The mean square discrepancy of scrambled (t, s)-sequences

Fred J. Hickernell*, Rong-Xian Yue

*Corresponding author for this work

Research output: Contribution to journalJournal articlepeer-review

39 Citations (Scopus)
3 Downloads (Pure)

Abstract

The discrepancy arises in the worst-case error analysis for quasi-Monte Carlo quadrature rules. Low discrepancy sets yield good quadrature rules. This article considers the mean square discrepancies for scrambled (λ, t, m, s)-nets and (i, s)-sequences in base b. It is found that the mean square discrepancy for scrambled nets and sequences is never more than a constant multiple of that under simple Monte Carlo sampling. If the reproducing kernel defining the discrepancy satisfies a Lipschitz condition with respect to one of its variables separately, then the asymptotic order of the root mean square discrepancy is O(n-1[log n](s-1)/2) for scrambled nets. If the reproducing kernel satisfies a Lipschitz condition with respect to both of its variables, then the asymptotic order of the root mean square discrepancy is O(n-3/2[log n](s-1)/2) for scrambled nets. For an arbitrary number of points taken from a (t, s)-sequence, the root mean square discrepancy appears to be no better than O(n-1[log n](s-1)/2), regardless of the smoothness of the reproducing kernel.

Original languageEnglish
Pages (from-to)1089-1112
Number of pages24
JournalSIAM Journal on Numerical Analysis
Volume38
Issue number4
DOIs
Publication statusPublished - Jan 2000
Externally publishedYes

Scopus Subject Areas

  • Numerical Analysis
  • Computational Mathematics
  • Applied Mathematics

User-Defined Keywords

  • (t, m, s)-nets
  • Multidimensional integration
  • Quadrature
  • Reproducing kernel Hilbert spaces

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