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- Michael Gnewuch
- Electr. J. Comb.
- 2005

The extreme or unanchored discrepancy is the geometric discrepancy of point sets in the d-dimensional unit cube with respect to the set system of axis-parallel boxes. For 2 ≤ p < ∞ we provide upper bounds for the average Lp-extreme discrepancy. With these bounds we are able to derive upper bounds for the inverse of the L∞-extreme discrepancy with optimal… (More)

- Michael Gnewuch
- J. Complexity
- 2008

In the first part of this paper we derive lower bounds and constructive upper bounds for the bracketing numbers of anchored and unanchored axis-parallel boxes in the d-dimensional unit cube. In the second part we apply these results to geometric discrepancy. We derive upper bounds for the inverse of the star and the extreme discrepancy with explicitly given… (More)

We provide a deterministic algorithm that constructs small point sets exhibiting a low star discrepancy. The algorithm is based on bracketing and on recent results on randomized roundings respecting hard constraints. It is structurally much simpler than the previous algorithm presented for this problem in [B. Doerr, M. Gnewuch, A. Srivastav. Bounds and… (More)

- Michael Gnewuch, Henryk Wozniakowski
- J. Complexity
- 2007

- Benjamin Doerr, Michael Gnewuch, Nils Hebbinghaus, Frank Neumann
- IEEE Congress on Evolutionary Computation
- 2007

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)

- Benjamin Doerr, Michael Gnewuch, Anand Srivastav
- J. Complexity
- 2005

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 lowdiscrepancy point sets. In this paper we give upper bounds for the star-discrepancy and its inverse for subsets of the d-dimensional unit… (More)

- Benjamin Doerr, Michael Gnewuch, Peter Kritzer, Friedrich Pillichshammer
- Monte Carlo Meth. and Appl.
- 2008

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)

- Michael Gnewuch
- Electr. J. Comb.
- 2008

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)

- Michael Gnewuch
- J. Complexity
- 2009

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)

- Michael Gnewuch
- 2010

The star discrepancy is a measure of how uniformly distributed a finite point set is in the d-dimensional unit cube. It is related to high-dimensional numerical integration of certain function classes as expressed by the Koksma-Hlawka inequality. A sharp version of this inequality states that the worst-case error of approximating the integral of functions… (More)