We consider the variant of stochastic homogenization theory introduced in [X. Blanc, C.Le Bris and P.-L. Lions, C. R. Acad. Sci. Série I 343 (2006)717–724.; X. Blanc, C. Le Bris and P.-L. Lions, J. Math. Pures Appl.88 (2007) 34–63.]. The equation under consideration is a standardlinear elliptic equation in divergence form, where the highly oscillatory coefficient isthe composition of a periodic matrix with a stochastic diffeomorphism. The homogenizedlimit of this problem has been identified in [X. Blanc, C. Le Bris and P.-L. Lions,C. R. Acad. Sci. Série I 343 (2006) 717–724.]. We firstestablish, in the one-dimensional case, a convergence result (with an explicit rate) onthe residual process, defined as the difference between the solution to the highlyoscillatory problem and the solution to the homogenized problem. We next return to themultidimensional situation. As often in random homogenization, the homogenized matrix isdefined from a so-called corrector function, which is the solution to a problem set on theentire space. We describe and prove the almost sure convergence of an approximationstrategy based on truncated versions of the corrector problem.

We study a Lagrangian numerical scheme for solution of a nonlinear drift diffusionequation on an interval. The discretization is based on the equation’s gradient flowstructure with respect to the Wasserstein distance. The scheme inherits various propertiesfrom the continuous flow, like entropy monotonicity, mass preservation, metric contractionand minimum/ maximum principles. As the main result, we give a proof of convergence in thelimit of vanishing mesh size under a CFL-type condition. We also present results fromnumerical experiments.

Several kinds of exact synchronizations and the generalized exact synchronization areintroduced for a coupled system of 1-D wave equations with various boundary conditions andwe show that these synchronizations can be realized by means of some boundarycontrols.

We generalize to the p-LaplacianΔp a spectral inequality proved by M.-T.Kohler−Jobin. As a particular case of such a generalization, we obtain a sharp lower boundon the first Dirichlet eigenvalue of Δp of aset in terms of its p-torsional rigidity. The result is valid in everyspace dimension, for every1 < p < ∞ and for every openset with finite measure. Moreover, it holds by replacing the first eigenvalue with moregeneral optimal Poincaré-Sobolev constants. The method of proof is based on ageneralization of the rearrangement technique introduced by Kohler−Jobin.

We establish an optimal, linear rate of convergence for the stochastic homogenization of discrete linear elliptic equations. We consider the model problem of independent and identically distributed coefficients on a discretized unit torus. We show that the difference between the solution to the random problem on the discretized torus and the first two terms of the two-scale asymptotic expansion has the same scaling as in the periodic case. In particular the L2-norm in probability of the H1-norm in space of this error scales like ε, where ε is the discretization parameter of the unit torus. The proof makes extensive use of previous results by the authors, and of recent annealed estimates on the Green’s function by Marahrens and the third author.

Compatible schemes localize degrees of freedom according to the physical nature of the underlying fields and operate a clear distinction between topological laws and closure relations. For elliptic problems, the cornerstone in the scheme design is the discrete Hodge operator linking gradients to fluxes by means of a dual mesh, while a structure-preserving discretization is employed for the gradient and divergence operators. The discrete Hodge operator is sparse, symmetric positive definite and is assembled cellwise from local operators. We analyze two schemes depending on whether the potential degrees of freedom are attached to the vertices or to the cells of the primal mesh. We derive new functional analysis results on the discrete gradient that are the counterpart of the Sobolev embeddings. Then, we identify the two design properties of the local discrete Hodge operators yielding optimal discrete energy error estimates. Additionally, we show how these operators can be built from local nonconforming gradient reconstructions using a dual barycentric mesh. In this case, we also prove an optimal L2-error estimate for the potential for smooth solutions. Links with existing schemes (finite elements, finite volumes, mimetic finite differences) are discussed. Numerical results are presented on three-dimensional polyhedral meshes.

In this paper, we consider a backward problem for a time-fractional diffusion equation with variable coefficients in a general bounded domain. That is to determine the initial data from a noisy final data. Based on a series expression of the solution, a conditional stability for the initial data is given. Further, we propose a modified quasi-boundary value regularization method to deal with the backward problem and obtain two kinds of convergence rates by using an a priori regularization parameter choice rule and an a posteriori regularization parameter choice rule. Numerical examples in one-dimensional and two-dimensional cases are provided to show the effectiveness of the proposed methods.

We present a high-resolution, non-oscillatory semi-discrete central scheme for one-dimensional shallow-water flows along channels with non uniform cross sections of arbitrary shape and bottom topography. The proposed scheme extends existing central semi-discrete schemes for hyperbolic conservation laws and enjoys two properties crucial for the accurate simulation of shallow-water flows: it preserves the positivity of the water height, and it is well balanced, i.e., the source terms arising from the geometry of the channel are discretized so as to balance the non-linear hyperbolic flux gradients. In addition to these, a modification in the numerical flux and the estimate of the speed of propagation, the scheme incorporates the ability to detect and resolve partially wet regions, i.e., wet-dry states. Along with a detailed description of the scheme and proofs of its properties, we present several numerical experiments that demonstrate the robustness of the numerical algorithm.

We discuss new MUSCL reconstructions to approximate the solutions of hyperbolic systems of conservations laws on 2D unstructured meshes. To address such an issue, we write two MUSCL schemes on two overlapping meshes. A gradient reconstruction procedure is next defined by involving both approximations coming from each MUSCL scheme. This process increases the number of numerical unknowns, but it allows to reconstruct very accurate gradients. Moreover a particular attention is paid on the limitation procedure to enforce the required robustness property. Indeed, the invariant region is usually preserved at the expense of a more restrictive CFL condition. Here, we try to optimize this condition in order to reduce the computational cost.

The central objective of this paper is to develop reduced basis methods for parameter dependent transport dominated problems that are rigorously proven to exhibit rate-optimal performance when compared with the Kolmogorov n-widths of the solution sets. The central ingredient is the construction of computationally feasible “tight” surrogates which in turn are based on deriving a suitable well-conditioned variational formulation for the parameter dependent problem. The theoretical results are illustrated by numerical experiments for convection-diffusion and pure transport equations. In particular, the latter example sheds some light on the smoothness of the dependence of the solutions on the parameters.

In this paper we study the realizability of a given smooth periodic gradient field ∇u defined in Rd, in the sense of finding when one can obtain a matrix conductivity σ such that σ∇u is a divergence free current field. The construction is shown to be always possible locally in Rd provided that ∇u is non-vanishing. This condition is also necessary in dimension two but not in dimension three. In fact the realizability may fail for non-regular gradient fields, and in general the conductivity cannot be both periodic and isotropic. However, using a dynamical systems approach the isotropic realizability is proved to hold in the whole space (without periodicity) under the assumption that the gradient does not vanish anywhere. Moreover, a sharp condition is obtained to ensure the isotropic realizability in the torus. The realizability of a matrix field is also investigated both in the periodic case and in the laminate case. In this context the sign of the matrix field determinant plays an essential role according to the space dimension. The present contribution essentially deals with the realizability question in the case of periodic boundary conditions.

A mathematical model, coupling the dynamics of short-term stem-like cells and matureleukocytes in leukemia with that of the immune system, is investigated. The model isdescribed by a system of seven delay differential equations with seven delays. Threeequilibrium points E0, E1,E2 are highlighted. The stability and the existence of theHopf bifurcation for the equilibrium points are investigated. In the analysis of themodel, the rate of asymmetric division and the rate of symmetric division are veryimportant.

In the present paper we study the reconstruction of a structured quadratic pencil fromeigenvalues distributed on ellipses or parabolas. A quadratic pencil is a square matrixpolynomial

QP(λ) = M λ2+Cλ +K,

where M,C, andK are realsquare matrices. The approach developed in the paper is based on the theory of orthogonalpolynomials on the real line. The results can be applied to more general distribution ofeigenvalues. The problem with added single eigenvector is also briefly discussed. As anillustration of the reconstruction method, the eigenvalue problem on linearized stabilityof certain class of stationary exact solution of the Navier-Stokes equations describingatmospheric flows on a spherical surface is reformulated as a simple mass-spring system bymeans of this method.

In this paper, we shall obtain the symmetries of the mathematical model describingspontaneous relaxation of eastward jets into a meandering state and use these symmetriesfor constructing the conservation laws. The basic eastward jet is a spectral parameter ofthe model, which is in geostrophic equilibrium with the basic density structure and whichguarantees the existence of nontrivial conservation laws.

This paper studies the periodic feedback stabilization of the controlled lineartime-periodic ordinary differential equation:ẏ(t) = A(t)y(t) + B(t)u(t),t ≥ 0, where [A(·), B(·)] is aT-periodic pair, i.e.,A(·) ∈ L∞(ℝ+;ℝn×n) andB(·) ∈ L∞(ℝ+;ℝn×m) satisfy respectivelyA(t + T) = A(t)for a.e. t ≥ 0 andB(t + T) = B(t)for a.e. t ≥ 0. Two periodic stablization criteria for aT-period pair [A(·), B(·)] areestablished. One is an analytic criterion which is related to the transformation over timeT associated with A(·); while another is a geometriccriterion which is connected with the null-controllable subspace of[A(·), B(·)]. Two kinds of periodic feedback lawsfor a T-periodically stabilizable pair [ A(·),B(·) ] are constructed. They are accordingly connected with two Cauchy problemsof linear ordinary differential equations. Besides, with the aid of the geometriccriterion, we find a way to determine, for a given T-periodicA(·), the minimal column number m, as well as atime-invariant n×m matrix B, suchthat the pair [A(·), B] isT-periodically stabilizable.

Darwin illustrated his theory about emergence and evolution of biological species with adiagram. It shows how species exist, evolve, appear and disappear. The goal of this workis to give a mathematical interpretation of this diagram and to show how it can bereproduced in mathematical models. It appears that conventional models in populationdynamics are not sufficient, and we introduce a number of new models which take intoaccount local, nonlocal and global consumption of resources, and models with space andtime dependent coefficients.

Scattering of electromagnetic (EM) waves by small (ka ≪ 1) impedance particleD of anarbitrary shape, embedded in a homogeneous medium, is studied. Analytic, closed form,formula for the scattered field is derived. The scattered field is of the orderO(a2 −κ), where κ ∈ [ 0,1)is a number. This field is much larger than in the case of Rayleigh-type scattering. Thenumerical results demonstrate a wide range of applicability of the analytic formula forthe scattered field. Comparison with Mie-type solution is carried out for various boundaryimpedances and radii of the particle.

In this paper we present a fluid-structure interaction model of neuron’s membranedeformation. The membrane-actin is considered as an elastic solid layer, while thecytoplasm is considered as a viscous fluid one. The membrane-actin layer is governed byelasticity equations while the cytoplasm is described by the Navier-Stokes equations. Atthe interface between the cytoplasm and the membrane we consider a match between the solidvelocity displacement and the fluid velocity as well as the mechanical equilibrium. Themembrane, which faces the extracellular medium, is free to move. This will change thegeometry in time. To take into account the deformation of the initial configuration, weuse the Arbitrary Lagrangian Eulerian method in order to take into account the meshdisplacement. The numerical simulations, show the emergence of a filopodium, a typicalstructure in cells undergoing deformation.

We study the structure of the spectrum of the infinite XXZ quantum spin chain, ananisotropic version of the Heisenberg model. The XXZ chain Hamiltonian preserves thenumber of down spins (or particle number), allowing to represent it as a direct sum ofN-particleinteracting discrete Schrödinger-type operators restricted to the fermionic subspace. Inthe Ising phase of the model we use this representation to give a detailed determinationof the band and gap structure of the spectrum at low energy. In particular, we show thatat sufficiently strong anisotropy the so-called droplet bands are separated from higherspectral bands uniformly in the particle number. Our presentation of all necessarybackground is self-contained and can serve as an introduction to the mathematical theoryof the Heisenberg and XXZ quantum spin chains.