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DTSTAMP:20210922T053201Z
UID:1610456400@ist.ac.at
DTSTART:20210112T140000
DTEND:20210112T153000
DESCRIPTION:Speaker: Nicolas Clozeau\nhosted by Julian Fischer\nAbstract: W
e consider a model problem of diffusion or conductivity in a random medium
(for instance\, an heterogeneous domain obtained by mixing randomly two d
ifferent phases\, one being the matrix and the other the inclusions). The
model that we use takes the form of an uniformly elliptic equation with hi
gh oscillatory (at scale S<<1) and random coefficient. Since the 70’ wit
h the work of Kozlov\, Papanicolaou and Varadhan\, it is well known that t
he solution\, provided the law of the coefficients is stationary and ergod
ic\, can be approximated by its two-scale expansion: that is a first-order
expansion in S taking account of the oscillation at scale S. The zero-ord
er term corresponds to the limit as S goes to zero and solves a nice ellip
tic PDE with deterministic constant coefficients (called the homogenized c
oefficients) for which we have an explicit formula. However\, the homogeni
zed coefficients depend on the so-called first-order corrector which solve
s an elliptic PDE posed in the whole space and for which numerical computa
tions are out of reach. The goal of this work is to analyse the approximat
ion of the homogenized coefficients by the representative volume element m
ethod\, that is when we replace the whole space by a large torus of size L
. This operation makes the corrector equation easy to solve numerically an
d provides a natural approximation of the homogenized coefficients by the
ones coming from the periodic homogenization theory. In the analysis of th
e error that we make in this approximation\, the variance suffers from two
types of error: a random one (that is the fluctuation around its expectat
ion) and a systematic one (that is the difference between the homogenized
coefficients and the expectation of the approximation). We focus in this w
ork on the systematic error and we characterize its asymptotic behaviour a
s L goes to infinity. We show that\, in the particular case where the law
is generated by a stationary Gaussian field\, the asymptotic of the system
atic error is characterized by a deterministic matrix depending on the fir
st and second-order correctors\, the gradient of the covariance function a
nd a fourth-order tensor involving the whole space Green function of the h
omogenized elliptic operator.
LOCATION:online via Zoom\, IST Austria
ORGANIZER:birgit.oosthuizen-noczil@ist.ac.at
SUMMARY:Nicolas Clozeau: Asymptotics of the systematic error in stochastic
homogenization
URL:https://talks-calendar.app.ist.ac.at/events/3002
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