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。在这些积分中，存在不能精确计算的积分，可能最常见的原因是函数的反导数不存在。因此，需要数值方法来近似这些计算，同时努力使近似误差小到积分值的实际值。蒙特卡罗积分法是这些数值方法中的一种，它将积分近似为在随机选择的一组点上计算的被积函数的平均值。相反，也存在准蒙特卡罗积分法，它通过在本质上是确定的而不是随机的点处平均被积分值来逼近积分。因此，问题产生于询问这组确定性点应该具有什么性质以确保最佳地逼近确切的积分值。著名的Koksma-Hlawka不等式指出，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有界。这两个因素可以单独研究，但我感兴趣的是量化序列分布均匀程度的术语，即序列的差异。这确保了准蒙特卡罗方法和均匀分布之间的联系。具体地说，该方法需要低偏差的序列以减少误差。经典背景下的均匀分布理论涉及d维单位立方中实数序列分布的不规则性。这一点以及一般空间的理论抽象概念和结果都是非常好理解的。例如，已有大量的结果已从经典环境推广到紧Hausdorff空间。然而，即使在经典的情形下，如Halton序列，已经有了低偏差序列的显式构造，但紧群上的一致分布序列没有具体的构造。因此，目前还不能用拟蒙特卡罗方法来逼近定义在这种抽象空间上的积分。因此，我在这个研究过程中的目标是将准蒙特卡罗方法扩展到更抽象的环境。具体地说，我计划在紧群中发展均匀分布序列的构造，例如正交群，它是欧氏空间的所有(保持距离的)变换的群。我将给出用准蒙特卡罗方法计算实现的具体例子，以近似定义在这些群上的数值积分。此外，我还需要形成一个差异的概念，因为还不清楚这应该是什么抽象意义上的差异。这将用于分析近似积分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.