Abstract

Quasi-Monte Carlo (QMC) methods are well known tools which have been developedfor approximating the value of the integral of a given function. Over the past decades, much progress has been made on numerical integration by means of QMC methods in various settings. The theory of uniform distribution modulo one and classical theory on QMC provide many nice problems and also satisfactory answers for numerical integration algorithms in moderate dimension. However, the question of how to deal with particularly high dimensional problems (i.e., the dimension might be in the hundreds or thousands) has become a major challenge and is a very active area of research. In our talk, we are going to discuss some of the key ideas of uniform distribution modulo one and QMC methods, and point out connections to related fields. After a brief overview of some classical concepts and results, we are going to present recent developments in the efficient construction and application of high dimensional quasi-Monte Carlo rules. The basic questions we would like to address are:

• What are examples of uniformly distributed point sets and QMC rules?

• How can we define quality measures for QMC rules?

• How can we deal with very high dimensional problems using QMC ?

In particular, we are going to present results on Niederreiter’s digital (t, m, s)-nets and (t, s)-sequences, and modifications thereof.