My dream quadrature rule

Journal of Complexity

Volumn 19, Issue 3, 2003, Pages 420-427

  Hickernell, Fred J ^a  

^a HONG KONG BAPTIST UNIVERSITY   (Hong Kong)

Author keywords

[No Author keywords available]

Indexed keywords

EID: 0038392439     PISSN: 0885064X     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0885-064X(02)00024-9     Document Type: Article

References (60)

1. R.E. Caflisch, W. Morokoff, A. Owen, Valuation of mortgage backed securities using Brownian bridges to reduce effective dimension, J. Comput. Finance 1 (1997) 27-46.
2. R, Cranley, T.N.L. Patterson, Randomization of number theoretic methods for multiple integration, SIAM J. Numer. Anal. 13 (1976) 904-914.
3. K. Entacher, Quasi-Monte Carlo methods for numerical integration of multivariate Haar series, BIT 37 (1997) 846-861.
4. H. Faure, Discrépance de suites associées à un système de numération (en dimension s), Acta Arith. 41 (1982) 337-351.
5. H. Faure, S. Tezuka, Another random scrambling of digital (t, s)-sequences, in: Fang et al. (Eds.), Monte Carlo and Quasi-Monte Carlo Methods 2000, Springer, Berlin, 2002, pp. 242-256.
6. I. Friedel, A. Keller, Fast generation of randomized low-discrepancy point sets, in: Fang et al. (Eds.), Monte Carlo and Quasi-Monte Carlo Methods 2000, Springer, Berlin, 2002, pp. 257-273.
7. S. Heinrich, Random approximation in numerical analysis, Functional analysis, in: K.D. Bierstedt, A. Pietsch, W.M. Ruess, D. Vogt (Eds.), Proceedings of the Essen Conference, held in Essen, Germany, November 24-30, 1991, Lecture Notes in Pure and Applied Mathematics, Vol. 150, Marcel Dekker, New York, 1993, pp. 123-171.
8. S. Heinrich, F.J. Hickernell, R.X. Yue, Optimal quadrature for Haar wavelet spaces, Math. Comput. (2003), to appear.
9. F.J. Hickernell, The mean square discrepancy of randomized nets, ACM Trans. Model. Comput. Simul. 6 (1996) 274-296.
10. F.J. Hickernell, Quadrature error bounds with applications to lattice rules, SIAM J. Numer. Anal. 33 (1996) 1995-2016 (corrected printing of Sections 3-6 in SIAM J. Numer. Anal. 34 (1997) 853-866).
11. F.J. Hickernell, Lattice rules: how well do they measure up?, in: P. Hellekalek, G. Larcher (Eds.), Random and Quasi-Random Point Sets, Lecture Notes in Statistics, Vol. 138, Springer, New York, 1998, pp. 109-166.
12. -2+E) convergence for lattice quadrature rules, in: Fang et al. (Eds.), Monte Carlo and Quasi-Monte Carlo Methods 2000, Springer, Berlin, 2002, pp. 274-289.
13. F.J. Hickernell, H.S. Hong, The asymptotic efficiency of randomized nets for quadrature, Math. Comput. 68 (1999) 767-791.
14. F.J. Hickernell, H.S. Hong, P. L'Écuyer, C. Lemieux, Extensible lattice sequences for quasi-Monte Carlo quadrature, SIAM J. Sci. Comput. 22 (2000) 1117-1138.
15. F.J. Hickernell, M.Q. Liu, Uniform experimental designs limit aliasing, Biometrika 89 (2002) 893-904.
16. F.J. Hickernell, H. Niederreiter, The existence of good extensible rank-1 lattices, J. Complexity 19 (2003) 286-300.
17. F.J. Hickernell, H. Woźniakowski, Integration and approximation in arbitrary dimensions, Adv. Comput. Math. 12 (2000) 25-58.
18. F.J. Hickernell, H. Woźniakowski, The price of pessimism for multidimensional quadrature, J. Complexity 17 (2001) 625-659.
19. F.J. Hickernell, H. Woźniakowski, Tractability of multivariate integration for periodic functions, J. Complexity 17 (2001) 660-682.
20. F.J. Hickernell, R.X. Yue, The mean square discrepancy of scrambled (t,s)-sequences, SIAM J. Numer. Anal. 38 (2000) 1089-1112.
21. H.S. Hong, F.J. Hickernell, Implementing scrambled digital nets, ACM TOMS, 2001, submitted for publication.
22. L.K. Hua, Y. Wang, Applications of Number Theory to Numerical Analysis, Springer, Science Press, Berlin, Beijing, 1981.
23. F.Y. Kuo, S. Joe, Component-by-component construction of good lattice rules with a composite number of points, J. Complexity (2002), to appear.
24. F.Y. Kuo, S. Joe, Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces, 2002, submitted for publication.
25. G. Larcher, Digital point sets: analysis and applications, in: P. Hellekalek, G. Larcher (Eds.), Random and Quasi-Random Point Sets, Lecture Notes in Statistics, Vol. 138, Springer, New York, 1998, pp. 167-222.
26. G. Larcher, W.Ch. Schmid, R. Wolf, Quasi-Monte Carlo methods for the numerical integration of multivariate Walsh series, Math. Comput. Modelling 23 (1996) 55-67.
27. G. Larcher, C. Traunfellner, On the numerical integration of Walsh series by number-theoretic methods, Math. Comput. 63 (1994) 277-291.
28. C. Lemieux, P. L'Ecuyer, Efficiency improvement by lattice rules for pricing asian options, Proceedings of the 1998 Winter Simulation Conference, IEEE Press, New York, 1998, pp. 579-586.
29. 2-discrepancy for anchored boxes, J. Complexity 14 (1998) 527-556.
30. H. Niederreiter, Low-discrepancy and low-dispersion sequences, J. Number Theory 30 (1988) 51-70.
31. H. Niederreiter, Random number generation and quasi-Monte Carlo methods, CBMS-NSF Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, PA, 1992.
32. H. Niederreiter, Constructions of (t, m, s)-nets, in: H. Niederreiter, J. Spanier (Eds.), Monte Carlo and Quasi-Monte Carlo Methods 1998, Springer, Berlin, 2000, pp. 70-85.
33. H. Niederreiter, C. Xing, Quasirandom points and global function fields, in: S. Cohen, H. Niederreiter (Eds.), Finite Fields and Applications, London Mathematical Society Lecture Notes Series, No. 233, Cambridge University Press, Cambridge, 1996, pp. 269-296.
34. S. Ninomiya, S. Tezuka, Toward real-time pricing of complex financial derivatives, Appl. Math. Finance 3 (1996) 1-20.
35. E. Novak, Deterministic and Stochastic Error Bounds in Numerical Analysis, in: Lectures Notes in Mathematics, Vol. 1349, Springer, Berlin, 1988.
36. E. Novak, H. Woźniakowski, Intractability results for integration and discrepancy, J. Complexity 17 (2001) 388-441.
37. E. Novak, H. Woźniakowski, When are integration and discrepancy tractable? Found. Comput. Math. (2001) 211-266.
38. A.B. Owen, The dimension distribution and quadrature test functions, submitted for publication.
39. A.B. Owen, Variance and discrepancy with alternative scramblings, submitted for publication.
40. A.B. Owen, Randomly permuted (t, m, s)-nets and (t, s)-sequences, in: H. Niederreiter, P.J.-S. Shiue (Eds.), Monte Carlo and Quasi-Monte Carlo Methods in Scientific Computing, Lecture Notes in Statistics, Vol. 106, Springer, New York, 1995, pp. 299-317.
41. A.B. Owen, Monte Carlo variance of scrambled net quadrature, SIAM J. Numer. Anal. 34 (1997) 1884-1910.
42. A.B. Owen, Scrambled net variance for integrals of smooth functions, Ann. Statist. 25 (1997) 1541-1562.
43. A.B. Owen, Latin supercube sampling for very high dimensional simulations, ACM Trans. Model. Comput. Simul. 8 (1998) 71-102.
44. A.B. Owen, Scrambling Sobol' and Niederreiter-Xing points, J. Complexity 14 (1998) 466-489.
45. A. Papageorgiou, J.F. Traub, Beating Monte Carlo, Risk 9 (6) (1996) 63-65.
46. S. Paskov, J. Traub, Faster valuation of financial derivatives, J. Portfolio Management 22 (1995) 113-120.
47. K. Ritter, Average-case Analysis of Numerical Problems, in: Lecture Notes in Mathematics, Vol. 1733, Springer, Berlin, 2000.
48. I.H. Sloan, S. Joe, Lattice Methods for Multiple Integration, Oxford University Press, Oxford, 1994.
49. I.H. Sloan, F.Y. Kuo, S. Joe, Constructing randomly shifted lattice rules in weighted Sobolev spaces, SIAM J. Numer. Anal. 40 (2002) 1650-1655.
50. I.H. Sloan, F.Y. Kuo, S. Joe, On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces, Math. Comput. (2002), to appear.
51. I.H. Sloan, H. Woźniakowski, An intractability result for multiple integration, Math. Comput. 66 (1997) 1119-1124.
52. I.H. Sloan, H. Woźniakowski, When are quasi-Monte Carlo algorithms efficient for high dimensional integrals, J. Complexity 14 (1998) 1-33.
53. I.H. Sloan, H. Woźniakowski, Tractability of multivariate integration for weighted Korobov classes, J. Complexity 17 (2001) 697-721.
54. I.M. Sobol', The distribution of points in a cube and the approximate evaluation of integrals, USSR Comput. Math. Math. Phys. 7 (1967) 86-112.
55. V.N. Temlyakov, Approximation of Functions with Bounded Mixed Derivative, Amer. Math. Soc., Providence, RI, 1989.
56. J.F. Traub, G.W. Wasilkowski, H. Woźniakowski, Information-Based Complexity, Academic Press, Boston, 1988.
57. H. Woźniakowski, Efficiency of quasi-Monte Carlo algorithms for high dimensions, in: H. Niederreiter, J. Spanier (Eds.), Monte Carlo and Quasi-Monte Carlo Methods 1998, Springer, Berlin, 2000, pp. 114-136.
58. R.X. Yue, Variance of quadrature over scrambled unions of nets, Statist. Sinica 9 (1999) 451-473.
59. R.X. Yue, F.J. Hickernell, The discrepancy and gain coefficients of scrambled digital nets, J. Complexity 18 (2002) 135-151.
60. R.X. Yue, S.S. Mao, On the variance of quadrature over scrambled nets and sequences, Statist. Probab. Lett. 44 (1999) 267-280.