- Habib AMMARI, DMA, Ecole normale supérieure, Paris
- Mourad CHOULLI, IECL, Université de Lorraine, Metz
- Yves COLIN DE VERDIÈRE, IF, Université Joseph Fourier, Grenoble
- Matias COURDURIER, Pontificia Universidad Catolica de Chile
- Charles DAPOGNY, LJK, Université Joseph Fourier, Grenoble
- Maher MOAKHER, ENIT, Tunisie
- Hoai-Minh NGUYEN, EPFL, Suisse
- Han WANG, Sivienn, Paris
The program.
The confirmed invited speakers:
Title: Mathematics of super-resolution in resonant media.
Abstract: We first briefly review existing super-resolution techniques in practise. We then focus on the particular approach of using resonant media. Two cases are analyzed. One is by using Helmholtz resonators and the other is by using high contrast material. We show that super-resolution in these cases are due to sub-wavelenghth resonant modes which can propagate into the far field. Our conclusion is based on the analysis of the Green’s function in the associated media. The work is motivated by the experiment [3].
References [1] H. Ammari and H. Zhang. Super-resolution in high contrast media, submitted, arX- iv:1412.2214. [2] H. Ammari and H. Zhang. A mathematical theory of super-resolution by using a system of sub-wavelength Helmholtz resonators, Comm. Math. Physics, accepted. [3] F. Lemoult, M. Fink and G. Lerosey. Acoustic resonators for far-field control of sound on a subwavelength scale, Phys. Rev. Lett., 107 (2011), 064301.
Title: Stability for the multi-dimensional Borg-Levinson theorem with partial spectral data.
Abstract: I will sketch the proof of a stability estimate related to the multi-dimensional Borg-Levinson theorem of determining a potential from spectral data: the Dirichlet eigenvalues λk and the normal derivatives ∂φk/∂ν of the eigenfunctions on the boundary of a bounded domain. The estimate is of H ̈lder type, and we allow o finitely many eigenvalues and normal derivatives to be unknown. I will also show that if the spectral data is known asymptotically only, up to O(k −α ) with α 1, then we still have Hölder stability. The results of this talk were obtained in collaboration with Plamen Stefanov.
Title: Weyl formulae for sub-Riemannian (sR) Laplacians.
Abstract: This is a work in progress with Luc Hillairet (Orléans) and Emmanuel Trélat (Paris 6). I plan to explain what is a sR Laplacian on a manifold and what kind of Weyl formulae are known or to be showed. I will focus on 3D contact manifolds and more generally to generic singularities of Pfaff distributions. I will also describe some ergodic properties of the eigenfunctions "à la Schnirelman". An inverse problem concerning Arnold-Hopf invariants will be presented in order to meet the main thema of the workshop.
Title: Image Reconstruction using Scattering in SPECT.
Abstract: In the medical imaging technique of Single-Photon Emission Computed Tomography (SPECT), the image of a source distributed inside the patient is obtained by measuring ballistic photons exiting the patient and assuming a known attenuation map. In our work we consider an extended model for SPECT obtained after a number of assumptions and simplifications in the Radiative Transfer Equation, and by measuring the ballistic and scattered photons we attempt to reconstruct an image of the source and attenuation maps at the same time. In this talk we will present the inverse problem arising in this extended model, results on the injectivity of the inverse problem and numerical experiments on synthetic and real data, with encouraging results about the feasibility of this approach. (joint work with CIB-UC, J.C. Quintana, F. Monard, A. Osses, F. Romero)
Title: Multi-phase optimization via a level set method.
Abstract: Optimizing the geometry of the repartition of several elastic materials among a fixed working domain has long been an issue of utmost important in structural design, and has led to significant mathematical developments (e.g. in the theory of homogenization). In this presentation, we investigate the sensitivity of the performance of a mixture of several linear elastic materials with respect to the geometry of the interface between the different phases. It turns out that the involved shape derivatives bring into play the jumps of discontinuous quantities at the interfaces, which are very difficult to evaluate numerically. To alleviate this difficulty, we propose a smoothed version of this model, in which the `sharp' interface between the different phases is smeared into a fixed band of (small) thickness. This change in perspective relies on the use of the notion of signed distance function, and its dependence on the domain itself. While it is more complicated at first glance, this new model gives rise to more usable shape derivatives in the numerical context, and it is proved to be mathematically consistant with the `sharp-interface model' it is meant to approximate. What's more, this model has its own interest when it comes to modeling non monotone interfaces. Eventually, several numerical examples are discussed to appraise the features of the proposed analysis. This is a joint work with G. Allaire (CMAP), G. Delgado (CMAP & EADS) and G. Michailidis (CMAP & Renault).
Title: A Slepian framework for a localized source identification problem in physical geodesy.
Abstract: In this talk, we are interested in the inverse problem that consists in the determination of a distribution of point masses (characterized by their intensities and positions), such that the potential generated by them best approximates a given potential field. On the whole Earth, a potential function is usually expressed in terms of spherical harmonics which are basis functions with global support. The identification of the two potentials is done by solving a least-squares problem. When only a limited area of the Earth is studied, the estimation of the point-mass parameters by means of spherical harmonics is prone to error, since they are no longer orthogonal over a partial domain of the sphere. The point-mass determination problem on a limited region is treated by the construction of a local spherical harmonic basis that is orthogonal over the specified limited domain of the sphere. We propose an iterative algorithm for the numerical solution of the local point mass determination problem and give some results on the robustness of this reconstruction process. We also study the stability of this problem against added noise. Some numerical tests are presented and commented.
Title: Reflecting complementary media and superlensing using complementary media for electromagnetic waves.
Abstract: Complementary media was suggested by Pendry and Ramakrishna and is an important notion in the study of negative index materials. A rich subclass of complementary media is the class of reflecting complementary media. This class has played a critical role in various applications of negative index materials such as superlensing and cloaking using complementary media, and cloaking via anomalous localized resonance. Reflecting complementary media and its applications has been so far studied in the acoustic setting. In this talk, I will discuss reflecting complementary media and superlensing using complementary media for electromagnetic waves. The analysis of these two problems is based on new results on the compactness, existence, and stability for the Maxwell equations. The removing of localized singularity technique plays an important role in the study of the superlensing problem.
Title: Shape Identification in electrolocation and echolocation.
Abstract: The aim of this talk is to exhibit possible mechanisms of shape identification observed in some biological phenomena. In electrolocation, the weakly electric fish orient themselves at night in complete darkness by employing their active electrosensing system. They generate a stable, high-frequency, weak electric field and perceive the transdermal potential modulations caused by a nearby target with different admittivity than the surrounding water. The electric fish would first locate the target using a specific location search algorithm. Then it could extract, from the perturbations of the electric field, features (generalized polarization tensors) of the target, and compute from the features shape descriptors invariant under rigid motions and scaling. When the data contain sufficient information, the fish could extract local features to perceive visually the geometric character of the target. This paradigm including feature extraction and construction of invariant shape descriptors can also be applied to other nonlinear inverse problems like the echolocation, a phenomenon that the bat use their bio sonar to identify targets.