Well-posedness for a modified bidomain model describing bioelectric activity in damaged heart tissues

We prove the existence and the uniqueness of a solution for a modified bidomain model, describing the electrical behaviour of the cardiac tissue in pathological situations. The main idea is to reduce the problem to an abstract parabolic setting, which requires to introduce several auxiliary differential systems and a non-standard bilinear form. The main difficulties are due to the degeneracy of the bidomain system and to its non-standard coupling with the diffusion equation

Asymptotic analysis for non-local problems in composites with different imperfect contact conditions

We consider a composite material made up of a hosting medium containing an \eps-periodic array of perfect thermal conductors. Comparing with the previous contributions in the literature, in the present paper, the inclusions are completely disconnected and form two families with dissimilar physical behaviour. More specifically, the imperfect contact between the hosting medium and the inclusions obeys two different laws, according to the two different types of inclusions. The contact conditions involve the small parameter \eps and two positive constants \contuno,\contdue. We investigate the homogenization limit \eps\to 0 and the limits for \contuno,\contdue going to $0$ or $+\infty$, taken in any order, with the aim to find out the cases in which the two limits commute

A degenerate pseudo-parabolic equation with memory

We prove the existence and uniqueness for a degenerate pseudo-parabolic problem with memory. This kind of problem arises in the study of the homogenization of some differential systems involving the Laplace-Beltrami operator and describes the effective behaviour of the electrical conduction in some composite materials

Learning a Mixture of Deep Networks for Single Image Super-Resolution

Single image super-resolution (SR) is an ill-posed problem which aims to recover high-resolution (HR) images from their low-resolution (LR) observations. The crux of this problem lies in learning the complex mapping between low-resolution patches and the corresponding high-resolution patches. Prior arts have used either a mixture of simple regression models or a single non-linear neural network for this propose. This paper proposes the method of learning a mixture of SR inference modules in a unified framework to tackle this problem. Specifically, a number of SR inference modules specialized in different image local patterns are first independently applied on the LR image to obtain various HR estimates, and the resultant HR estimates are adaptively aggregated to form the final HR image. By selecting neural networks as the SR inference module, the whole procedure can be incorporated into a unified network and be optimized jointly. Extensive experiments are conducted to investigate the relation between restoration performance and different network architectures. Compared with other current image SR approaches, our proposed method achieves state-of-the-arts restoration results on a wide range of images consistently while allowing more flexible design choices. The source codes are available in http://www.ifp.illinois.edu/~dingliu2/accv2016

Effective macroscopic dynamics of stochastic partial differential equations in perforated domains

An effective macroscopic model for a stochastic microscopic system is derived. The original microscopic system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes or heterogeneities. The homogenized effective model is still a stochastic partial differential equation but defined on a unified domain without holes. The solutions of the microscopic model is shown to converge to those of the effective macroscopic model in probability distribution, as the size of holes diminishes to zero. Moreover, the long time effectivity of the macroscopic system in the sense of \emph{convergence in probability distribution}, and the effectivity of the macroscopic system in the sense of \emph{convergence in energy} are also proved