Well-posedness and inverse Robin estimate for a multiscale elliptic/parabolic system

Abstract

We establish the well-posedness of a coupled micro–macro parabolic–elliptic system modeling the interplay between two pressures in a gas–liquid mixture close to equilibrium that is filling a porous media with distributed microstructures. Additionally, we prove a local stability estimate for the inverse micro–macro Robin problem, potentially useful in identifying quantitatively a micro–macro interfacial Robin transfer coefficient given microscopic measurements on accessible fixed interfaces. To tackle the solvability issue we use two-scale energy estimates and two-scale regularity/compactness arguments cast in the Schauder’s fixed point theorem. A number of auxiliary problems, regularity, and scaling arguments are used in ensuring the suitable Fréchet differentiability of the solution and the structure of the inverse stability estimate.     

Coupled heat and moisture transport in paper with application to a warm print surface

In this work we present a mathematical model describing the coupled heat and moisture transport in paper. The model is solved numerically and the numerical solution is used to study the interdependency of the moisture and temperature distribution in paper. The results show that variation with temperature of the saturated water vapor concentration and the sorption isotherm parameters are both important for inducing moisture desorption. It is also found that for steep relative humidity ramps moisture sorption generates temperature increments that slow down the sorption process itself. The model is also used to study the moisture gradients in a paper sheet inside a printer from Océ Technologies, which contains a warm print surface. The results predict changes in moisture content of only 0.2%, which suggests that no deformations are induced on the printed sheet.     

A two-scale reaction-diffusion system with time-dependent microstructure

We present a two-scale model for reactive transport in a unsaturated porous medium. Reaction and diffusion of a gas dissolved in the pore water is modelled at the pore-scale, while the fast transport of the gas in the air phase is described by an averaged equation. The pore geometry is allowed to vary in time to account for temporal changes of the domain occupied by the pore water. The resulting semilinear parabolic system of PDEs is coupled in a non-standard way. After a transformation to a fixed domain, the existence of a unique weak W^1,p -solution is established by semigroup methods. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

A two-scale model for the periodic homogenization of the wave equation

We present a new mathematical object designed to analyze the oscillations occurring on both microscopic and macroscopic scales in a wave equation with oscillating coefficients and data. Through a Bloch wave homogenization method, our study addresses typical problems of two-scale convergence in the interior of the domain, and sheds some light on the behavior near the boundary. A decoupled system of (systems of) transport equations is derived in each energy band, and the total energy field is approximated. We also recover previously known results in homogenization as a restricted part of our model.     

Modelling and simulation of heat exchange and moisture content in a cereal storage silo

ABSTRACT

In this paper, a mathematical model is developed based on the heat transfer of stored grains aerated in a cylindrical silo. This work is a part of study that aims to model the whole process of cereal storage system including a dehumidifier. The use of dehumidifier is intended to remove excess moisture from the airflow injected by the ventilator system in the silo filled with wheat, and to keep hygroscopic properties of grain in safe level during the storage period. Temperature and humidity are the two important variables coupled to control the process and to preserve grain quality. The laboratory device permitted us to achieve several tests for different conditions of grain stored in silo without aeration. A simulation of the airflow through the thermal space of the silo and grain parameters has been carried out. The results are reasonably in agreement with experiments and other published data. The system performance is evaluated at critical conditions of storage boundaries.     

Homogenization for rate-independent systems

This paper is devoted to the homogenization for a class of rate-independent systems described by the energetic formulation. The associated nonlinear partial differential system has periodically oscillating coefficients, but has the form of a standard evolutionary variational inequality. Thus, the model applies to standard linearized elastoplasticity with hardening. Using the recently developed methods of two-scale convergence, periodic unfolding and the new introduced one, periodic folding, we show that the homogenized problem can be represented as a two-scale limit which is again an energetic formulation, but now involving the macroscopic variable in the physical domain as well as the microscopic variable in the periodicity cell. (© 2006 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim)     

Scale-integration and scale-disintegration in nonlinear homogenization

This work is devoted to scale transformations of stationary nonlinear problems. A class of coarse-scale problems is first derived by integrating a family of two-scale minimization problems (scale-integration), in presence of appropriate orthogonality conditions. The equivalence between the two formulations is established by showing that conversely any solution of the coarse-scale problem can be represented as the fine-scale average of a solution of the two-scale problem (scale-disintegration). This procedure may be applied to the homogenization of several quasilinear problems, and is related to De Giorgi’s notion of Γ-convergence. As an example the homogenization of a simple nonlinear model of magnetostatics is illustrated: a two-scale minimization problem is first derived via Nguetseng’s notion of two-scale convergence, and afterwards the equivalence with a coarse-scale problem is proved.     

On two-scale convergence and related sequential compactness topics

A general concept of two-scale convergence is introduced and two-scale compactness theorems are stated and proved for some classes of sequences of bounded functions in L ²(Ω) involving no periodicity assumptions. Further, the relation to the classical notion of compensated compactness and the recent concepts of two-scale compensated compactness and unfolding is discussed and a defect measure for two-scale convergence is introduced.     

Two-scale convergence of some integral functionals

Nguetseng’s notion of two-scale convergence is reviewed, and some related properties of integral functionals are derived. The coupling of two-scale convergence with convexity and monotonicity is then investigated, and a two-scale version is provided for compactness by strict convexity. The div-curl lemma of Murat and Tartar is also extended to two-scale convergence, and applications are outlined.     

A note on admissibility of closed Galois structures

Abstract

We present a new admissibility theorem for Galois structures in the sense of G. Janelidze. It applies to relative exact categories satisfying a suitable relative modularity condition, and extends the known admissibility theorem in the theory of generalized central extensions. We also show that our relative modularity condition holds in every relative exact Goursat category.     

Two-scale sparse finite element approximations

To reduce computational cost,we study some two-scale finite element approximations on sparse grids for elliptic partial differential equations of second order in a general setting.Over any tensor product domain ?R~d with d = 2,3,we construct the two-scale finite element approximations for both boundary value and eigenvalue problems by using a Boolean sum of some existing finite element approximations on a coarse grid and some univariate fine grids and hence they are cheaper approximations.As applications,we obtain some new efficient finite element discretizations for the two classes of problem:The new two-scale finite element approximation on a sparse grid not only has the less degrees of freedom but also achieves a good accuracy of approximation.     

Softening Bilevel Problems Via Two-scale Gibbs Measures

We introduce a new, and elementary, approximation method for bilevel optimization problems motivated by Stackelberg leader-follower games. Our technique is based on the notion of two-scale Gibbs measures. The first scale corresponds to the cost function of the follower and the second scale to that of the leader. We explain how to choose the weights corresponding to these two scales under very general assumptions and establish rigorous Γ-convergence results. An advantage of our method is that it is applicable both to optimistic and to pessimistic bilevel problems.

     

Three-scale convergence for processes in heterogeneous media

In this article, we propose a new notion of multiscale convergence, called ‘three-scale’, which aims to give a topological framework in which to assess complex processes occurring at three different scales or levels within a heterogeneous medium. This generalizes and extends the notion of two-scale convergence, a well-established concept that is now commonly used for obtaining an averaged, asymptotic value (homogenization) of processes that exist on two different spatial scales. The well-posedness of this new concept is justified via a compactness theorem which ensures that all bounded sequences in L ²(Ω) are relative compact with respect to the three-scale convergence. This is taken further by giving a boundedness characterization of three-scale convergent sequences and is then continued with the introduction of the notion of ‘strong three-scale convergence’ whose well-posedness is also discussed. Finally, the three-scale convergence of the gradients is established.     

On the number of cyclic subgroups of a group

Abstract

There has been some interest in understanding the relationship between the number of cyclic subgroups of a group and its order. This relationship is controlled in many cases by the size of the set of non-squares of the group. We improve upon previously established bounds and classify groups that obtain our new bound.     

A two-scale model for the wave equation with oscillating coefficients and data

We introduce a time-space two-scale transform designed to capture the high and low frequency waves in the asymptotics of the periodic homogenization of the wave equation. The asymptotical solution is the sum of the solution of known homogenized equations and of Bloch waves. We also derive the transport equations satisfied by the Bloch wave coefficients. To cite this article: M. Brassart, M. Lenczner, C. R. Acad. Sci. Paris, Ser. I 347 (2009).     

A mathematical analysis of quantum transport in three-dimensional crystals

Summary We present an existence and uniqueness result for a quantum transport model in three dimensional crystals. The model consists of a quantum transport (Wigner) equation posed on the phase space consisting of a discrete position variable and a «continuous» wave vector, which is restricted to a bounded domain inR ³ (first Brillouin zone of the crystal). The potential is modeled self-consistently by a discrete Poisson equation (Coulomb interaction). Also we investigate the limits of solutions of this model as the grid spacing tends to zero and show that they converge to the solution of a quantum transport model posed on the «fully continuous» phase space. The transport model derived by this limiting procedure treats the band diagram of the crystal in a semi-classical way and the potential energy term quantum mechanically.     

Weighted Statistical Relative Invariant Mean in Modular Function Spaces with Related approximation Results

Abstract

The idea of statistical relative convergence on modular spaces has been introduced by Orhan and Demirci. The notion of σ-statistical convergence was introduced by Mursaleen and Edely and further extended based on a fractional order difference operator by Kadak. The concern of this paper is to define two new summability methods for double sequences by combining the concepts of statistical relative convergence and σ-statistical convergence in modular spaces. Furthermore, we give some inclusion relations involving the newly proposed methods and present an illustrative example to show that our methods are nontrivial generalizations of the existing results in the literature. We also prove a Korovkin-type approximation theorem and estimate the rate of convergence by means of the modulus of continuity. Finally, using the bivariate type of Stancu-Schurer-Kantorovich operators, we display an example such that our approximation results are more powerful than the classical, statistical, and relative modular cases of Korovkin-type approximation theorems.     

Homogenization of linear and nonlinear transport equations

We study the homogenization of the linear and nonlinear transport equations with oscillatory velocity fields. Two types of homogenized equations are derived. For general n-dimensional linear and nonlinear problems, we derive homogenized equations by introducing additional independent variables to represent the small scales. For the two-dimensional linear transport equations, we derive effective equations for the averaged quantities. Such equations take the form of either a degenerate non-local diffusion equation with memory or a higher order hyperbolic equation. To study the nonlinear transport equations we introduce the concept of two-scale Young measure and extend DiPerna's method to prove that it reduces to a family of Dirac measures.     

Drift Estimation of Multiscale Diffusions Based on Filtered Data

We study the problem of drift estimation for two-scale continuous time series. We set ourselves in the framework of overdamped Langevin equations, for which a single-scale surrogate homogenized equation exists. In this setting, estimating the drift coefficient of the homogenized equation requires pre-processing of the data, often in the form of subsampling; this is because the two-scale equation and the homogenized single-scale equation are incompatible at small scales, generating mutually singular measures on the path space. We avoid subsampling and work instead with filtered data, found by application of an appropriate kernel function, and compute maximum likelihood estimators based on the filtered process. We show that the estimators we propose are asymptotically unbiased and demonstrate numerically the advantages of our method with respect to subsampling. Finally, we show how our filtered data methodology can be combined with Bayesian techniques and provide a full uncertainty quantification of the inference procedure.

     

Two-scale symbol and autocorrelation symbol for B-splines with multiple knots

The Fourier transforms of B-splines with multiple integer knots are shown to satisfy a simple recursion relation. This recursion formula is applied to derive a generalized two-scale relation for B-splines with multiple knots. Furthermore, the structure of the corresponding autocorrelation symbol is investigated. In particular, it can be observed that the solvability of the cardinal Hermite spline interpolation problem for spline functions of degree 2m+1 and defectr, first considered by Lipow and Schoenberg [9], is equivalent to the Riesz basis property of our B-splines with degreem and defectr. In this way we obtain a new, simple proof for the assertion that the cardinal Hermite spline interpolation problem in [9] has a unique solution.