  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.
