We develop gradient schemes for the approximation of the Perona-Malik equations and nonlinear tensor-diffusion equations. We prove the convergence of these methods to the weak solutions of the corresponding nonlinear PDEs. A particular gradient scheme on rectangular meshes is then studied numerically with respect to experimental order of convergence which shows its second order accuracy. We present also numerical experiments related to image filtering by time-delayed Perona-Malik and tensor diffusion...

In this paper we study an approximation scheme for a class of control problems involving an ordinary control v, an impulsive control u and its derivative $\dot{u}$. Adopting a space-time reparametrization of the problem which adds one variable to the state space we overcome some difficulties connected to the presence of $\dot{u}$. We construct an approximation scheme for that augmented system, prove that it converges to the value function of the augmented problem and establish an error estimates in L∞ for this approximation....

We consider in ${ℝ}^2$ all curvature equation $\frac{\partial u}{\partial t}=\left|Du\right|G\left(\mathrm{curv}\left(u\right)\right)$ where G is a nondecreasing function and curv(u) is the curvature of the level line passing by x. These equations are invariant with respect to any contrast change u → g(u), with g nondecreasing. Consider the contrast invariant operator $T_t:u_o\to u\left(t\right)$. A Matheron theorem asserts that all contrast invariant operator T can be put in a form $\left(Tu\right)\left(𝐱\right)={inf}_{B\in ℬ}{sup}_{𝐲\in B}u(𝐱+𝐲)$. We show the asymptotic equivalence of both formulations. More precisely, we show that all curvature equations can be obtained...

This paper is devoted to the study of the approximation problem for the abstract hyperbolic differential equation u'(t) = A(t)u(t) for t ∈ [0,T], where A(t):t ∈ [0,T] is a family of closed linear operators, without assuming the density of their domains.

This work is concerned with the asymptotic analysis of a time-splitting scheme for the Schrödinger equation with a random potential having weak amplitude, fast oscillations in time and space, and long-range correlations. Such a problem arises for instance in the simulation of waves propagating in random media in the paraxial approximation. The high-frequency limit of the Schrödinger equation leads to different regimes depending on the distance of propagation, the oscillation pattern of the initial...