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Due to quasicrystals having long-range orientational order but withouttranslational symmetry, traditional numerical methods usually suffer whenapplied as is. In the past decade, the projection method has emerged as aprominent solver for quasiperiodic problems. Transforming them into a higherdimensional but periodic ones, the projection method facilitates theapplication of the fast Fourier transform. However, the computationalcomplexity inevitably becomes high which significantly impedes e.g. thegeneration of the phase diagram since a high-fidelity simulation of a problemwhose dimension is doubled must be performed for numerous times.  To address the computational challenge of quasiperiodic problems based on theprojection method, this paper proposes a multi-component multi-state reducedbasis method (MCMS-RBM). Featuring multiple components with each providingreduction functionality for one branch of the problem induced by one part ofthe parameter domain, the MCMS-RBM does not resort to the parameter domainconfigurations (e.g. phase diagrams) a priori. It enriches each component in agreedy fashion via a phase-transition guided exploration of the multiple statesinherent to the problem. Adopting the empirical interpolation method, theresulting online-efficient method vastly accelerates the generation of adelicate phase diagram to a matter of minutes for a parametrized two-turn-fourdimensional Lifshitz-Petrich model with two length scales. Moreover, itfurnishes surrogate and equally accurate field variables anywhere in theparameter domain.

MCMS-RBM: Multi-Component Multi-State Reduced Basis Method toward Efficient Transition Pathway Identification for Crystals and Quasicrystals