Due to quasicrystals having long-range orientational order but without translational symmetry, traditional numerical methods usually suffer when applied as is. In the past decade, the projection method has emerged as a prominent solver for quasiperiodic problems. Transforming them into a higher dimensional but periodic ones, the projection method facilitates the application of the fast Fourier transform. However, the computational complexity inevitably becomes high which significantly impedes e.g. the generation of the phase diagram since a high-fidelity simulation of a problem whose dimension is doubled must be performed for numerous times. To address the computational challenge of quasiperiodic problems based on the projection method, this paper proposes a multi-component multi-state reduced basis method (MCMS-RBM). Featuring multiple components with each providing reduction functionality for one branch of the problem induced by one part of the parameter domain, the MCMS-RBM does not resort to the parameter domain configurations (e.g. phase diagrams) a priori. It enriches each component in a greedy fashion via a phase-transition guided exploration of the multiple states inherent to the problem. Adopting the empirical interpolation method, the resulting online-efficient method vastly accelerates the generation of a delicate phase diagram to a matter of minutes for a parametrized two-turn-four dimensional Lifshitz-Petrich model with two length scales. Moreover, it furnishes surrogate and equally accurate field variables anywhere in the parameter domain.

翻译：由于准晶体具有长程取向序但缺乏平移对称性，传统数值方法通常难以直接应用。过去十年中，投影法已成为拟周期问题的主要求解工具。通过将问题转化为高维但周期性的形式，投影法促进了快速傅里叶变换的应用。然而，其计算复杂度不可避免地升高，尤其严重阻碍了相图的生成——因为必须对维度加倍的问题进行多次高保真模拟。为应对基于投影法的拟周期问题所面临的计算挑战，本文提出了一种多组分多状态降基方法（MCMS-RBM）。该方法包含多个组分，每个组分为参数域中某一部分所诱导的问题分支提供降维功能，无需预先依赖参数域构型（例如相图）。通过采用相变引导的多状态探索，MCMS-RBM以贪婪方式丰富每个组分。结合经验插值方法，由此产生的在线高效方法将参数化双转向四维Lifshitz-Petrich模型（含两个长度尺度）的精细相图生成时间大幅缩短至数分钟。此外，该方法还能在参数域任意位置提供等精度场变量的替代模型。