Uniform Distribution Theory in Compact Groups with Application to Quasi-Monte Carlo Integration

Many problems in the applications of mathematics can be reduced to the computation of a multi-dimensional integral. Of these, there exist integrals that cannot be evaluated precisely with perhaps the most common reason being the non-existence of an antiderivative of the function. Numerical methods are therefore required to approximate these calculations, while striving for a small approximation error to the actual value of the integral. The Monte Carlo integration method is one of these numerical methods which approximates an integral as the average of the integrand evaluated at a randomly selected set of points. In contrast, there also exists the quasi-Monte Carlo integration method which approximates the integral by averaging integrand values at points which are deterministic in nature, not random.

Questions therefore arise from asking what properties this set of deterministic points should possess to ensure best approximation to the exact integral value. The remarkable 'Koksma- Hlawka inequality' states that the error between the approximated value by QMC methods and the exact integral value can be bounded by a product of two independent factors. Both factors can be investigated separately, however I am interested in the term which quantifies how well a sequence is distributed, namely the discrepancy of a sequence. This ensures the link between quasi-Monte Carlo methods and uniform distribution. In particular, the method desires sequences of low discrepancy to reduce the error.

The theory of uniform distribution in the classical setting is concerned with the irregularity of the distribution of sequences of real numbers in the d-dimensional unit cube. This, along with theoretical abstract concepts and results for general spaces are very well understood. For example, there exists a wealth of results which have been generalised from the classical setting to compact Hausdorff spaces. However, even though there are explicit constructions for low discrepancy sequences in the classical setting such as the Halton sequence, there are no concrete constructions of uniformly distributed sequences over compact groups. Hence, currently one cannot approximate integrals defined over such abstract spaces via quasi-Monte Carlo methods.

Therefore, my goal during this course of study is to extend the quasi-Monte Carlo method to a more abstract setting. Specifically, I plan to develop constructions of uniformly distributed sequences in compact groups such as the orthogonal group which is the group of all (distance-preserving) transformations of Euclidean space. I will arrive at specific examples to computationally implement in quasi-Monte Carlo methods to approximate numerical integrals defined over these groups. In addition, I will need to form a concept of discrepancy since it is not yet clear what this should be in the abstract sense. This will be used to analyse the error values between the approximation and exact integral value.