Ana Carpio

Solving inhomogeneous inverse problems by topological derivative methods

We introduce new iterative schemes to reconstruct scatterers buried in a medium and their physical properties. The inverse scattering problem is reformulated as a constrained optimization problem involving transmission boundary value problems in heterogeneous media. Our first step consists in developing a reconstruction scheme assuming that the properties of the objects are known. In a second step, we combine iterations to reconstruct the objects with iterations to recover the material parameters. This hybrid method provides reasonable guesses of the parameter values and the number of scatterers, their location and size. Our schemes to reconstruct objects knowing their nature rely on an extended notion of topological derivative. Explicit expressions for the topological derivatives of the corresponding shape functionals are computed in general exterior domains. Small objects, shapes with cavities and poorly illuminated obstacles are easily recovered. To improve the predictions of the parameters in the successive guesses of the domains we use a gradient method.

Large-time behavior in incompressible Navier-Stokes equations

We give a development up to the second order for strong solutions u of incompressible Navier–Stokes equations in

Asymptotic behavior for the vorticity equations in dimensions two and three

\mathbb{R}^n

DEPINNING TRANSITIONS IN DISCRETE REACTION-DIFFUSION EQUATIONS ∗

,

Oscillatory wave fronts in chains of coupled nonlinear oscillators.

n \geq 2

Periodized discrete elasticity models for defects in graphene

. By combining estimates obtained from the integral equation with a scaling technique, we prove that, for initial data satisfying some integrability conditions (and small enough, if

Edge Dislocations in Crystal Structures Considered as Traveling Waves in Discrete Models

n \geq 3

Topological Derivatives for Shape Reconstruction

), u behaves like the solution of the heat equation taking the same initial data as u plus a corrector term that we compute explicitely.

Wave Front Depinning Transition in Discrete One-Dimensional Reaction-Diffusion Systems

We establish the selfsimilar behavior of the solutions of the two and three dimensional vorticity equations for some classes of initial data. More precisely, any solution v of the two dimensional vorticity equation taking as initial data a finite Radon measure v0 is shown to be asymptotically equivalent to the fundamental solution of the heat equation with mass M = ∫ R2 v0 provided that |M | is small enough, in the following sense: t1− 1 p ‖v(t)−MG(t)‖Lp(R2) → 0 when t tends to ∞ for all 1 ≤ p ≤ ∞. This extends a result due to Giga and Kambe where the total variation of v0 was assumed to be small. If v is a solution of the three dimensional vorticity equation with initial data v0 in the Morrey space (M 3 2 (R))3 with zero divergence and small norm, such that λv0(λx) → μ in the sense of measures as λ →∞ and ‖v0‖ M 3 2 (|x|>R) → 0 when R →∞ then lim t→∞ t1− 3 2p ‖v(t)− ν(t)‖p = 0 for all 32 ≤ p ≤ ∞, where ν is the unique solution of the vorticity equation with initial data μ, which has been proved to be selfsimilar by Giga and Miyakawa. AMS Classification: 35B40, 35K05, 35K55, 76Rxx, 35Q35, 76D05.

Long-time Behaviour for Solutions of the Vlasov-Poisson-Fokker-Planck Equation

We consider spatially discrete bistable reaction-diffusion equations that admit wave front solutions. Depending on the parameters involved, such wave fronts appear to be pinned or to glide at a certain speed. We study the transition of traveling waves to steady solutions near threshold and give conditions for front pinning (propagation failure). The critical parameter values are characterized at the depinning transition, and an approximation for the front speed just beyond threshold is given.