Felix Otto

Antoine Gloria, Stefan Neukamm, Felix Otto

We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the $L^2$-norm in probability of the \mbox{$H^1$-norm} in space of this error scales like $ε$, where $ε$ is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Green's function by Marahrens and the third author.

Venera Khoromskaia, Boris N. Khoromskij, Felix Otto

We describe the numerical scheme for the discretization and solution of 2D elliptic equations with strongly varying piecewise constant coefficients arising in the stochastic homogenization of multiscale composite materials. An efficient stiffness matrix generation scheme based on assembling the local Kronecker product matrices is introduced. The resulting large linear systems of equations are solved by the preconditioned CG iteration with a convergence rate that is independent of the grid size and the variation in jumping coefficients (contrast). Using this solver we numerically investigate the convergence of the Representative Volume Element (RVE) method in stochastic homogenization that extracts the effective behavior of the random coefficient field. Our numerical experiments confirm the asymptotic convergence rate of systematic error and standard deviation in the size of RVE rigorously established in [6]. The asymptotic behavior of covariances of the homogenized matrix in the form of a quartic tensor is also studied numerically. Our approach allows laptop computation of sufficiently large number of stochastic realizations even for large sizes of the RVE.

Jianfeng Lu, Felix Otto

We are given a uniformly elliptic coefficient field that we regard as a realization of a stationary and finite-range (say, range unity) ensemble of coefficient fields. Given a (deterministic) right-hand-side supported in a ball of size $\ell\gg 1$ and of vanishing average, we are interested in an algorithm to compute the (gradient of the) solution near the origin, just using the knowledge of the (given realization of the) coefficient field in some large box of size $L\gg\ell$. More precisely, we are interested in the most seamless (artificial) boundary condition on the boundary of the computational domain of size $L$. Motivated by the recently introduced multipole expansion in random media, we propose an algorithm. We rigorously establish an error estimate (on the level of the gradient) in terms of $L\gg\ell\gg 1$, using recent results in quantitative stochastic homogenization. More precisely, our error estimate has an a priori and an a posteriori aspect: With a priori overwhelming probability, the (random) prefactor can be bounded by a constant that is computable without much further effort, on the basis of the given realization in the box of size $L$. We also rigorously establish that the order of the error estimate in both $L$ and $\ell$ is optimal, where in this paper we focus on the case of $d=2$. This amounts to a lower bound on the variance of the quantity of interest when conditioned on the coefficients inside the computational domain, and relies on the deterministic insight that a sensitivity analysis wrt a defect commutes with (stochastic) homogenization. Finally, we carry out numerical experiments that show that this optimal convergence rate already sets in at only moderately large $L$, and that more naive boundary conditions perform worse both in terms of rate and prefactor.

Antoine Gloria, Felix Otto

In the present contribution we establish quantitative results on the periodic approximation of the corrector equation for the stochastic homogenization of linear elliptic equations in divergence form, when the diffusion coefficients satisfy a spectral gap estimate in probability, and for $d>2$. The main difference with respect to the first part of [Gloria-Otto, arXiv:1409.0801] is that we avoid here the use of Green's functions and more directly rely on the De Giorgi-Nash-Moser theory.