对称快速傅里叶变换的数学基础

中文摘要 AI 辅助

本研究针对具有晶体对称性的离散数据的快速傅里叶变换（FFT）展开探讨，标准FFT算法在此类场景下会产生大量冗余计算。我们建立了n维晶体群的对称快速傅里叶变换（SFFT）的严格数学理论，并提出SFFT与对称逆快速傅里叶变换（SIFFT）算法。我们推导了线性与平移对称性的对称约化公式，并将其统一为递归框架；分析了算术复杂度以得到相对于标准FFT的理论加速因子，还通过对代表性三维空间群的数值实验验证了所提方法的准确性与效率。

英文摘要

In this work, we study the fast Fourier transform (FFT) of discrete data with crystallographic symmetry, for which standard FFT algorithms incur substantial redundant computation. We establish a rigorous mathematical theory of the symmetric fast Fourier transform (SFFT) for $n$-dimensional crystallographic groups and propose the SFFT and symmetric inverse fast Fourier transform (SIFFT) algorithms. We derive symmetry-reduction formulas for both linear and translational symmetries and unify them into a recursive framework. We analyze the arithmetic complexity to obtain theoretical speedup factors relative to the standard FFT, and we confirm the accuracy and efficiency of the proposed methods by numerical experiments on representative three-dimensional space groups.

发表机构

• Hunan Key Laboratory for Computation and Simulation in Science and Engineering, Key Laboratory of Intelligent Computing and Information Processing of Ministry of Education, School of Mathematics and Computational Science, Xiangtan University（湘潭大学数学与计算科学学院）
• Wuhan Institute for Math & AI, Wuhan University（武汉大学武汉数学与人工智能研究院）

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