  In this paper, we propose and analyze the numerical algorithms for fast
solution of periodic elliptic problems in random media in $\mathbb{R}^d$,
$d=2,3$. We consider the stochastic realizations using checkerboard
configuration of the equation coefficients built on a large $L \times L \times
L$ lattice, where $L$ is the size of representative volume elements. The
Kronecker tensor product scheme is introduced for fast generation of the
stiffness matrix for FDM discretization on a tensor grid. We describe tensor
techniques for the construction of the low Kronecker rank spectrally equivalent
preconditioner in periodic setting to be used in the framework of PCG
iteration. In our construction the diagonal matrix of the discrete Laplacian
inverse represented in the Fourier basis is reshaped into a 3D tensor, which is
then approximated by a low-rank canonical tensor, calculated by the multigrid
Tucker-to-canonical tensor transform. The FDM discretization scheme on a tensor
grid is described in detail, and the computational characteristics in terms of
$L$ for the 3D Matlab implementation of the PCG iteration are illustrated. The
present work continues the developments in [22], where the numerical primer to
study the asymptotic convergence rate vs. $L$ for the homogenized matrix for 2D
elliptic PDEs with random coefficients was investigated numerically. The
presented elliptic problem solver can be applied for calculation of long
sequences of stochastic realizations in numerical analysis of 3D stochastic
homogenization problems, for solving 3D quasi-periodic geometric homogenization
problems, as well as in the numerical simulation of dynamical many body
interaction processes and multi-particle electrostatics.
