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

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

Homogenized dynamics of stochastic partial differential equations with dynamical boundary conditions

A microscopic heterogeneous system under random influence is considered. The randomness enters the system at physical boundary of small scale obstacles as well as at the interior of the physical medium. This system is modeled by a stochastic partial differential equation defined on a domain perforated with small holes (obstacles or heterogeneities), together with random dynamical boundary conditions on the boundaries of these small holes. A homogenized macroscopic model for this microscopic heterogeneous stochastic system is derived. This homogenized effective model is a new stochastic partial differential equation defined on a unified domain without small holes, with static boundary condition only. In fact, the random dynamical boundary conditions are homogenized out, but the impact of random forces on the small holes' boundaries is quantified as an extra stochastic term in the homogenized stochastic partial differential equation. Moreover, the validity of the homogenized model is justified by showing that the solutions of the microscopic model converge to those of the effective macroscopic model in probability distribution, as the size of small holes diminishes to zero.Comment: Communications in Mathematical Physics, to appear, 200

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

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

The diversity, evolution and ecology of Salmonella in venomous snakes

BACKGROUND: Reptile-associated Salmonella bacteria are a major, but often neglected cause of both gastrointestinal and bloodstream infection in humans globally. The diversity of Salmonella enterica has not yet been determined in venomous snakes, however other ectothermic animals have been reported to carry a broad range of Salmonella bacteria. We investigated the prevalence and diversity of Salmonella in a collection of venomous snakes and non-venomous reptiles. METHODOLOGY/PRINCIPLE FINDINGS: We used a combination of selective enrichment techniques to establish a unique dataset of reptilian isolates to study Salmonella enterica species-level evolution and ecology and used whole-genome sequencing to investigate the relatedness of phylogenetic groups. We observed that 91% of venomous snakes carried Salmonella, and found that a diverse range of serovars (n = 58) were carried by reptiles. The Salmonella serovars belonged to four of the six Salmonella enterica subspecies: diarizonae, enterica, houtanae and salamae. Subspecies enterica isolates were distributed among two distinct phylogenetic clusters, previously described as clade A (52%) and clade B (48%). We identified metabolic differences between S. diarizonae, S. enterica clade A and clade B involving growth on lactose, tartaric acid, dulcitol, myo-inositol and allantoin. SIGNIFICANCE: We present the first whole genome-based comparative study of the Salmonella bacteria that colonise venomous and non-venomous reptiles and shed new light on Salmonella evolution. Venomous snakes examined in this study carried a broad range of Salmonella, including serovars which have been associated with disease in humans such as S. Enteritidis. The findings raise the possibility that venomous snakes could be a reservoir for Salmonella serovars associated with human salmonellosis

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