分形测度的离散逼近与最优分划的几何结构

The discrete approximation problem for measures is to approximate a given probability measure with discrete probability measures of finite support in Wasserstein metrics. The aim of discrete approximation is to help people to treat real-world problems by using computers. In this project, we will deeply investigate the geometric structure of optimal discrete measures; based on this, we study the asymptotic uniformity of the contributions made by each supporting point. Once such uniformity is confirmed, people will be able to search for asymptotic optimal measures more conveniently. It is known that, scaling rates in discrete approximations are closely connected with various fractal dimensions. Therefore, we will explore the connections between the quantization problem of a measure and its fractal properties. In particular, we will also study self-affine measures, Moran measures and invariant measures of condensation systems. As integrals for a discrete measure is usually easy to estimate, this project will hopefully lead to significant extensions of results in numerical integrations.

测度的离散逼近问题是用有限支撑的离散测度在Wasserstein度量下逼近给定的测度。离散逼近的目的是方便人们利用计算机技术处理实际问题。本课题拟深入研究最佳逼近的离散测度的支撑点集的几何结构，由此分析各个支撑点对离散逼近误差的贡献的渐近均匀性。一旦此种均匀性得到证实，人们将能更方便地寻找渐近的最佳逼近测度，而渐近最佳逼近测度亦能反映出原测度的分布性质。研究结果表明，离散逼近的标度率与分形中各种维数密切相关，因此我们亦将探索测度的离散逼近与其局部性态及其支撑的分形性质之间的本质关系。作为应用，我们将集中研究莫朗测度、自仿测度以及凝聚系统对应的不变测度的离散逼近问题。由于离散测度的积分容易估算，本课题的研究结果在测度积分的数值计算等数学课题中的应用极有希望得到拓展。