Tensor-based techniques for fast discretization and solution of 3D elliptic equations with random coefficients

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.

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