  Fast Fourier transform (FFT) based methods have turned out to be an effective
computational approach for numerical homogenisation. In particular,
Fourier-Galerkin methods are computational methods for partial differential
equations that are discretised with trigonometric polynomials. Their
computational effectiveness benefits from efficient FFT based algorithms as
well as a favourable condition number. Here these kind of methods are
accelerated by low-rank tensor approximation techniques for a solution field
using canonical polyadic, Tucker, and tensor train formats. This reduced order
model also allows to efficiently compute suboptimal global basis functions
without solving the full problem. It significantly reduces computational and
memory requirements for problems with a material coefficient field that admits
a moderate rank approximation. The advantages of this approach against those
using full material tensors are demonstrated using numerical examples for the
model homogenisation problem that consists of a scalar linear elliptic
variational problem defined in two and three dimensional settings with
continuous and discontinuous heterogeneous material coefficients. This approach
opens up the potential of an efficient reduced order modelling of large scale
engineering problems with heterogeneous material.
