Fatiha Alabau (Lorraine)

Control questions and full source reconstruction by partial measurements for a class of hyperbolic systems of PDE'

We present some recent results on inverse problems concerned with source reconstructions for systems of PDE's. We consider situations for which we have only measurements on a subset of the unknowns. The challenging questions are then to determine if one can reconstruct all the sources, in spite of this partial information. We present several results in this direction. This work is in collaboration with Piermarco Cannarsa and Masahiro Yamamoto.

Giovanni Alberti (Università di Pisa):

On the structure of minimizing N-partitions for large N

By minimizing N-partitions I mean partitions of a given bounded d-dimensional domain E which consist of N sets (chambers) with equal volumes, and which minimize the total area of the interfaces between chambers. Among other results, T. C. Hales proved in 2001 that if E is the flat 2-dimensional torus then the regular hexagonal partition (when it exists) is the only minimizing N-partition. Apart from that result, not much is known about the structure of N-partitions when N is large. In particular when Eis a planar domain we expect that minimizing N-partitions should look hexagonal at least in some asymptotic sense.
In this talk I will give some partial results in this direction, and describe some open problems.
This is a work in progress with M. Caroccia (Carnegie Mellon University)
and Giacomo Del Nin (University of Pisa). 

Anne-Sophie Bonnet BenDhia (Ensta-Paris):

A perturbative method for the design of invisible obstacles

The question of whether an object can be “invisible" to acoustic or electromagnetic waves is of great interest for applications. Here, we suppose that the experimental setup is such that only a finite number of measures are available, and we say that an object is invisible if its presence does not modify these measures. The idea that we have developed is to use theoretical results of asymptotic analysis in order to choose a clever parametrization of the object (its shape and/or its material coefficients), with as many parameters as measures. Then the invisibility can be simply obtained by the implicit function theorem and numerical solutions by a fixed point algorithm.
Various illustrations will be presented and degenerate cases (where hypotheses of the implicit function theorem are not satisfied) will be discussed.

Maya de Buhan (CNRS & MAP5):

Convergent algorithm based on Carleman estimates for the recovery of a coefficient in the wave equation (with Lucie Baudouin and Sylvain Ervedoza)

We are interested in an inverse problem for the wave equation. More precisely, it consists in the determination of an unknown time-independent coefficient from a single measurement of the Neumann derivative of the solution on a part of the boundary. While its uniqueness and stability properties are already well known, we propose an original reconstruction algorithm and prove its global convergence thanks to Carleman estimates for the wave operator. The numerical implementation of this strategy presents some challenges that we propose to address in this talk. Several numerical examples will illustrate the efficiency of the algorithm.

Sylvain Ervedoza (CNRS & Institut de Mathématiques de Toulouse):

On the controllability of viscous fluid equations for non-homogeneous fluids.

In this talk, I will report on recent works on the local exact controllability to trajectories of viscous fluids when the density is not assumed to be constant. This includes in particular the case of compressible Navier-Stokes equations (results obtained in collaboration with O. Glass and S. Guerrero) or density dependent incompressible Navier-Stokes equations (results obtained in collaboration with M. Badra and S. Guerrero). In both cases, the main trick is to develop Carleman estimates adapted to the parabolic equation satisfied by the velocity field and to the transport equation satisfied by the density. One then needs to analyze the coupling of the equations and to decouple the parabolic and hyperbolic effects within the system of equation.

Yannick Privat (CNRS & Laboratoire Jacques-Louis Lions):

On the asymptotic of observability constants for wave like equations

In this talk, we investigate asymptotic properties of the observability constants for wave equations as the observation time T tends to infinity. We first consider a simplified one-dimensional wave equation involving a Sturm-Liouville operator and provide upper and lower estimates of the time observability constant in terms of the observation set Lebesgue measure. More generally, given a wave equation on a manifold without boundary, and given an arbitrary observation subset, we prove that the time-asymptotic observability constant is the minimum of two quantities: the first is a purely spectral one, and contains information on the quantum ergodic properties of the manifold; the second is purely geometric and gives an account for rays propagating in the manifold. We discuss some applications to control theory and shape optimization.

Carola-Bibiane Schönlieb (Cambridge):

Bilevel learning of variational regularisation models

When assigned with the task of reconstructing an image from imperfect data the first challenge one faces is the derivation of a truthful image and data model. In the context of regularised reconstructions, some of this task amounts to selecting an appropriate regularisation term for the image, as well as an appropriate distance function for the data fit. This can be determined by the a-priori knowledge about the image, the data and their relation to each other. The source of this knowledge is either our understanding of the type of images we want to reconstruct and of the physics behind the acquisition of the data or we can thrive to learn parametric models from the data itself. The common question arises: how can we optimise our model choice? 
In this talk we discuss a bilevel optimization method for learning optimal variational regularisation models. Parametrising regularisation and data fidelity terms, we will learn optimal total variation type regularisation models for image and video de-noising, and optimal data fidelity functions for pure and mixed noise corruptions.
 
This is joint work with M. Benning, L. Calatroni, C. Chung, J. C. De Los Reyes, T. Valkonen, and V. Vlacic 

Michael Vogelius (Rutgers University):