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

数学应用中的许多问题都可以归结为多维积分的计算。其中，存在不能精确求值的积分，最常见的原因可能是函数的不定积分不存在。因此，需要数值方法来近似这些计算，同时力求对积分的实际值有一个小的近似误差。蒙特卡罗积分法是一种数值方法，它将积分近似为随机选择的一组点的被积量的平均值。与此相反，也存在准蒙特卡罗积分方法，该方法通过在本质上是确定性的而不是随机的点上取被积值的平均值来逼近积分。因此，问题就产生了，这组确定性点应该具有什么性质，以确保最接近确切的积分值。值得注意的“Koksma- Hlawka不等式”表明，QMC方法的近似值与精确积分值之间的误差可以由两个独立因素的乘积限定。这两个因素都可以单独研究，但我感兴趣的是量化序列分布的好坏的术语，即序列的差异。这保证了准蒙特卡罗方法与均匀分布之间的联系。特别地，该方法需要低差异序列以减小误差。经典条件下的均匀分布理论研究的是d维单位立方体中实数序列分布的不规则性。这一点，以及理论上的抽象概念和一般空间的结果都是很容易理解的。例如，存在大量的结果，这些结果已经从经典环境推广到紧化的Hausdorff空间。然而，尽管在Halton序列等经典环境中存在低差序列的明确构造，但在紧群上却没有均匀分布序列的具体构造。因此，目前还不能用拟蒙特卡罗方法逼近在这些抽象空间上定义的积分。因此，在这门课程的学习中，我的目标是将拟蒙特卡罗方法扩展到更抽象的环境中。具体地说，我计划发展紧群中均匀分布序列的构造，例如正交群，它是欧几里德空间中所有（距离保持）变换的群。我将给出具体的例子，用拟蒙特卡罗方法计算实现在这些群上定义的近似数值积分。此外，我还需要形成一个差异的概念，因为目前还不清楚这在抽象意义上应该是什么。这将用于分析近似值和精确积分值之间的误差值。

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.