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Indispensable for students, invaluable for researchers, this comprehensive treatment of contemporary quasiâ€“Monte Carlo methods, digital nets and sequences, and discrepancy theory starts from scratchâ€¦ (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 spacesâ€¦ (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-caseâ€¦ (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-componentâ€¦ (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 weightedâ€¦ (More)

- Josef Dick
- SIAM J. Numerical Analysis
- 2008

We define a Walsh space which contains all functions whose partial mixed derivatives up to order Î´ â‰¥ 1 exist and have finite variation. In particular, for a suitable choice of parameters, thisâ€¦ (More)

- Josef Dick, Friedrich Pillichshammer
- J. Complexity
- 2005

We introduce a weighted reproducing kernel Hilbert space which is based on Walsh functions. The worst-case error for integration in this space is studied, especially with regard to (t, m, s)-nets. Itâ€¦ (More)

We develop a convergence analysis of a multi-level algorithm combining higher order quasi-Monte Carlo (QMC) quadratures with general Petrov-Galerkin discretizations of countably affine parametricâ€¦ (More)

- Josef Dick
- J. Complexity
- 2007

It was shown by Heinrich et al. [The inverse of the star-discrepancy depends linearly on the dimension, Acta Arith. 96 (2001) 279â€“302] that there exist point sets for which the inverse of the starâ€¦ (More)

- Josef Dick, Ian H. Sloan, Xiaoqun Wang, Henryk Wozniakowski
- J. Complexity
- 2004

A partial answer to why quasi-Monte Carlo algorithms work well for multivariate integration was given in [15] by introducing weighted spaces. In these spaces the importance of successive coordinateâ€¦ (More)