Local Fourier analysis is a commonly used tool to assess the quality and aid in the construction of geometric multigrid methods for translationally invariant operators. In this paper we automate the process of local Fourier analysis and present a framework that can be applied to arbitrary, including non-orthogonal, repetitive structures. To this end we introduce the notion of crystal structures and a suitable definition of corresponding wave functions, which allow for a natural representation of almost all translationally invariant operators that are encountered in applications, e.g., discretizations of systems of PDEs, tight-binding Hamiltonians of crystalline structures, colored domain decomposition approaches and last but not least two- or multigrid hierarchies. Based on this definition we are able to automate the process of local Fourier analysis both with respect to spatial manipulations of operators as well as the Fourier analysis back-end. This automation most notably simplifies the user input by removing the necessity for compatible representations of the involved operators. Each individual operator and its corresponding structure can be provided in any representation chosen by the user.

中文翻译：

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自动局部傅里叶分析 (aLFA)

局部傅里叶分析是一种常用的工具，用于评估质量并帮助构建平移不变算子的几何多重网格方法。在本文中，我们自动化了局部傅立叶分析的过程，并提出了一个框架，该框架可以应用于任意的，包括非正交的重复结构。为此，我们引入了晶体结构的概念和相应波函数的合适定义，这允许对应用中遇到的几乎所有平移不变算子进行自然表示，例如 PDE 系统的离散化、紧束缚哈密顿量晶体结构、彩色域分解方法以及最后但并非最不重要的两个或多重网格层次结构。基于这个定义，我们能够自动化局部傅里叶分析过程，包括操作符的空间操作以及傅里叶分析后端。这种自动化最显着地简化了用户输入，因为不需要相关运算符的兼容表示。每个单独的运算符及其相应的结构可以以用户选择的任何表示形式提供。