Quasiperiodic systems, related to irrational numbers, are space-filling structures without decay nor translation invariance. How to accurately recover these systems, especially for non-smooth cases, presents a big challenge in numerical computation. In this paper, we propose a new algorithm, finite points recovery (FPR) method, which is available for both smooth and non-smooth cases, to address this challenge. The FPR method first establishes a homomorphism between the lower-dimensional definition domain of the quasiperiodic function and the higher-dimensional torus, then recovers the global quasiperiodic system by employing 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 FPR approach in recovering both smooth quasiperiodic functions and piecewise constant Fibonacci quasicrystals. While existing spectral methods encounter difficulties in accurately recovering non-smooth quasiperiodic functions.

翻译：准周期系统与无理数相关，是空间填充结构，既无衰减性也无平移不变性。如何精确恢复这些系统（尤其是非光滑情形）是数值计算中的重大挑战。本文提出一种适用于光滑与非光滑情形的新算法——有限点恢复（FPR）方法以应对此挑战。FPR方法首先建立准周期函数低维定义域与高维环面之间的同态映射，随后在定义域内通过有限点插值技术实现全局准周期系统的恢复，无需进行维度提升。进一步地，我们根据无理数的算术性质发展出精确高效的有限点选取策略，并给出关于有限点选取的相应数学理论、收敛性分析及计算复杂度分析。数值实验表明，FPR方法在恢复光滑准周期函数与分段常数Fibonacci准晶时均展现出有效性与优越性，而现有谱方法在精确恢复非光滑准周期函数时面临困难。