École normale supérieure de Cachan

About: École normale supérieure de Cachan is a education organization based out in Cachan, Île-de-France, France. It is known for research contribution in the topics: Decidability & Nonlinear system. The organization has 2717 authors who have published 5585 publications receiving 175925 citations.

IMITATOR 2.5: A Tool for Analyzing Robustness in Scheduling Problems

Abstract: The tool Imitator implements the Inverse Method (IM) for Timed Automata (TAs). Given a TA \(\mathcal{A}\) and a tuple π 0 of reference valuations for timings, IM synthesizes a constraint around π 0 where \(\mathcal{A}\) behaves in the same discrete manner. This provides us with a quantitative measure of robustness of the behavior of \(\mathcal{A}\) around π 0. The new version Imitator 2.5 integrates the new features of stopwatches (in addition to standard clocks) and updates (in addition to standard clock resets), as well as powerful algorithmic improvements for state space reduction. These new features make the tool well-suited to analyze the robustness of solutions in several classes of preemptive scheduling problems.

Nonintrusive reduced‐order modeling of parametrized time‐dependent partial differential equations

Abstract: We propose a nonintrusive reduced-order modeling method based on the notion of space-time-parameter proper orthogonal decomposition (POD) for approximating the solution of nonlinear parametrized time-dependent partial differential equations. A two-level POD method is introduced for constructing spatial and temporal basis functions with special properties such that the reduced-order model satisfies the boundary and initial conditions by construction. A radial basis function approximation method is used to estimate the undetermined coefficients in the reduced-order model without resorting to Galerkin projection. This nonintrusive approach enables the application of our approach to general problems with complicated nonlinearity terms. Numerical studies are presented for the parametrized Burgers' equation and a parametrized convection-reaction-diffusion problem. We demonstrate that our approach leads to reduced-order models that accurately capture the behavior of the field variables as a function of the spatial coordinates, the parameter vector and time. © 2013 Wiley Periodicals, Inc. Numer Methods Partial Differential Eq 2013

Magnetic imaging with an ensemble of nitrogen-vacancy centers in diamond

Abstract: The nitrogen-vacancy (NV) color center in diamond is an atom-like system in the solid-state which specific spin properties can be efficiently used as a sensitive magnetic sensor. An external magnetic field induces Zeeman shifts of the NV center levels which can be measured using optically detected magnetic resonance (ODMR). In this work, we quantitatively map the vectorial structure of the magnetic field produced by a sample close to the surface of a CVD diamond hosting a thin layer of NV centers. The magnetic field reconstruction is based on a maximum-likelihood technique which exploits the response of the four intrinsic orientations of the NV center inside the diamond lattice. The sensitivity associated to a 1 μm2 area of the doped layer, equivalent to a sensor consisting of approximately 104 NV centers, is of the order of 2 μT/√Hz. The spatial resolution of the imaging device is 480 nm, limited by the numerical aperture of the optical microscope which is used to collect the photoluminescence of the NV layer. The effectiveness of the method is illustrated by the accurate reconstruction of the magnetic field created by a DC current inside a copper wire deposited on the diamond sample.

The weakly nonlinear large-box limit of the 2D cubic nonlinear Schrödinger equation

Abstract: We consider the cubic nonlinear Schrodinger (NLS) equation set on a two dimensional box of size $L$ with periodic boundary conditions. By taking the large box limit $L \to \infty$ in the weakly nonlinear regime (characterized by smallness in the critical space), we derive a new equation set on $\R^2$ that approximates the dynamics of the frequency modes. This nonlinear equation turns out to be Hamiltonian and enjoys interesting symmetries, such as its invariance under Fourier transform, as well as several families of explicit solutions. A large part of this work is devoted to a rigorous approximation result that allows to project the long-time dynamics of the limit equation into that of the cubic NLS equation on a box of finite size.