Hostname: page-component-586b7cd67f-tf8b9 Total loading time: 0 Render date: 2024-11-22T07:01:02.487Z Has data issue: false hasContentIssue false
  • English
  • Français

Constructions of general polynomial lattices for multivariate integration

Published online by Cambridge University Press:  17 April 2009

Peter Kritzer  and
Friedrich Pillichshammer
Peter Kritzer
Affiliation:
Fachbereich Mathematik, Universität Salzburg, Hellbrunnerstr. 34, A-5020 Salzburg, Austria, e-mail: [email protected]
Friedrich Pillichshammer
Affiliation:
Institut für Finanzmathematik, Universität Linz, Altenbergerstr. 69, A-4040 Linz, Austria, e-mail: [email protected]
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We study a construction algorithm for certain polynomial lattice rules modulo arbitrary polynomials. The underlying polynomial lattices are special types of digital nets as introduced by Niederreiter. Dick, Kuo, Pillichshammer and Sloan recently introduced construction algorithms for polynomial lattice rules modulo irreducible polynomials which yield a small worst-case error for integration of functions in certain weighted Hilbert spaces. Here, we generalize these results to the case where the polynomial lattice rules are constructed modulo arbitrary polynomials.

Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 76 , Issue 1 , August 2007 , pp. 93 - 110
Copyright
Copyright © Australian Mathematical Society 2007

References

[1]Chrestenson, H.E., ‘A class of generalized Walsh functions’, Pacific J. Math. 5 (1955), 17–31.Google Scholar
[2]Dick, J., Kritzer, P. and Kuo, F.Y., ‘Approximation of functions using digital nets’, (submitted).Google Scholar
[3]Dick, J., Kritzer, P., Leobacher, G. and Pillichshammer, F., ‘Constructions of general polynomial lattice rules based on the weighted star discrepancy’, Finite Fields Appl. (2007) (to appear).Google Scholar
[4]Dick, J., Kritzer, P., Pillichshammer, F. and Schmid, W. Ch., ‘On the existence of higher order polynomial lattices based on a generalized figure of merit’, J. Complexity (2007) (to appear).Google Scholar
[5]Dick, J., Kuo, F.Y., Pillichshammer, F. and Sloan, I.H., ‘Construction algorithms for polynomial lattice rules for multivariate integration’, Math. Comp. 74 (2005), 1895–1921.Google Scholar
[6]Dick, J., Leobacher, G. and Pillichshammer, F., ‘Construction algorithms for digital nets with small weighted star discrepancy’, SIAM J. Numer. Anal. 43 (2005), 76–95.CrossRefGoogle Scholar
[7]Dick, J. and Pillichshammer, F., ‘Multivariate integration in weighted Hilbert spaces based on Walsh functions and weighted Sobolev spaces’, J. Complexity 21 (2005), 149–195.CrossRefGoogle Scholar
[8]Kuo, F.Y., ‘Component-by-component constructions achieve the optimal rate of convergence for multivariate integration in weighted Korobov and Sobolev spaces’, J. Complexity 19 (2003), 301–320.CrossRefGoogle Scholar
[9]Larcher, G., ‘Digital point sets: Analysis and applications’, in Random and Quasi-Random point sets, (Hellekalek, P. and Larcher, G., Editors), Lecture Notes in Statistics 138 (Springer-Verlag, New York, 1998), pp. 167–222.CrossRefGoogle Scholar
[10]Larcher, G., Lauss, A., Niederreiter, N. and Schmid, W.Ch., ‘Optimal polynomials for (t, m, s)-nets and numerical integration of multivariate Walsh series’, SIAM J. Numer. Anal. 33 (1996), 2239–2253.CrossRefGoogle Scholar
[11]L'Ecuyer, P., ‘Polynomial integration lattices’, in Monte Carlo and Quasi-Monte Carlo Methods 2002, (Niederreiter, H., Editor) (Springer-Verlag, Berlin, 2004), pp. 73–98.Google Scholar
[12]L'Ecuyer, P. and Lemieux, C., ‘Recent advances in randomized quasi-Monte Carlo methods’, in Modeling Uncertainty: An Examination of Stochastic Theory, Methods, and Applications, (Dror, M., L'Ecuyer, P. and Szidarovszky, F., Editors) (Kluwer Academic Publishers, Boston, 2002), pp. 419–474.CrossRefGoogle Scholar
[13]Lemieux, C. and L'Ecuyer, P., ‘Randomized polynomial lattice rules for multivariate integration and simulation’, SIAM J. Sci. Comput. 24 (2003), 1768–1789.Google Scholar
[14]Niederreiter, H., ‘Point sets and sequences with small discrepancy’, Monatsh. Math. 104 (1987), 273–337.Google Scholar
[15]Niederreiter, H., ‘Low-discrepancy point sets obtained by digital constructions over finite fields’, Czechoslovak Math. J. 42 (1992), 143–166.Google Scholar
[16]Niederreiter, H., Random number generation and quasi-Monte Carlo methods, CBMS-NSF Series in Applied Mathematics 63 (SIAM, Philadelphia, 1992).CrossRefGoogle Scholar
[17]Niederreiter, H., ‘The existence of good extensible polynomial lattice rules’, Monatsh. Math. 139 (2003), 295–307.CrossRefGoogle Scholar
[18]Niederreiter, H., ‘Constructions of (t, m, s)-nets and (t, s)-sequences’, Finite Fields Appl. 11 (2005), 578–600.CrossRefGoogle Scholar
[19]Nuyens, D. and Cools, R., Fast component-by-component constructions, a reprise for different kernels, (Niederreiter, H. and Talay, D., Editors), Monte Carlo and Quasi-Monte Carlo Methods 2004 (Springer-Verlag, Berlin, 2006), pp. 373–378.Google Scholar
[20]Sloan, I.H., Kuo, F.Y. and Joe, S., ‘Constructing randomly shifted lattice rules in weighted Sobolev spaces’, SIAM J. Numer. Anal. 40 (2002), 1650–1655.CrossRefGoogle Scholar
[21]Sloan, I.H., Kuo, F.Y. and Joe, S., ‘On the step-by-step construction of quasi-Monte Carlo integration rules that achieve strong tractability error bounds in weighted Sobolev spaces’, Math. Comp. 71 (2002), 1609–1640.Google Scholar
[22]Sloan, I.H. and Woźniakowski, H., ‘When are quasi-Monte Carlo algorithms efficient for high dimensional integrals?’, J. Complexity 14 (1998), 1–33.Google Scholar