Special Research Area (SFB) F55 Quasi-Monte Carlo Methods: Theory and Applications

By "Quasi-Monte Carlo (QMC) methods" we understand all methods in which most carefully chosen quasi-random-point sets are used to carry out simulations in the framework of sophisticated and highly developed modeling environments, to obtain quantitative information in different branches of applications. The further study and development of QMC methods therefore requires the generation, investigation, and analysis of distribution properties of finite or infinite sequences in all kinds of regions the development, investigation, and analysis of suitable theoretical models on which the applications of the QMC methods are based, and in particular the derivation of error bounds for QMC methods in these models the ecient implementation of the theoretical models and of the algorithms for the generation of the (sometimes very large and high-dimensional) quasi-random point sets, and the provision of sophisticated software the concrete application of the QMC methods in different areas, the discussion of the implications and of the performance of the applied QMC methods.
Consequently, many different branches of mathematics are involved in the comprehensive investigation and development of QMC methods, most notably number theory, discrete mathematics, combinatorics, harmonic analysis, functional analysis, stochastics, complexity theory, theory of algorithms, and numerical analysis. Furthermore, profound knowledge of the branches of applications in which the QMC methods are intended to be used is necessary. The theory and application of QMC methods is a modern and extremely lively branch of mathematics. This is demonstrated by an enormous output of research papers on this topic in the last decades, and by the great and growing success of the series of the international conferences on Monte Carlo andQuasi-Monte Carlo Methods in Scientifc Computing" (MCQMC), which started in 1994 in Las Vegas and was most recently held in Sydney in 2012. The Austrian research groups initiating this SFB play leading roles in the development of QMC methods. It is the aim of this SFB to intensify the cooperation both between these research groups and with their international partners, to promote new directions and new developments within the theory of QMC methods and their applications, and to train a new generation of highly talented young researchers to carry out research work in the field of QMC methods.

Subprojects
FWF - DEDIFI; Diophantine equations, discrepancy and finance (Direction: Tichy, Robert; O.Univ.-Prof. Dr.phil.)
Quasi-Monte Carlo (QMC) methods are frequently used in applied mathematics, in particular for the evaluation of multivariate integrals and for simulations in mathematical physics and finance. In the last decade probabilistic methods have been fruitfully used to achieve QMC error bounds which depend linearly in the dimension. Furthermore, a combination of such probabilistic tools with deep machinery from Diophantine analysis can be used to obtain precise limit laws for discrepancy functions and related objects of QMC theory. This subproject consists of save parts: 1) Lacunarity, symmetry and Diophantine equations. 2) Discrepancy and lacunarity. 3) Pseudorandomness and discrepancy. 4) Uniform distribution and dynamical systems. 5) QMC methods, copulas and finance. In part 1) we will investigate permutation invariant limit laws for discrepancies and for averages of lacunary functions. The structure of these limit laws signifcantly depends on the number of solutions of certain Diophantine equations. With the help of Schmidt's subspace theorem, it is possible to derive strong results for lacunary and even for classes of sublacunary sequences. Part 2) is devoted to geometric discrepancies related to lacunary sets of directions in Euclidean plane and space. This connects the first part of this subproject with modern developments in the theory of irregularities of distribution. In part 3) we want to investigate various measures of pseudorandomness, mainly the well-distribution measure introduced by Mauduit and Sark}ozy and correlation measures. In particular, we will study paircorrelations of multivariate sequences from a probabilistic point of view. Part 4) is concerned with uniform distribution, asymptotic distribution functions and dynamical systems. We will focus on the von Neumann-Kakutani transformation and its applications to low discrepancy sequences. The final part 5) is devoted to applications in mathematical finance where we are mainly interested in the study of copulas and in multi-level Monte- Carlo methods.

FWF - Minimale Energie und sphärische De
This sub-project of the Special Research Program Quasi-Monte Carlo methods: Theory and Applications'' is concerned with constructions of well-distributed point sets on compact manifolds, especially the d-sphere. Especially, two construction principles will be investigated: Minimal energy point sets: For a given compact manifold M and a set of N distinct points, the Riesz s-energy is the sum of the negative s-th powers of the mutual distances of the points. A configuration, which minimises this energy amongst all N-point configurations, is called a minimal energy configuration. The motivation for studying such configurations comes from Chemistry and Physics, where self-organisation of mutually repelling particles under inverse power laws occur. For fixed s and N tending to infinity the distribution of the minimal energy point set approaches a continuous limiting measure, which can be described by classical potential theory, if sdim(M) by a recent result of D. P. Hardin and E. B. Saff. Spherical designs: A spherical t-design is a finite set of points such that equal weight quadrature with these points is exact for polynomials of degree up to t. Only recently A. V. Bondarenko et al. could show that t-designs with O(t^d) points exist. This is the same order of magnitude as previously known lower bounds. The discrepancy, integration error, and separation properties of minimal energy point sets and designs shall be analysed. Furthermore, number-theoretic constructions for well-distributed point sets on the sphere shall be investigated further.

FWF - NTPCA; Number theoretic, probabilistic and computational aspects of uniform distribution theory; (Project Leader: Aistleitner, Christoph; Assoc.Prof. Dipl.-Ing. Dr.techn.)
The topics addressed in this subproject can be grouped into questions concerning number-theoretic, metric and analytic aspects of uniform distribution theory and discrepancy theory on the one hand, and questions concerning potential problems in the implementation of QMC methods for problems from quantitative finance and insurance mathematics on the other hand. The problems from the first part pertain to the "classical" theory of uniform distribution modulo one, as it was developed in the early 20th century by Hermann Weyl and others. In our project we want to exploit the powerful new method linking correlations of dilated sums to so-called GCD sums, which was developed in recent years and which should enable us to obtain significant progress on some long-standing open problems. In the "applied" part of the subproject we want to investigate problems which were not in the focus of classical discrepancy theory, such as questions concerning the feasibility of implementing QMC integration in the high-dimensional setting, or questions concerning the application of QMC methods in the case of dependencies between coordinates or in the case of irregular integrand functions. These questions address problems which have been neglected by researchers for a long time, despite the fact that they represent crucial issues in applications of the QMC method.