Nicolas Forcadel

Reachability and Minimal Times for State Constrained Nonlinear Problems without Any Controllability Assumption

We consider a target problem for a nonlinear system under state constraints. We give a new continuous level-set approach for characterizing the optimal times and the backward-reachability sets. This approach leads to a characterization via a Hamilton-Jacobi equation, without assuming any controllability assumption. We also treat the case of time-dependent state constraints, as well as a target problem for a two-player game with state constraints. Our method gives a good framework for numerical approximations, and some numerical illustrations are included in the paper.

Generalized fast marching method: applications to image segmentation

In this paper, we propose a segmentation method based on the generalized fast marching method (GFMM) developed by Carlini et al. (submitted). The classical fast marching method (FMM) is a very efficient method for front evolution problems with normal velocity (see also Epstein and Gage, The curve shortening flow. In: Chorin, A., Majda, A. (eds.) Wave Motion: Theory, Modelling and Computation, 1997) of constant sign. The GFMM is an extension of the FMM and removes this sign constraint by authorizing time-dependent velocity with no restriction on the sign. In our modelling, the velocity is borrowed from the Chan–Vese model for segmentation (Chan and Vese, IEEE Trans Image Process 10(2):266–277, 2001). The algorithm is presented and analyzed and some numerical experiments are given, showing in particular that the constraints in the initialization stage can be weakened and that the GFMM offers a powerful and computationally efficient algorithm.

Convergence of a non-local eikonal equation to anisotropic mean curvature motion. Application to dislocations dynamics.

In this paper we prove the convergence at a large scale of a non-local first order equation to an anisotropic mean curvature motion. This first order equation is an eikonal-type equation with a velocity depending in a non-local way on the solution itself, that arises in the theory of dislocations dynamics. We show that if an anisotropic mean curvature motion is approximated by this type of equations then it is always of variational type, whereas the converse is true only in dimension two

Convergence of a Generalized Fast-Marching Method for an Eikonal Equation with a Velocity-Changing Sign

We present a new fast-marching algorithm for an eikonal equation with a velocity-changing sign. This first order equation models a front propagation in the normal direction. The algorithm is an extension of the fast-marching method in two respects. The first is that the new scheme can deal with a time-dependent velocity, and the second is that there is no restriction on its change in sign. We analyze the properties of the algorithm, and we prove its convergence in the class of discontinuous viscosity solutions. Finally, we present some numerical simulations of fronts propagating in

A convergent scheme for a non-local coupled system modelling dislocations densities dynamics

\mathbb{R}^2

A Generalized Fast Marching Method for Dislocation Dynamics

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Homogenization of accelerated Frenkel-Kontorova models with n types of particles

In this paper, we study a non-local coupled system that arises in the theory of dislocations densities dynamics. Within the framework of viscosity solutions, we prove a long time existence and uniqueness result for the solution of this model. We also propose a convergent numerical scheme and we prove a Crandall-Lions type error estimate between the continuous solution and the numerical one. As far as we know, this is the first error estimate of Crandall-Lions type for Hamilton-Jacobi systems. We also provide some numerical simulations.

Comparison Principle for a Generalized Fast Marching Method

In this paper, we consider a generalized fast marching method (GFMM) as a numerical method to compute dislocation dynamics. The dynamics of a dislocation hypersurface in

Dislocation Dynamics: a Non-local Moving Boundary

\mathbb{R}^N

L^1-error estimate for numerical approximations of Hamilton-Jacobi-Bellman equations in dimension 1.

(with