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Higher order polynomial lattice point sets are special types of digital higher order nets which are known to achieve almost optimal convergence rates when used in a quasi-Monte Carlo algorithm to approximate high-dimensional integrals over the unit cube. Recently it has been shown that higher order polynomial lattice point sets of " good " quality must… (More)

- Josef Dick, Jan Baldeaux
- 2009

Generalized digital nets and sequences have been introduced for the numerical integration of smooth functions using quasi-Monte Carlo rules. In this paper we study geometrical properties of such nets and sequences. The definition of these nets and sequences does not depend on linear algebra over finite fields, it only requires that the point set or sequence… (More)

We show how to obtain a fast component-by-component construction algorithm for higher order polynomial lattice rules. Such rules are useful for multivariate quadrature of high-dimensional smooth functions over the unit cube as they achieve the near optimal order of convergence. The main problem addressed in this paper is to find an efficient way of… (More)

In this paper we study an approximation algorithm which firstly approximates certain Walsh coefficients of the function under consideration and consequently uses a Walsh polynomial to approximate the function. A similar approach has previously been used for approximating periodic functions, using lattice rules (and Fourier polynomials), and for… (More)

Generalized nets and sequences are used in quasi-Monte Carlo rules for the approximation of high dimensional integrals over the unit cube. Hence one wants to have generalized nets and sequences of high quality. In this paper we introduce a duality theory for generalized nets whose construction is not necessarily based on linear algebra over finite fields.… (More)

- Jan Baldeaux, Michael Gnewuch, J Hickernell, T Müller-Gronbach, B Niu, K Ritter
- 2012

In this paper, we consider the infinite-dimensional integration problem on weighted reproducing kernel Hilbert spaces with norms induced by an underlying function space decomposition of ANOVA-type. The weights model the relative importance of different groups of variables. We present new randomized multilevel algorithms to tackle this integration problem… (More)

- Jan Baldeaux, Josef Dick, Julia Greslehner, Friedrich Pillichshammer
- 2009

Recently, generalized digital nets were introduced and shown to achieve almost optimal convergence rates when used in a quasi-Monte Carlo algorithm to approximate high-dimensional integrals over the unit cube. Since their inception, one method of constructing generalized digital nets from classical digital nets has been known. However, the desirable… (More)