准晶结构投影算法的机理和数学分析

Quasicrystals are important space-filling ordered structures. How to simulate quasicrystals in an efficient way and evaluate their energy density to high precision is a fundamental problem for studying the stability and properties of quasicrystals. There have been some researches on computational methods for quasicrystals, but nothing on theoretical and fast algorithm researches. This project will focus on the mathematical research on the latest projection method for quasicrystals, and the development of efficient algorithms. We will study the mathematical features of the projective operator and the basis function of the projection method with different symmetry of quasicrystals, and analyze the overcomplete frame in projection method by frame theory. This project will develop a systematic and efficient numerical method by making use of the above theoretical analysis and the symmetry of quasicrystals. This numerical method system includes the fast algorithm for high-dimensional spatial discretization, the strategy of screening reasonable initial values based on the symmetric group, and the iterative method for optimizing the computational domain of quasicrystals adaptively. These developed theories and numerical methods will be applied to related physical problems, which could help us to promote the theoretical research and further optimize the numerical algorithms.

准晶结构是一类重要的全空间有序结构。如何高效地模拟出这种全空间的结构，高精度计算其能量密度是理论研究准晶结构稳定性及其性质的基础。目前在计算准晶结构的算法方面上已经有了一些研究，但缺乏相应的数学理论和高效算法研究。本项目将基于最新提出的计算准晶结构的投影算法，开展算法的数学理论研究，设计适合准晶结构计算的高效投影算法。主要研究投影算子和基函数在不同准晶结构对称性下的数学性质，引入框架理论，研究算法中过度完备化框架的数学性质。本项目将充分利用理论研究和准晶结构的对称性，发展一套系统、高效的投影算法来计算准晶结构。这套算法包括高维空间离散的快速算法，通过对称群初值选取策略，自适应优化结构计算区域的迭代法。本项目也将运用发展的理论和算法到相关的物理问题，进一步促进算法的理论研究和优化算法。

准晶结构是一类重要的全空间有序结构。本项目基于最新计算准晶结构的投影算法，开展算法的数学理论研究，设计适合准晶结构计算的高效投影算法，研究投影算子和基函数在不同准晶结构对称性下的数学性质。本项目充分利用理论研究和准晶结构的对称性，发展一套系统、高效的投影算法来计算准晶结构。这套算法包括高维空间离散的快速算法，通过对称群初值选取策略，自适应优化结构计算区域的迭代法。本项目将发展的理论和算法到两尺度粗粒化模型，数值结果表明了算法的优越性，也能和部分实验体系吻合。