## 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 language | English |
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Pages (from-to) | 1089-1112 |

Number of pages | 24 |

Journal | SIAM Journal on Numerical Analysis |

Volume | 38 |

Issue number | 4 |

DOIs | |

Publication status | Published - Jan 2000 |

Externally published | Yes |

## Scopus Subject Areas

- Numerical Analysis
- Computational Mathematics
- Applied Mathematics

## User-Defined Keywords

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