The powerful Hamilton-Jacobi theory is used for constructing regularizations and error estimates for optimal design problems. The constructed Pontryagin method is a simple and general method for optimal design and reconstruction: the first, analytical, step is to regularize the Hamiltonian; next the solution to its stationary Hamiltonian system, a nonlinear partial differential equation, is computed with the Newton method. The method is efficient for designs where the Hamiltonian function can be explicitly formulated and when the Jacobian is sparse, but becomes impractical otherwise (e.g. for non local control constraints). An error estimate for the difference between exact and approximate objective functions is derived, depending only on the difference of the Hamiltonian and its finite dimensional regularization along the solution path and its L2 projection, i.e. not on the difference of the exact and approximate solutions to the Hamiltonian systems.

From the fundamental laws of elasticity, we write a model for the contact between two membranes and we perform the analysis of the corresponding system of variational inequalities. We propose a finite element discretization of this problem and prove its well-posedness. We also establish a priori and a posteriori error estimates.

We propose here a model and a numerical scheme to compute the motionof rigid particles interacting through the lubrication force. In thecase of a particle approaching a plane, we propose an algorithm andprove its convergence towards the solutions to the gluey particle modeldescribed in [B. Maury, ESAIM: Proceedings18 (2007)133–142]. We propose a multi-particle version ofthis gluey model which is based on the projection of the velocitiesonto a set of admissible velocities. Then, we describe a multi-particle algorithmfor the simulation of such systems and present numerical results.

This paper is devoted to the numerical solution of stationarylaminar Bingham fluids by path-following methods. By using duality theory, asystem that characterizes the solution of the original problem is derived.Since this system is ill-posed, a family of regularized problems is obtainedand the convergence of the regularized solutions to the original one is proved.For the update of the regularization parameter, a path-following method isinvestigated. Based on the differentiability properties of the path, a model ofthe value functional and a correspondent algorithm are constructed. For thesolution of the systems obtained in each path-following iteration a semismoothNewton method is proposed. Numerical experiments are performed in order toinvestigate the behavior and efficiency of the method, and a comparison with apenalty-Newton-Uzawa-conjugate gradient method, proposed in [Dean et al., J. Non-Newtonian Fluid Mech.142 (2007) 36–62], iscarried out.

While making informed decisions regardinginvestments in customer retention and acquisition becomes apressing managerial issue, formal models and analysis, which mayprovide insight into this topic, are still scarce. In this studywe examine two dynamic models for optimal acquisition andretention models of a monopoly, the total cost and the cost percustomer models.These models are analytically analyzed using classical, direct,methods and asymptotic expansions (for the total cost model).In order to numerically simulated the models, an innovativenumerical method was developed for solving ODE systems withinitial/final value problems.


This article deals with the numerical computation of the Cheeger constant and the approximation of the maximal Cheeger set of a given subset of ${\mathbb R}^d$ . This problem is motivated by landslide modelling as well as by the continuous maximal flow problem. Using the fact that the maximal Cheeger set can be approximated by solving a rather simple projection problem, we propose a numerical strategy to compute maximal Cheeger sets and Cheeger constants.


We show that the entropy method, that has been used successfully in orderto prove exponential convergence towards equilibrium with explicit constants in many contexts,among which reaction-diffusion systems coming out of reversible chemistry, can also be usedwhen one considers a reaction-diffusion system corresponding to an irreversible mechanism ofdissociation/recombination, for which no natural entropy is available.


In this paper, we present numerical methodsfor the determination of solitons, that consist in spatially localizedstationary states of nonlinear scalar equations or coupled systemsarising in nonlinear optics.We first use the well-known shooting method in order to findexcited states (characterized by the number k of nodes) for theclassical nonlinear Schrödinger equation. Asymptotics can thenbe derived in the limits of either large k are large nonlinear exponents σ.In a second part, we compute solitons for a nonlinearsystem governing the propagation of two coupled waves in a quadratic media in anyspatial dimension, starting from one-dimensional states obtained with a shooting method and considering the dimension as a continuation parameter. Finally, we investigate the case of three wavemixing, for which the shooting method is not relevant.