FWF - Probabilistische Untersuchung - Probabilistic analysis of multivariate problems

The development of the theory of uniform distribution modulo one and discrepancy the- ory started with Hermann Weyl's seminal paper of 1916. Since then, it turned out that these notions are not only remarkably important concepts, but can be conveniently used for many problems of applied mathematics. For example, the so-called Quasi-Monte Carlo (QMC) method for numerical integration is based on the fact that the difference between the average value of a function, evaluated at certain sampling points, and its integral can be bounded by the variation of this function and the discrepancy of the set of sampling points. This observation, whose precise version is called the Koksma-Hlawka inequality , suggests that low-discrepancy point sets provide small integration errors in numerical integration. Since there exist constructions of point sets whose discrepancy is almost of asymptotic order N -1 , this means that QMC integration can perform significantly better than Monte Carlo (MC) integration (where the sampling points are randomly drawn, and the probabilistic error is of asymptotic order N -1/2 ). Although discrepancy theory is an intensively investigated field of mathematics, a num- ber of important problems are still open. In particular, many known results only provide reasonable error bounds if the number of sampling points is very large, and do not imply that QMC integration should work also for a moderate number of points (in compari- son with the dimension). This observation recently led to an increased interest in error bounds for QMC integration in high dimensions, when the number of sampling points cannot be chosen large enough for an application of the classical discrepancy bounds. While classical constructions of low-discrepancy sequences are usually of purely de- terministic nature, a good deal of probability theory is used in these new methods, either by randomizing classical constructions, or by proving the existence of appropri- ate points sets in a random environment. Our research project will focus on applications of probabilistic methods for high- dimensional problems, particularly in the context of discrepancy theory. In the first year we will investigate so-called lacunary sequences of functions, which by a classical heuristics imitate many properties of independent random variables. The behavior of such lacunary sequences has never been investigated in detail in a multidimensional setting, which is surprising, since it can be hoped that the similarity between such sys- tems and independent random variables extends to the multivariate case, and therefore lacunary sequences could serve as an alternative for MC integration. In the second year we will study the probabilistic properties of multivariate random sequences in detail, in particular the deviation between the empirical distribution function and the under- lying distribution. Finally, in the last year, we will mainly focus on discrepancy theory, and use probabilistic methods to prove error bounds for the discrepancy of randomized QMC sequences and to investigate the existence of low-discrepancy points sets with "few" elements in comparison with the dimension.