Discrepancy of Digital Nets and Sequences

Our project is devoted to an analysis of various notions of discrepancy of digital nets and sequences over a finite field. At the moment, the concept of digital nets and sequences provides the most efficient method to generate point sets with small discrepancy. Today, such point sets are most frequently used for quasi-Monte Carlo quadrature rules, which are successfully used in many applications as in financial mathematics, for example. On the one hand, our analysis of such point sets will focus on the classical notions of discrepancy, as for example the star discrepancy. We will generalize and improve several existing results and analyze different types of nets and sequences as for example symmetrized digital sequences or digital shift-nets. On the other hand, however, a main part of our projected work will deal with a new notion of discrepancy, the so-called weighted discrepancy. Weighted discrepancy was introduced quite recently by Sloan and Wozniakowski to give a general form of the Koksma-Hlawka inequality which takes more explicitly imbalances in the importance of the projections of the integrand into account.

Point sets which are extremely well distributed (in the unit cube) are needed in many fields of applied mathematics, e.g. simulations of complex physical processes or simulations related to financial mathematics. Currently, the best methods known for constructing point sets with good distribution properties are based on the concepts of so-called (t,m,s)-nets and (t,s)-sequences. In our project we analyzed such point sets and studied various aspects of their quality of distribution. The most important way of measuring the quality of distribution of a point set is known as the star discrepancy, since it can be directly applied to error estimation (e.g, in the field of numerical integration). However, other measures of equidistribution were considered as well within our work. In a number of cases, we were able to improve and generalize existing results. In particular, we were able to show best possible results for a large class of point sets (so-called digital nets). In many applications it is useful to have a random element in the point sets under consideration, which are then referred to as randomized point sets. So far, a major drawback in applying the known effective methods for randomization was their cost. We managed to develop a way of randomizing point sets which can easily be implemented on the one hand, and yields results comparable to those of earlier methods on the other hand.