  A recently introduced representation by a set of Wang tiles -- a
generalization of the traditional Periodic Unit Cell based approach -- serves
as a reduced geometrical model for materials with stochastic heterogeneous
microstructure, enabling an efficient synthesis of microstructural
realizations. To facilitate macroscopic analyses with a fully resolved
microstructure generated with Wang tiles, we develop a reduced order modelling
scheme utilizing pre-computed characteristic features of the tiles. In the
offline phase, inspired by the computational homogenization, we extract
continuous fluctuation fields from the compressed microstructural
representation as responses to generalized loading represented by the first-
and second-order macroscopic gradients. In the online phase, using the ansatz
of the Generalized Finite Element Method, we combine these fields with a coarse
finite element discretization to create microstructure-informed reduced modes
specific for a given macroscopic problem. Considering a two-dimensional scalar
elliptic problem, we demonstrate that our scheme delivers less than a 3% error
in both the relative $L_2$ and energy norms with only 0.01% of the unknowns
when compared to the fully resolved problem. Accuracy can be further improved
by locally refining the macroscopic discretization and/or employing more
pre-computed fluctuation fields. Finally, unlike the standard snapshot-based
reduced-order approaches, our scheme handles significant changes in the
macroscopic geometry or loading without the need for recalculating the offline
phase, because the fluctuation fields are extracted without any prior knowledge
on the macroscopic problem.
