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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)

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)

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)

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)

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)

A duality theory for digital nets, i.e., for finite points sets with strong uniformity properties, was introduced by Niederreiter and Pirsic. This duality theory is based on the concept of the dual space M of a digital net. In this paper we extend the duality theory from (finite) digital nets to (infinite) digital sequences. The analogue of the dual space… (More)