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Quasi-Monte Carlo Methods: Theory and Applications

FWF Special Research Program (SFB)

Project Part 12 (C.Aistleitner):Number-theoretic, probabilistic and computational aspects of uniform distribution theory

SFB funding period 2 (2018-2022)

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-ethods 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.