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

Abstract

In this paper, we propose and analyze the numerical algorithms for fast solution of periodic elliptic problems in random media in R d , d = 2, 3. We consider the stochastic realizations using checkerboard configuration of the equation coefficients built on a large L × L × 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 ergodic processes, 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.

Representative volume elements for matrix-inclusion composites -- a computational study on periodizing the ensemble

“…From the viewpoint of mathematical analysis, this article is a natural continuation of previous work by Khoromskaia et al [26,27], where quantitative homogenization results for random media were confirmed by numerical simulations. Indeed, we go beyond the cited work by investigating microstructure models of higher complexity, as is required for engineering applications.…”

“…We carry out numerical experiments to quantify the systematic and the random error for composite microstructures. In contrast to previous works [26,27], we use microstructures of industrial relevance and complexity, utilizing real physical parameters, investigating spherical and cylindrical fillers. While the ensembles considered here do not fall into the class considered in the theory -they are neither finite range nor a suitable functional inequality is known, see, however, Duerinckx & Gloria [28,29] -we expect that they belong to the same universality class.…”

“…Indeed, the latter images naturally correspond to snapshots of ensembles with "improperly" treated particles at the boundaries. The effort of extracting such a specimen precludes computing the apparent properties corresponding to this cell size, as the necessary number of samples, which is typically on the order of 10 5 for engineering accuracy [26,27], is both too expensive and excessively arduous. For this purpose, we prepare a similar protocol under "laboratory conditions".…”

SOTT: Greedy Approximation of a Tensor as a Sum of Tensor Trains

“…We refer the reader to [14] and [18] for extensive reviews on tensor theory and extended analysis of tensor decompositions and their numerous applications. Tensor formats are also used for solving time-dependent and stochastic/parametric PDEs ( [16], [1]). Other families of methods called Proper Generalized Decomposition (PGD) methods use tensor decomposition in high dimensional problems [26], [25].…”