, November 24-30, 1991. (K. D. Bierstedt, A. Pietsch, W. M. Ruess, and D. Vogt, eds.), Lecture Notes in Pure and Applied Mathematics, vol. 150, Marcel Dekker, New York, 1993, pp. 123-171. MR 94i:65015
4. [HH99] F. J. Hickernell and H. S. Hong, The asymptotic efficiency of randomized nets for quadrature, Math. Comp. 68 (1999), 767-791. MR 99i:65021
5. [Hic96] F. J. Hickernell, The mean square discrepancy of randomized nets, ACM Trans. Model. Comput. Simul. 6 (1996), 274-296.
6. [HW01] F. J. Hickernell and H. Woźniakowski, The price of pessimism for multidimensional quadrature, J. Complexity 17 (2001), 625-659.
7. [HY00] F. J. Hickernell and R. X. Yue, The mean square discrepancy of scrambled (t,s)-sequences, SIAM J. Numer. Anal. 38 (2000), 1089-1112. MR 2002c:65009
8. [Lar98] G. Larcher, Digital point sets: Analysis and applications, Random and Quasi-Random Point Sets (P. Hellekalek and G. Larcher, eds.), Lecture Notes in Statistics, vol. 138, Springer-Verlag, New York, 1998, pp. 167-222. MR 99m:11085
9. [LLNS96] G. Larcher, A. Lauss, H. Niederreiter, and W. Ch. Schmid, Optimal polynomials for (t, m, s}-nets and numerical integration of Walsh series, SIAM J. Numer. Anal. 33 (1996), 2239-2253. MR 97m:65046
10. [Loh01] W.-L. Loh, On the asymptotic distribution of scrambled net quadrature, 2001, submitted for publication.
11. [LP99] G. Larcher and G. Pirsic, Base change problems for generalized Walsh series and multivariate numerical integration, Pac. J. Math. 189 (1999), 75-105. MR 2000f:42017
12. [LS95a] G. Larcher and W. Ch. Schmid, Multivariate Walsh-series, digital nets, and quasi-Monte Carlo integration, In Niederreiter and Shiue [NS95], pp. 252-262. MR 97j:65002
13. [LS95b] G. Larcher and W. Ch. Schmid, On the numerical integration of high dimensional multivariate Walsh-series by quasi-Monte Carlo methods, Math. Comput. Simul. 38 (1995), 127-134. MR 96c:65009
14. [LSW96] G. Larcher, W. Ch. Schmid, and R. Wolf, Quasi-Monte Carlo methods for the numerical integration of multivariate Walsh series, Mathl. Comput. Modelling 23 (1996), 55-67. MR 97j:65044
15. [LT94] G. Larcher and C. Traunfellner, On the numerical integration of Walsh series by number-theoretic methods, Math. Comp. 63 (1994), 277-291. MR 94j:65030
16. [Nie92] H. Niederreiter, Random number generation and quasi-Monte Carlo methods, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1992. MR 93h:65008
17. [Nov88] E. Novak, Deterministic and stochastic error bounds in numerical analysis, Lectures Notes in Math., no. 1349, Springer-Verlag, Berlin, 1988. MR 90a:65004
18. [NP01] H. Niederreiter and G. Pirsic, The microstructure of (t, m, s)-nets, J. Complexity 17 (2001), 683-696.
19. [NS95] H. Niederreiter and P. J.-S. Shiue (eds.), Monte Carlo and quasi- Monte Carlo methods in scientific computing, Lecture Notes in Statistics, vol. 106, Springer-Verlag, New York, 1995. MR 97j:65002
20. [Owe95] A. B. Owen, Randomly permuted (t, m, s)-nets and (t, s)-sequences, In Niederreiter and Shiue [NS95], pp. 299-317. MR 97k:65013
21. [Owe97a] A. B. Owen, Monte Carlo variance of scrambled net quadrature, SIAM J. Numer. Anal. 34 (1997), 1884-1910. MR 98h:65006
22. [Owe97b] A. B. Owen, Scrambled net variance for integrals of smooth functions, Ann. Stat. 25 (1997), 1541-1562. MR 98j:65004
23. [Owe98] A. B. Owen, Scrambling Sobol' and Niederreiter-Xing points, J. Complexity 14 (1998), 466-489. MR 2000c:65005
24. [Owe00] A. B. Owen, Monte Carlo, quasi-Monte Carlo, and randomized quasi-Monte Carlo, Monte Carlo and Quasi-Monte Carlo Methods 1998 (H. Niederreiter and J. Spanier, eds.), Springer-Verlag, Berlin, 2000, 86-97. MR 2002e:65013
25. [Rit00] K. Ritter, Average-case analysis of numerical problems, Lecture Notes in Mathematics, vol. 1733, Springer-Verlag, Berlin, 2000. MR 2001i:65001
26. [YH01] R. X. Yue and F. J. Hickernell, Integration and approximation based on scramble sampling in arbitrary dimensions, J. Complexity 17 (2001), 881-897.
27. [YH02] R. X. Yue and F. J. Hickernell, The discrepancy and gain coefficients of scrambled digital nets, J. Complexity 18 (2002), 135-151.
28. [YM99] R. X. Yue and S. S. Mao, On the variance of quadrature over scrambled nets and sequences, Statist. Probab. Lett. 44 (1999), 267-280. MR 2000i:65008
29. [Yue99] R. X. Yue, Variance of quadrature over scrambled unions of nets, Statist. Sinica 9 (1999), 451-473. MR 2000h:65022