Mathematical Sciences: Algorithms and Programs for Fourier Transforms and Crystallography

这项探索性研究小额赠款将支持快速傅里叶变换算法的一般数学结构的数学研究。重点是计算一般情况下的离散傅里叶变换。将应用于具有晶体对称性的数据。并将对相关的快速卷积算法进行研究。将产生数据转换和排列算法。这些对于在傅里叶变换的各个阶段之间转换数据以及对输入和输出的数据进行化简和扩展是必要的。算法将适应机器架构，包括那些向量，并行和分层存储机器。他们将在大规模计算中考虑数据流、数据排列和数据访问。群体理论方法将在这一努力中发挥重要作用。这项工作的目标是对所有晶体学群的对称性进行系统的分类。算法和程序的结晶学计算的其他部分，特别是细化问题，将产生。在一个大型的通用晶体学程序包中实现这项工作，并计划报告效率。

This Small Grant for Exploratory Research will support mathematical research on general mathematical structures for fast Fourier transforms algorithms. The focus will be on computing the discrete Fourier transform for the general case. Applications to data possessing crystallographic symmetries will be made. Studies on related fast convolution algorithms will also be made. Data transformation and permutation algorithms will be produced. These be necessary for transforming data between stages of the Fourier transforms and for reducing and expanding data for input and output. Algorithms will be adapted to machine architectures including those of vector, parallel and hierarchical memory machines. They will take account of data flow, data permutations and data access in large scale computations. Group theoretical methods will play an important part in this effort. Included in the work is the goal of a systematic classification of the symmetries of all crystallographic groups. Algorithms and programs for others parts of the crystallographic calculation, particularly the refinement problem, will be produced. Implementation of this work in one large general crystallographic program package and report on the efficiency is planned.