Friedrich Pillichshammer (Principal Investigator)
Wolfgang Ch. Schmid (Co-Investigator)
Ligia-Loretta Cristea (until Sept. 2007)
Peter Kritzer (until Feb. 2008)
Gottlieb Pirsic (since Jan. 2008)
This project is devoted to the analysis of various notions of discrepancy of digital nets
and sequences over a finite field. Digital nets were introduced by Niederreiter
and at the moment they provide the most efficient method to generate point sets with
small discrepancy. Today such point sets are most frequently used for quasi-Monte Carlo
(QMC) quadrature rules, which are successfully used in many applications in financial
mathematics, physics and engineering.
Firstly we will focus on the classical notions of discrepancy, as for example the star
discrepancy or the L2 discrepancy. For example, we want to extend the method of Walsh
series, where estimates for digital (0,m,2)-nets in base 2 were considerably improved in the
context of symmetrised digital sequences or digital shift-nets.
Secondly we will will deal with a new notion of discrepancy, the so-called weighted
discrepancy. Our aim is to give good and effectively useful bounds for the weighted
discrepancy of digital nets and to construct digital nets with “small” weighted discrepancy.
Further we are interested in the worst-case error for integration with QMC rules using
digital point sets as nodes.