We study the discrete version of a family of ill-posed, nonlinear diffusion equations of order $2n$. The fourth order $(n=2)$ version of these equations constitutes our main motivation, as it appears prominently in image processing and computer vision literature. It was proposed by You and Kaveh as a model for denoising images while maintaining sharp object boundaries (edges). The second order equation $(n=1)$ corresponds to another famous model from image processing, namely Perona and Malik’s anisotropic diffusion, and was studied in earlier papers. The equations studied in this paper are high order analogues of the Perona-Malik equation, and like the second order model, their continuum versions violate parabolicity and hence lack well-posedness theory. We follow a recent technique from Kohn and Otto, and prove a weak upper bound on the coarsening rate of the discrete in space version of these high order equations in any space dimension, for a large class of diffusivities. Numerical experiments indicate that the bounds are close to being optimal, and are typically observed.

In this paper we introduce two novel numerical integration schemes within the framework of the heterogeneous multiscale method (HMM), when the finite element method is used as the macroscopic solver, to resolve the elliptic problem with a multiscale coefficient. For nonself-adjoint elliptic problems, optimal convergence rate is proved for the proposed methods, which naturally yields a new strategy for refining the macro-micro meshes and a criterion for determining the size of the microcell. Numerical results following this strategy show that the new methods significantly reduce the computational cost without loss of accuracy.

The need for service rules, or policies, in supply chains arises if not all the parts processed in the chain are considered identical, but are distinguished by certain attributes. We develop and analyze a methodology to model arbitrary service rules in large supply chains based on a kinetic (traffic flow like) theory and a level set approach. The final result is a system of hyperbolic conservation laws for the densities of parts, grouped by their attributes. The validity of the model is verified against discrete event simulations for several test cases.

This paper deals with the existence of optimal transport maps for some optimal transport problems with a convex but non-strictly convex cost. We give a decomposition strategy to address this issue. As a consequence of our procedure, we have to treat some transport problems, of independent interest, with a convex constraint on the displacement. To illustrate possible results obtained through this general approach, we prove existence of optimal transport maps in the case where the source measure is absolutely continuous with respect to the Lebesgue measure and the transportation cost is of the form $h║x-y║$, with h strictly convex increasing and $║.║$ an arbitrary norm in $R^2$.

We consider a magnetohydrodynamic-α model with kinematic viscosity and magnetic diffusivity for an incompressible fluid in a three-dimensional periodic box (torus). Similar models are useful to study the turbulent behavior of fluids in presence of a magnetic field because of the current impossibility to handle non-regularized systems neither analytically nor via numerical simulations.

We prove the existence of a global solution and a global attractor. Moreover, we provide an upper bound for the Hausdorff and the fractal dimension of the attractor. This bound can be interpreted in terms of degrees of freedom of the system. In some sense, this result provides an intermediate bound between the number of degrees of freedom for the simplified Bardina model and the Navier–Stokes-α equation.

We propose an asymptotic numerical method called the Gaussian beam approach for the boundary value problem of high frequency Helmholtz equation. The basic idea is to approximate the traveling waves with a summation of Gaussian beams by the least squares algorithm. Gaussian beams are asymptotic solutions of linear wave equations in the high frequency regime. We deduce the ODE systems satisfied by the Gaussian beams up to third order. The key ingredient of the proposed method is the construction of a finite-dimensional beam space which has a good approximating property. If the exact solutions of boundary value problems contain some strongly evanescent wave modes, the Gaussian beam approach might fail. To remedy this problem, we resort to the domain decomposition technique to separate the domain of definition into a boundary layer region and its complementary interior region. The former is handled by a domain-based discretization method, and the latter by the Gaussian beam approach. Schwarz iterations should then be performed based on suitable transmission boundary conditions at the interface of two regions. Numerical tests demonstrate that the proposed method is very promising.

G-equations are popular front propagation models in combustion literature and describe the front motion law of normal velocity equal to a constant plus the normal projection of fluid velocity. G-equations are Hamilton-Jacobi equations with convex but non-coercive Hamiltonians. We prove homogenization of the inviscid G-equation for space periodic incompressible flows. This extends a two space dimensional result in "Periodic homogenization of G-equations and viscosity effects," Nonlinearity, to appear. We construct approximate correctors to bypass the lack of compactness due to the non-coercive Hamiltonian. The existence of approximate correctors rely on a local reachability property of the controlled flow trajectory as well as incompressibility of the flow. Homogenization then follows from the comparison principle and the perturbed test function method. The effective Hamiltonian is convex and homogeneous of degree one. It is also coercive if we further assume that the flow is mean zero.

We show that for a particular linear Fokker-Planck operator, the explicit Chang and Cooper scheme is positive and entropy satisfying under a CFL criterion when the initial condition is positive. Then, we deduce that the distribution given by the explicit Chang and Cooper scheme converges toward a discrete Maxwellian equilibrium.