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Tractability of multivariate problems has become nowadays a popular research subject. Polynomial tractability means that the solution of a d-variate problem can be solved to within ε with polynomial cost in ε −1 and d. Unfortunately, many multivariate problems are not polynomially tractable. This holds for all non-trivial unweighted linear tensor product… (More)

—Motivated by neutrality observed in natural evolution often redundant encodings are used in evolutionary algorithms. Many experimental studies have been carried out on this topic. In this paper we present a first rigorous runtime analysis on the effect of using neutrality. We consider a simple model where a layer of constant fitness is distributed in the… (More)

For numerical integration in higher dimensions, bounds for the star-discrepancy with polynomial dependence on the dimension d are desirable. Furthermore , it is still a great challenge to give construction methods for low-discrepancy point sets. In this paper we give upper bounds for the star-discrepancy and its inverse for subsets of the d-dimensional unit… (More)

We investigate the problem of constructing small point sets with low star discrepancy in the s-dimensional unit cube. The size of the point set shall always be polynomial in the dimension s. Our particular focus is on extending the dimension of a given low-discrepancy point set. This results in a deterministic algorithm that constructs N-point sets with… (More)

We construct explicit δ-bracketing covers with minimal cardinality for the set system of (anchored) rectangles in the two dimensional unit cube. More precisely, the cardinality of these δ-bracketing covers are bounded from above by δ −2 +o(δ −2). A lower bound for the cardinality of arbitrary δ-bracketing covers for d-dimensional anchored boxes from [M.… (More)

We study the numerical integration problem for functions with infinitely many variables. The functions we want to integrate are from a reproducing kernel Hilbert space which is endowed with a weighted norm. We study the worst case ε-complexity which is defined as the minimal cost among all algorithms whose worst case error over the Hilbert space unit ball… (More)

The well-known star discrepancy is a common measure for the uniformity of point distributions. It is used, e.g., in multivariate integration, pseudo random number generation, experimental design, statistics, or computer graphics. We study here the complexity of calculating the star discrepancy of point sets in the d-dimensional unit cube and show that this… (More)

In many applications it has been observed that hybrid-Monte Carlo sequences perform better than Monte Carlo and quasi-Monte Carlo sequences, especially in difficult problems. For a mixed s-dimensional sequence m, whose elements are vectors obtained by concatenating d-dimensional vectors from a low-discrepancy sequence q with (s − d)-dimensional random… (More)