Accurately recover global quasiperiodic systems by finite points

TL;DR

This paper introduces a novel finite points recovery (FPR) algorithm that accurately reconstructs both smooth and non-smooth quasiperiodic systems by leveraging a homomorphism to a higher-dimensional torus and strategic finite point selection.

Contribution

The paper presents a new FPR method that avoids dimensional lifting and improves recovery accuracy for non-smooth quasiperiodic systems, supported by theoretical analysis and numerical validation.

Findings

FPR effectively recovers smooth quasiperiodic functions.

FPR accurately reconstructs non-smooth Fibonacci quasicrystals.

The method outperforms existing spectral approaches in non-smooth cases.

Abstract

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.