Abstract
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 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 multicomponent multistate 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.