  In this paper, we assess the performance of four iterative algorithms for
solving non-symmetric rank-deficient linear systems arising in the FFT-based
homogenization of heterogeneous materials defined by digital images. Our
framework is based on the Fourier-Galerkin method with exact and approximate
integrations that has recently been shown to generalize the Lippmann-Schwinger
setting of the original work by Moulinec and Suquet from 1994. It follows from
this variational format that the ensuing system of linear equations can be
solved by general-purpose iterative algorithms for symmetric positive-definite
systems, such as the Richardson, the Conjugate gradient, and the Chebyshev
algorithms, that are compared here to the Eyre-Milton scheme - the most
efficient specialized method currently available. Our numerical experiments,
carried out for two-dimensional elliptic problems, reveal that the Conjugate
gradient algorithm is the most efficient option, while the Eyre-Milton method
performs comparably to the Chebyshev semi-iteration. The Richardson algorithm,
equivalent to the still widely used original Moulinec-Suquet solver, exhibits
the slowest convergence. Besides this, we hope that our study highlights the
potential of the well-established techniques of numerical linear algebra to
further increase the efficiency of FFT-based homogenization methods.
