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- 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)

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
- 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 coordinate directions is quantified by a sequence of weights. However, to be able to make use of weighted spaces for a particular application one has to make a choice of… (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)

- Josef Dick
- J. Complexity
- 2007

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

- Josef Dick, Peter Kritzer, Frances Y. Kuo, Ian H. Sloan
- J. Complexity
- 2007

Dedicated to Henryk Wo´zniakowski on the occasion of his 60th birthday. Abstract We consider Fredholm integral equations of the second kind of the form f (x) = g(x) + k(x − y)f (y) dy, where g and k are given functions from weighted Korobov spaces. These spaces are characterized by a smoothness parameter α > 1 and weights γ 1 ≥ γ 2 ≥ · · ·. The weight γ j… (More)

- Josef Dick, Gerhard Larcher, Friedrich Pillichshammer, Henryk Wozniakowski
- Math. Comput.
- 2011

In this paper we study multivariate integration for a weighted Korobov space for which the Fourier coefficients of the functions decay exponentially fast. This implies that the functions of this space are infinitely times differentiable. Weights of the Korobov space monitor the influence of each variable and each group of variables. We show that there are… (More)

- Josef Dick, Frances Y. Kuo, Quoc Thong Le Gia, Dirk Nuyens, Christoph Schwab
- SIAM J. Numerical Analysis
- 2014

We construct quasi-Monte Carlo methods to approximate the expected values of linear functionals of Petrov-Galerkin discretizations of parametric operator equations which depend on a possibly infinite sequence of parameters. Such problems arise in the numerical solution of differential and integral equations with random field inputs. We analyze the… (More)