

The plenary lectures of the congress are the following:
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.
By minimizing Npartitions I mean partitions of a given bounded ddimensional 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 2dimensional torus then the regular hexagonal partition (when it exists) is the only minimizing Npartition. Apart from that result, not much is known about the structure of Npartitions when N is large. In particular when Eis a planar domain we expect that minimizing Npartitions 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).
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.
We are interested in an inverse problem for the wave equation. More precisely, it consists in the determination of an unknown timeindependent 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.
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 NavierStokes equations (results obtained in collaboration with O. Glass and S. Guerrero) or density dependent incompressible NavierStokes 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.
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 onedimensional wave equation involving a SturmLiouville 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 timeasymptotic 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.
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 apriori 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 denoising, 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