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Abstract:
For numerical integration in higher dimensions, bounds for the
star-dis\-cre\-pan\-cy 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
$d$-di\-men\-sio\-nal unit cube. They improve known results.
In particular, we determine the usually only implicitly given
constants.
The bounds are based on the construction of nearly optimal $\delta$-covers
of anchored boxes in the $d$-dimensional unit cube.
We give an explicit construction of low-discrepancy points with a derandomized
algorithm.
The running time of the algorithm, which is exponentially in $d$, is discussed
in
detail and comparisons with other methods are given.
