138 research outputs found
Periodic homogenization of non-local operators with a convolution type kernel
The paper deals with homogenization problem for a non-local linear operator
with a kernel of convolution type in a medium with a periodic structure. We
consider the natural diffusive scaling of this operator and study the limit
behaviour of the rescaled operators as the scaling parameter tends to 0. More
precisely we show that in the topology of resolvent convergence the family of
rescaled operators converges to a second order elliptic operator with constant
coefficients. We also prove the convergence of the corresponding semigroups
both in space and the space of continuous functions, and show that for
the related family of Markov processes the invariance principle holds
Gibbs point field models for extraction problems in image analysis
International audienceThis paper is a review of some probabilistic methods, based on Gibbs fields theory, applied to solve image analysis tasks. We present the mathematical background and show different applications
Homogenization of biased convolution type operators
The final publication is available at IOS Press through http://dx.doi.org/10.3233/ASY-191533This paper deals with homogenization of parabolic problems for integral convolution type operators with a non-symmetric jump kernel in a periodic elliptic medium. It is shown that the homogenization result holds in moving coordinates. We determine the corresponding effective velocity and prove that the limit operator is a second order parabolic operator with constant coefficients. We also consider the behaviour of the effective velocity in the case of small antisymmetric perturbations of a symmetric kernel, in particular we show that the Einstein relation holds for the studied periodic environment
Mathematical multi-scale model of water purification
In this work, we consider a mathematical model of the water treatment process and determine the effective characteristics of this model. At the microscopic
length scale, we describe our model in terms of a lattice random walk in a
high-contrast periodic medium with absorption. Applying then the upscaling
procedure, we obtain the macroscopic model for total mass evolution. We discuss both the dynamic and the stationary regimes and show how the efficiency
of the purification process depends on the characteristics of the macroscopic
model
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