Generalized fast marching method for computing highest threatening trajectories with curvature constraints and detection ambiguities in distance and radial speed

Work presented at the 9th Conference on Curves and Surfaces, 2018, ArcachonWe present a recent numerical method devoted to computing curves that globally minimize an energy featuring both a data driven term, and a second order curvature penalizing term. Applications to image segmentation are discussed. We then describe in detail recent progress on radar network configuration, in which the optimal curves represent an opponent's trajectories

Cloaking and anamorphism for light and mass diffusion

We first review classical results on cloaking and mirage effects for electromagnetic waves. We then show that transformation optics allows the masking of objects or produces mirages in diffusive regimes. In order to achieve this, we consider the equation for diffusive photon density in transformed coordinates, which is valid for diffusive light in scattering media. More precisely, generalizing transformations for star domains introduced in [Diatta and Guenneau, J. Opt. 13, 024012, 2011] for matter waves, we numerically demonstrate that infinite conducting objects of different shapes scatter diffusive light in exactly the same way. We also propose a design of external light-diffusion cloak with spatially varying sign-shifting parameters that hides a finite size scatterer outside the cloak. We next analyse non-physical parameter in the transformed Fick's equation derived in [Guenneau and Puvirajesinghe, R. Soc. Interface 10, 20130106, 2013], and propose to use a non-linear transform that overcomes this problem. We finally investigate other form invariant transformed diffusion-like equations in the time domain, and touch upon conformal mappings and non-Euclidean cloaking applied to diffusion processes.Comment: 42 pages, Latex, 14 figures. V2: Major changes : some formulas corrected, some extra cases added, overall length extended from 21 pages (V1) to 42 pages (present version V2). The last version will appear at Journal of Optic

On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials

The set $GU_f$ of possible effective elastic tensors of composites built from two materials with elasticity tensors \BC_1>0 and \BC_2=0 comprising the set U=\{\BC_1,\BC_2\} and mixed in proportions $f$ and $1-f$ is partly characterized. The material with tensor \BC_2=0 corresponds to a material which is void. (For technical reasons \BC_2 is actually taken to be nonzero and we take the limit \BC_2\to 0). Specifically, recalling that $GU_f$ is completely characterized through minimums of sums of energies, involving a set of applied strains, and complementary energies, involving a set of applied stresses, we provide descriptions of microgeometries that in appropriate limits achieve the minimums in many cases. In these cases the calculation of the minimum is reduced to a finite dimensional minimization problem that can be done numerically. Each microgeometry consists of a union of walls in appropriate directions, where the material in the wall is an appropriate $p$-mode material, that is easily compliant to $p\leq 5$ independent applied strains, yet supports any stress in the orthogonal space. Thus the material can easily slip in certain directions along the walls. The region outside the walls contains "complementary Avellaneda material" which is a hierarchical laminate which minimizes the sum of complementary energies.Comment: 39 pages, 11 figure

Fractal Weyl law for the Ruelle spectrum of Anosov flows

On a closed manifold $M$, we consider a smooth vector field $X$ that generates an Anosov flow. Let $V\in C^{\infty}\left(M;\mathbb{R}\right)$ be a smooth potential function. It is known that for any $C>0$, there exists some anisotropic Sobolev space $\mathcal{H}_{C}$ such that the operator $A=-X+V$ has intrinsic discrete spectrum on $\mathrm{Re}\left(z\right)>-C$ called Ruelle-Pollicott resonances. In this paper, we show that the density of resonances is bounded by $O\left(\left\langle \omega\right\rangle ^{\frac{n}{1+\beta_{0}}}\right)$ where $\omega=\mathrm{Im}\left(z\right)$, $n=\mathrm{dim}M-1$ and $0<\beta_{0}\leq1$ is the H\"older exponent of the distribution $E_{u}\oplus E_{s}$ (strong stable and unstable). We also obtain some more precise results concerning the wave front set of the resonances states and the group property of the transfer operator. We use some semiclassical analysis based on wave packet transform associated to an adapted metric on $T^{*}M$ and construct some specific anisotropic Sobolev spaces