## Quadrature Error Bounds With Applications To Lattice Rules (1996)

Venue: | SIAM J. Numer. Anal |

Citations: | 26 - 6 self |

### BibTeX

@ARTICLE{Hickernell96quadratureerror,

author = {Fred J. Hickernell},

title = {Quadrature Error Bounds With Applications To Lattice Rules},

journal = {SIAM J. Numer. Anal},

year = {1996},

volume = {33},

pages = {1995--2016}

}

### Years of Citing Articles

### OpenURL

### Abstract

. Reproducing kernel Hilbert spaces are used to derive error bounds and worst-case integrands for a large family of quadrature rules. In the case of lattice rules applied to periodic integrands these error bounds resemble those previously derived in the literature. However, the theory developed here does not require periodicity and is not restricted to lattice rules. An ANOVA decomposition is employed in defining the inner product. It is shown that imbedded rules are superior when integrating functions with large high order ANOVA effects. Key words. ANOVA decomposition, good lattice points, imbedded rules, multidimensional integration, Monte Carlo, number-theoretic, quasirandom, periodic functions, reproducing kernel Hilbert spaces AMS subject classifications. 65D30, 65D32 1. Introduction. Many multidimensional integrals can be written as I(f) = Z C d f(x) dx; where C d = [0; 1) d is the d-dimensional unit cube. A simple approximation to I(f) is the arithmetic mean of the va...

### Citations

1 |
Theoretic Methods in Statistics
- Number
- 1994
(Show Context)
Citation Context ...N . Quadrature rules based on rank-1 lattices (also called good lattice points) were proposed independently by Korobov [14] and Hlawka [10]. They have been studied by Hua and Wang [11], Fang and Wang =-=[6]-=- and others. Because integration lattices have period 1 in each coordinate direction, it is convenient to define f on R d by a periodic extension of its definition on C d . If f is Department of Mathe... |

1 |
angenaherten berechnung mehrfacher integrale, Monatsh
- Zur
- 1962
(Show Context)
Citation Context ...is a generating vector normally chosen to be relatively prime to N . Quadrature rules based on rank-1 lattices (also called good lattice points) were proposed independently by Korobov [14] and Hlawka =-=[10]-=-. They have been studied by Hua and Wang [11], Fang and Wang [6] and others. Because integration lattices have period 1 in each coordinate direction, it is convenient to define f on R d by a periodic ... |

1 |
lattice rules for multidimensional integration
- Imbedded
- 1992
(Show Context)
Citation Context ... +Gn )"C d consists of n 1 \Delta \Delta \Delta n d translated copies of the set S 0 . Quadrature rules based on S are called imbedded rules. Sloan and Walsh [26], Disney and Sloan [4], Joe and S=-=loan [13] and -=-Joe and Disney [12] studied imbedded lattice rules. They recommended choosing S 0 = L 0 " C d for some rank-1 lattice L 0 and choosing n j = 1 or 2. Their computations and theoretical results sug... |

1 |
sequences and their discrepancies
- Quasi-random
- 1994
(Show Context)
Citation Context ...e as P ff . Furthermore the worstcase integrand, as constructed in [16, x2.2] does not have as simple a form as (1.5). Other error bounds, including those based on an L 2 discrepancy, may be found in =-=[15]-=-. The approach we use to derive new quadrature error bounds differs from those outlined above in that it is based on the theory of reproducing kernel Hilbert spaces. Unlike the standard error analysis... |

1 |
Carlo methods with modified vertex weights
- Quasi-Monte
- 1993
(Show Context)
Citation Context ...drature rules with non-uniform weights, provided that the weights depend only on the points in S and not on the integrand. Some rules of this type have recently been studied by Niederreiter and Sloan =-=[18, 19]-=-. Acknowledgments. The author wishes to thank Kai-Tai Fang, James Lyness, Art Owen, Ian Sloan and two anonymous referees for valuable discussions and suggestions regarding this manuscript. The author ... |

1 |
arrays for computer experiments, integration and visualization
- Orthogonal
- 1992
(Show Context)
Citation Context ...product on X it is convenient to consider analysis of variance (ANOVA) decompositions of functions in X. These decompositions have been used in the statistical literature by Efron and Stein [5], Owen =-=[20]-=- and others. Let D = f1; : : : ; dg be the set of coordinate indices. For any index set u ` D, let juj denote its cardinality. Let x u denote the juj-vector containing the components of x indexed by u... |