### Abstract

The computational complexity of the integration problem in terms of the expected error has recently been an important topic in Information-Based Complexity. In this setting, we assume some sample space of integration rules from which we randomly choose one. The most popular sample space is based on Owen's random scrambling scheme whose theoretical advantage is the fast convergence rate for certain smooth functions. This paper considers a reduction of randomness required for Owen's random scrambling by using the notion of i-binomial property. We first establish a set of necessary and sufficient conditions for digital (0,s)-sequences to have the i-binomial property. Then based on these conditions, the left and right i-binomial scramblings are defined. We show that Owen's key lemma (Lemma 4, SIAM J. Numer. Anal. 34 (1997) 1884) remains valid with the left i-binomial scrambling, and thereby conclude that all the results on the expected errors of the integration problem so far obtained with Owen's scrambling also hold with the left i-binomial scrambling.

Original language | English |
---|---|

Pages (from-to) | 744-757 |

Number of pages | 14 |

Journal | Journal of Complexity |

Volume | 19 |

Issue number | 6 |

DOIs | |

Publication status | Published - Jan 1 2003 |

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### All Science Journal Classification (ASJC) codes

- Algebra and Number Theory
- Statistics and Probability
- Numerical Analysis
- Control and Optimization
- Applied Mathematics

### Cite this

*Journal of Complexity*,

*19*(6), 744-757. https://doi.org/10.1016/S0885-064X(03)00035-9

**I-binomial scrambling of digital nets and sequences.** / Tezuka, Shu; Faure, Henri.

Research output: Contribution to journal › Article

*Journal of Complexity*, vol. 19, no. 6, pp. 744-757. https://doi.org/10.1016/S0885-064X(03)00035-9

}

TY - JOUR

T1 - I-binomial scrambling of digital nets and sequences

AU - Tezuka, Shu

AU - Faure, Henri

PY - 2003/1/1

Y1 - 2003/1/1

N2 - The computational complexity of the integration problem in terms of the expected error has recently been an important topic in Information-Based Complexity. In this setting, we assume some sample space of integration rules from which we randomly choose one. The most popular sample space is based on Owen's random scrambling scheme whose theoretical advantage is the fast convergence rate for certain smooth functions. This paper considers a reduction of randomness required for Owen's random scrambling by using the notion of i-binomial property. We first establish a set of necessary and sufficient conditions for digital (0,s)-sequences to have the i-binomial property. Then based on these conditions, the left and right i-binomial scramblings are defined. We show that Owen's key lemma (Lemma 4, SIAM J. Numer. Anal. 34 (1997) 1884) remains valid with the left i-binomial scrambling, and thereby conclude that all the results on the expected errors of the integration problem so far obtained with Owen's scrambling also hold with the left i-binomial scrambling.

AB - The computational complexity of the integration problem in terms of the expected error has recently been an important topic in Information-Based Complexity. In this setting, we assume some sample space of integration rules from which we randomly choose one. The most popular sample space is based on Owen's random scrambling scheme whose theoretical advantage is the fast convergence rate for certain smooth functions. This paper considers a reduction of randomness required for Owen's random scrambling by using the notion of i-binomial property. We first establish a set of necessary and sufficient conditions for digital (0,s)-sequences to have the i-binomial property. Then based on these conditions, the left and right i-binomial scramblings are defined. We show that Owen's key lemma (Lemma 4, SIAM J. Numer. Anal. 34 (1997) 1884) remains valid with the left i-binomial scrambling, and thereby conclude that all the results on the expected errors of the integration problem so far obtained with Owen's scrambling also hold with the left i-binomial scrambling.

UR - http://www.scopus.com/inward/record.url?scp=0348233983&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0348233983&partnerID=8YFLogxK

U2 - 10.1016/S0885-064X(03)00035-9

DO - 10.1016/S0885-064X(03)00035-9

M3 - Article

VL - 19

SP - 744

EP - 757

JO - Journal of Complexity

JF - Journal of Complexity

SN - 0885-064X

IS - 6

ER -