## Abstract

We study the multivariate integration problem ∫_{Rd } f (x) ρ (x) d x, with ρ being a product of univariate probability density functions. We assume that f belongs to a weighted tensor-product reproducing kernel Hilbert space of functions whose mixed first derivatives, when multiplied by a weight function ψ, have bounded L_{2}-norms. After mapping into the unit cube [0, 1]^{d}, the transformed integrands are typically unbounded or have huge derivatives near the boundary, and thus fail to lie in the usual function space setting where many good results have been established. In our previous work, we have shown that randomly shifted lattice rules can be constructed component-by-component to achieve a worst case error bound of order O (n^{- 1 / 2}) in this new function space setting. Using a more clever proof technique together with more restrictive assumptions, in this article we improve the results by proving that a rate of convergence close to the optimal order O (n^{- 1}) can be achieved with an appropriate choice of parameters for the function space. The implied constants in the big-O bounds can be independent of d under appropriate conditions on the weights of the function space.

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

Pages (from-to) | 135-160 |

Number of pages | 26 |

Journal | Journal of Complexity |

Volume | 26 |

Issue number | 2 |

DOIs | |

State | Published - Apr 2010 |

### Bibliographical note

Funding Information:The support of the Australian Research Council under its Centres of Excellence and Linkage Programs is gratefully acknowledged. The first author is supported by an Australian Research Council Queen Elizabeth II Research Fellowship. The third author was partially supported by the National Science Foundation under grant DMS-0609703.

### Funding

The support of the Australian Research Council under its Centres of Excellence and Linkage Programs is gratefully acknowledged. The first author is supported by an Australian Research Council Queen Elizabeth II Research Fellowship. The third author was partially supported by the National Science Foundation under grant DMS-0609703.

Funders | Funder number |
---|---|

Australian Research Council Queen Elizabeth II | |

National Science Foundation (NSF) | DMS-0609703 |

Directorate for Mathematical and Physical Sciences | 0609703 |

Australian Research Council |

## Keywords

- Quasi-Monte Carlo methods
- Randomly shifted lattice rules
- Unbounded integrands
- Worst case error

## ASJC Scopus subject areas

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