SIAM Journal on Numerical Analysis, Volume 62, Issue 4, Page 1713-1735, August 2024.
Abstract. Quasiperiodic systems, related to irrational numbers, are space-filling structures without decay or translation invariance. How to accurately recover these systems, especially for low-regularity cases, presents a big challenge in numerical computation. In this paper, we propose a new algorithm, the finite points recovery (FPR) method, which is available for both continuous and low-regularity cases, to address this challenge. The FPR method first establishes a homomorphism between the lower-dimensional definition domain of quasiperiodic function and the higher-dimensional torus, and then recovers the global quasiperiodic system by employing an interpolation technique with finite points in the definition domain without dimensional lifting. Furthermore, we develop accurate and efficient strategies of selecting finite points according to the arithmetic properties of irrational numbers. The corresponding mathematical theory, convergence analysis, and computational complexity analysis on choosing finite points are presented. Numerical experiments demonstrate the effectiveness and superiority of the FPR approach in recovering both continuous quasiperiodic functions and piecewise constant Fibonacci quasicrystals while existing spectral methods encounter difficulties in recovering piecewise constant quasiperiodic functions.

中文翻译：

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通过有限点精确恢复全局准周期系统


《SIAM 数值分析杂志》，第 62 卷，第 4 期，第 1713-1735 页，2024 年 8 月。

抽象的。与无理数相关的准周期系统是没有衰变或平移不变性的空间填充结构。如何准确地恢复这些系统，特别是对于低规律性情况，对数值计算提出了巨大的挑战。在本文中，我们提出了一种新算法，即有限点恢复（FPR）方法，该方法可用于连续和低规律性情况，以应对这一挑战。 FPR方法首先在准周期函数的低维定义域和高维环面之间建立同态，然后通过在定义域内采用有限点插值技术而不进行降维来恢复全局准周期系统。此外，我们根据无理数的算术性质开发了准确有效的选择有限点的策略。给出了相应的数学理论、收敛性分析以及有限点选择的计算复杂度分析。数值实验证明了FPR方法在恢复连续准周期函数和分段常数斐波那契准晶体方面的有效性和优越性，而现有谱方法在恢复分段常数准周期函数方面遇到了困难。