准晶结构的数学理论研究和高效算法设计

Quasicrystals are important ordered structures in materials. Their theoretical studies are important both in science and in mathematics. Since the lack of translational symmetry, the theoretical studies of quasicrystals cannot be reduced to a finite domain. It makes the traditional theories and numerical methods not be applicable for the study of quasicrystals. An important mathematical tool to study quasicrystals is the cut-and-project method which can be mainly used to generate the quasi-lattices. However, this approach does not provide an accurate representation of quasicrystals. Meanwhile, there are not reliable and efficient numerical methods to compute scientific problems for quasicrystals. Based on the idea of high-dimensional projection and the frame theory, the project will study the mathematical theory of quasicrystals in a suitable function space under an appropriate norm. Within the new developed mathematical framework, we will design efficient methods for computing various quasicrystals, and establish corresponding numerical analysis theory. The main research context of this project includes the study of the accurate mathematical representation, the stable decomposition and reconstruction system, and the development of reliable, stable, and efficient numerical methods for quasicrystals. We will also apply the developed theories and methods to the computation of quasicrystals in related physical models. After completing this program, we will provide an accurate mathematical representation and efficient numerical methods for studying quasicrystals.

准晶结构是材料中一类重要的有序结构，其理论研究既是重要的科学问题，也是重要的数学问题。准晶结构是缺乏平移对称性的一种全空间结构，其理论研究不能约化到一个有限的区域内，这使得许多传统的理论和算法都不适用于准晶结构的研究。研究准晶结构的一个重要数学工具是切割-投影方法，用于生成准周期格点。但这类方法在数学上没有给出准晶结构函数的准确表示。同时，在计算关于准晶结构的科学问题中，目前尚没有高效可靠的计算方法。本项目将基于高维投影的思想，引入框架理论，在合适的函数空间和范数下，研究准晶结构的数学理论。在新发展的数学理论框架下，构造适合各类准晶结构计算的高效算法，并建立相应的数值分析理论。本项目将主要研究准晶结构的准确数学表示，稳定的分解和重构系统，发展可靠、稳定、高效的计算方法，并对相关物理模型进行数值求解。本项目的完成将为准晶结构的研究建立准确的数学表示以及高效的计算方法。

理论上，给出了任意维周期逼近算法的数学分析。进一步，我们的理论指出：在给定准周期函数，如何得到最佳逼近的周期函数。在算法上，给出了计算准周期系统的统一算法框架。我们研究了多元组分下三维二十面体等准晶结构稳定性，十二重对称性准晶和周期晶体的相变路径，以及准周期量子系统的计算。同时，结合微分方程数值格式和现代优化算法，给出了求解单序参量和多序参量郎道模型稳态结构的高效算法，能够极大地提高迭代效率，并给出了算法的收敛性。基于这套算法，我们给出了自适应计算相图的软件。