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Indispensable for students, invaluable for researchers, this comprehensive treatment of contemporary quasi–Monte Carlo methods, digital nets and sequences, and discrepancy theory starts from scratch with detailed explanations of the basic concepts and then advances to current methods used in research. As deterministic versions of the Monte Carlo method,… (More)

This paper is a contemporary review of QMC (" Quasi-Monte Carlo ") methods , i.e., equal-weight rules for the approximate evaluation of high dimensional integrals over the unit cube [0, 1] s , where s may be large, or even infinite. After a general introduction, the paper surveys recent developments in lattice methods, digital nets, and related themes.… (More)

- Josef Dick, Ian H. Sloan, Xiaoqun Wang, Henryk Wozniakowski
- Numerische Mathematik
- 2006

We study the problem of multivariate integration and the construction of good lattice rules in weighted Korobov spaces with general weights. These spaces are not necessarily tensor products of spaces of univariate functions. Sufficient conditions for tractability and strong tractability of multivariate integration in such weighted function spaces are found.… (More)

1. The functional properties of tibialis anterior muscles of normal adult (C57BL/10) and age-matched dystrophin-deficient (C57BL/10 mdx) mice have been investigated in situ. Comparisons were made between tibialis anterior muscle strength, rates of force development and relaxation, force-frequency responses and fatiguability. Subjecting mdx and C57 muscles… (More)

- Josef Dick
- J. Complexity
- 2004

We prove error bounds on the worst-case error for integration in certain Korobov and Sobolev spaces using rank-1 lattice rules with generating vectors constructed by the component-by-component algorithm. For a prime number of points n a rate of convergence of the worst-case error for multivariate integration in Korobov spaces of O (n−α/2+δ), where α > 1 is… (More)

- P S Sharp, J R T Dick, L Greensmith
- Neuroscience
- 2005

Around 20% of familial cases of amyotrophic lateral sclerosis have been shown to carry mutations in Cu/Zn superoxide dismutase 1 (Cu/Zn SOD1). Transgenic mice over-expressing human mutant SOD1 genes have been developed and in this study we examined the effect of nerve injury on disease progression in these mice. Firstly, disease progression in uninjured… (More)

- Josef Dick
- SIAM J. Numerical Analysis
- 2008

We define a Walsh space which contains all functions whose partial mixed derivatives up to order δ ≥ 1 exist and have finite variation. In particular, for a suitable choice of parameters, this implies that certain reproducing kernel Sobolev spaces are contained in these Walsh spaces. For this Walsh space we then show that quasi-Monte Carlo rules based on… (More)

- Josef Dick
- SIAM J. Numerical Analysis
- 2007

In this paper we give first explicit constructions of point sets in the s dimensional unit cube yielding quasi-Monte Carlo algorithms which achieve the optimal rate of convergence of the worst-case error for numerically integrating high dimensional periodic functions. In the classical measure Pα of the worst-case error introduced by Korobov in 1959 the… (More)

- Josef Dick, Frances Y. Kuo
- Math. Comput.
- 2004

The construction of randomly shifted rank-1 lattice rules, where the number of points n is a prime number, has recently been developed by Sloan, Kuo and Joe for integration of functions in weighted Sobolev spaces and was extended by Kuo and Joe and by Dick to composite numbers. To construct d-dimensional rules, the shifts were generated randomly and the… (More)

- Christoph Aistleitner, Johann S. Brauchart, Josef Dick
- Discrete & Computational Geometry
- 2012