A numerical primer in 2D stochastic homogenization: CLT scaling in the Representative Volume Element

Venera Khoromskaia, Boris N. Khoromskij, and Felix Otto

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Submission date: 03. Aug. 2017 (revised version: March 2018)
Pages: 30
Bibtex
MSC-Numbers: 65F30, 65F50, 65F10, 65N35
Keywords and phrases: stochastic homogenization, Representative Volume Element, elliptic problem solvers, PCG iteration, empirical variance, kronecker products
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Abstract:
We describe the numerical scheme for the discretization and solution of 2D elliptic equations with strongly varying piecewise constant coefficients arising in stochastic homogenization. An efficient stiffness matrix generation scheme based on assembling of the local Kronecker product matrices is introduced. Spectral properties of the discrete stochastic operators are studied by estimation of the density of spectrum for the family of stochastic realizations. The resulting large linear systems of equations are solved by the preconditioned CG iteration with the convergence rate that is independent of the grid size and the variation in jumping coefficients (contrast). The numerical analysis on the convergence rates in stochastic homogenization theory is provided. This includes the calculation of the homogenized coefficient matrix and the subsequent estimation on the limit of empirical average/variance. The Central Limit Theorem (CLT) scaling in the size of Representative Volume Element L to derive the homogenized coefficient, rigorously established in [2], is reproduced by the numerical experiments. The proposed tensor-based numerical method allows to compute descriptive series of stochastic realizations for a large size of the representative volume element, L, using MATLAB on a moderate computer cluster. The tensor-based numerical scheme can be extended to a 3D case.