Distribution Properties of Quasi-Monte Carlo Point Sets

In this project, we study the distribution properties of (high dimensional) point sets stemming from quasi-Monte Carlo algorithms, which are frequently used in numerical integration. The success of such algorithms depends to a considerable extent on the way the elements of a finite or infinite sequence are distributed in a given domain - in most cases, the domain is assumed to be the s-dimensional unit cube. To be more precise, it is known that uniform distribution of the point sets underlying quasi-Monte Carlo rules yields very good results with respect to the integration error. Accordingly, due to the important role of quasi-Monte Carlo methods and related algorithms in applications, notably finance, the theory of uniform distribution of sequences has been a very active area of mathematical research during the past decades, with many contributions by groups of researchers from all over the globe. Our project continues research on this topic and is aimed at advancing recent progress on finding and constructing point sets with excellent distribution properties. In particular, this project is dedicated to the study of polynomial lattices, an important subclass of (t,m,s)-nets, hybrid sequences, which are obtained by mixing the coordinates of different quasi-random sequences, and further, special, point sets which are designed to have excellent distribution properties and a relatively small number of points. The overall goal of the project is to show new results regarding different notions of the discrepancy, a quantity frequently used for measuring the quality of distribution of sequences, of the aforementioned classes of point sets. In our project, we would like to show upper as well as lower bounds on the discrepancy. Among the different types of discrepancy, the so-called star discrepancy and extreme discrepancy will be of most interest in our work, though other types of discrepancy or related measures of uniformity, such as the diaphony, might be considered as well. Regarding methodology, the project will make use of results in and is closely linked to the areas of number theory, (linear) algebra, finite fields, numerical integration, harmonic analysis, and exponential sums.

Quasi-Monte Carlo (QMC) methods are mathematical algorithms for numerically evaluating complicated integrals that arise in various fields of mathematics, including areas of applied mathematics such as finance or computer graphics. QMC algorithms frequently make use of well distributed point sets, i.e., points that are distributed very evenly in a given domain. In this project, we derived new results on certain classes of such uniformly distributed point sets. On the one hand, we studied new types of uniformly distributed point sets, which are so-called hybrid point sets. These are a combination of different previously known QMC point sets. Hybrid point sets have attracted much interest during the past years and may be of relevance for certain applications. In addition, we dealt with QMC algorithms based on such uniformly distributed point sets and studied how they can be used to effectively deal with high dimensional problems of numerically integrating or approximating functions. Here, we studied classes of particularly smooth functions for which we could find algorithms with a very low error.