24,883 research outputs found
Homogenization of a singular random one dimensional parabolic PDE with time varying coefficients
The paper studies homogenization problem for a non-autonomous parabolic
equation with a large random rapidly oscillating potential in the case of one
dimensional spatial variable. We show that if the potential is a statistically
homogeneous rapidly oscillating function of both temporal and spatial variables
then, under proper mixing assumptions, the limit equation is deterministic and
the convergence in probability holds. To the contrary, for the potential having
a microstructure only in one of these variables, the limit problem is
stochastic and we only prove the convergence in law
Homogenization of lateral diffusion on a random surface
We study the problem of lateral diffusion on a static, quasi-planar surface
generated by a stationary, ergodic random field possessing rapid small-scale
spatial fluctuations. The aim is to study the effective behaviour of a particle
undergoing Brownian motion on the surface viewed as a projection on the
underlying plane. By formulating the problem as a diffusion in a random medium,
we are able to use known results from the theory of stochastic homogenization
of SDEs to show that, in the limit of small scale fluctuations, the diffusion
process behaves quantitatively like a Brownian motion with constant diffusion
tensor . While will not have a closed-form expression in general, we are
able to derive variational bounds for the effective diffusion tensor, and using
a duality transformation argument, obtain a closed form expression for in
the special case where is isotropic. We also describe a numerical scheme
for approximating the effective diffusion tensor and illustrate this scheme
with two examples.Comment: 25 pages, 7 figure
The corrector in stochastic homogenization: optimal rates, stochastic integrability, and fluctuations
We consider uniformly elliptic coefficient fields that are randomly
distributed according to a stationary ensemble of a finite range of dependence.
We show that the gradient and flux of the
corrector , when spatially averaged over a scale decay like the
CLT scaling . We establish this optimal rate on the level of
sub-Gaussian bounds in terms of the stochastic integrability, and also
establish a suboptimal rate on the level of optimal Gaussian bounds in terms of
the stochastic integrability. The proof unravels and exploits the
self-averaging property of the associated semi-group, which provides a natural
and convenient disintegration of scales, and culminates in a propagator
estimate with strong stochastic integrability. As an application, we
characterize the fluctuations of the homogenization commutator, and prove sharp
bounds on the spatial growth of the corrector, a quantitative two-scale
expansion, and several other estimates of interest in homogenization.Comment: 114 pages. Revised version with some new results: optimal scaling
with nearly-optimal stochastic integrability on top of nearly-optimal scaling
with optimal stochastic integrability, CLT for the homogenization commutator,
and several estimates on growth of the extended corrector, semi-group
estimates, and systematic error
Multi-scale homogenization with bounded ratios and Anomalous Slow Diffusion
We show that the effective diffusivity matrix for the heat operator
in a periodic potential
obtained as a superposition of Holder-continuous
periodic potentials (of period \T^d:=\R^d/\Z^d, ,
) decays exponentially fast with the number of scales when the
scale-ratios are bounded above and below. From this we deduce the
anomalous slow behavior for a Brownian Motion in a potential obtained as a
superposition of an infinite number of scales: Comment: 29 pages, 1 figure, submitted versio
Sublinear growth of the corrector in stochastic homogenization: Optimal stochastic estimates for slowly decaying correlations
We establish sublinear growth of correctors in the context of stochastic
homogenization of linear elliptic PDEs. In case of weak decorrelation and
"essentially Gaussian" coefficient fields, we obtain optimal (stretched
exponential) stochastic moments for the minimal radius above which the
corrector is sublinear. Our estimates also capture the quantitative
sublinearity of the corrector (caused by the quantitative decorrelation on
larger scales) correctly. The result is based on estimates on the Malliavin
derivative for certain functionals which are basically averages of the gradient
of the corrector, on concentration of measure, and on a mean value property for
-harmonic functions
A jigsaw puzzle framework for homogenization of high porosity foams
An approach to homogenization of high porosity metallic foams is explored.
The emphasis is on the \Alporas{} foam and its representation by means of
two-dimensional wire-frame models. The guaranteed upper and lower bounds on the
effective properties are derived by the first-order homogenization with the
uniform and minimal kinematic boundary conditions at heart. This is combined
with the method of Wang tilings to generate sufficiently large material samples
along with their finite element discretization. The obtained results are
compared to experimental and numerical data available in literature and the
suitability of the two-dimensional setting itself is discussed.Comment: 11 pages, 7 figures, 3 table
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