23 research outputs found
Correlation structure of the corrector in stochastic homogenization
Recently, the quantification of errors in the stochastic homogenization of
divergence-form operators has witnessed important progress. Our aim now is to
go beyond error bounds, and give precise descriptions of the effect of the
randomness, in the large-scale limit. This paper is a first step in this
direction. Our main result is to identify the correlation structure of the
corrector, in dimension and higher. This correlation structure is similar
to, but different from that of a Gaussian free field.Comment: Published at http://dx.doi.org/10.1214/15-AOP1045 in the Annals of
Probability (http://www.imstat.org/aop/) by the Institute of Mathematical
Statistics (http://www.imstat.org
Corrector Analysis of a Heterogeneous Multi-scale Scheme for Elliptic Equations with Random Potential
This paper analyzes the random fluctuations obtained by a heterogeneous
multi-scale first-order finite element method applied to solve elliptic
equations with a random potential. We show that the random fluctuations of such
solutions are correctly estimated by the heterogeneous multi-scale algorithm
when appropriate fine-scale problems are solved on subsets that cover the whole
computational domain. However, when the fine-scale problems are solved over
patches that do not cover the entire domain, the random fluctuations may or may
not be estimated accurately. In the case of random potentials with short-range
interactions, the variance of the random fluctuations is amplified as the
inverse of the fraction of the medium covered by the patches. In the case of
random potentials with long-range interactions, however, such an amplification
does not occur and random fluctuations are correctly captured independent of
the (macroscopic) size of the patches.
These results are consistent with those obtained by the authors for more
general equations in the one-dimensional setting and provide indications on the
loss in accuracy that results from using coarser, and hence less
computationally intensive, algorithms
Pointwise two-scale expansion for parabolic equations with random coefficients
We investigate the first-order correction in the homogenization of linear
parabolic equations with random coefficients. In dimension and higher and
for coefficients having a finite range of dependence, we prove a pointwise
version of the two-scale expansion. A similar expansion is derived for elliptic
equations in divergence form. The result is surprising, since it was not
expected to be true without further symmetry assumptions on the law of the
coefficients.Comment: 25 pages. Minor revisions, to appear in PTR
Fluctuations of Parabolic Equations with Large Random Potentials
In this paper, we present a fluctuation analysis of a type of parabolic
equations with large, highly oscillatory, random potentials around the
homogenization limit. With a Feynman-Kac representation, the Kipnis-Varadhan's
method, and a quantitative martingale central limit theorem, we derive the
asymptotic distribution of the rescaled error between heterogeneous and
homogenized solutions under different assumptions in dimension . The
results depend highly on whether a stationary corrector exits.Comment: 44 pages; reorganized the structure and extended the results; to
appear in SPDE: Analysis and Computation