2020 - 2021 Program

Title : A duality between scattering poles and transmission eigenvalues in scattering theory (slides).

Houssem Haddar: Ecole Polytechnique. France.

Abstract: We develop a conceptually unified approach for characterizing and determining scattering poles and interior eigenvalues for a given scattering problem.

Our approach explores a duality stemming from interchanging the roles of incident and scattered fields in our analysis. Both sets are related to the kernel of the relative scattering operator mapping incident fields to scattered fields, corresponding to the exterior scattering problem for the interior eigenvalues, and the interior scattering

problem for scattering poles. Our discussion includes the scattering problem for a Dirichlet obstacle where duality is between scattering poles and Dirichlet eigenvalues, and the inhomogeneous scattering problem where the duality is between scattering poles and transmission eigenvalues. Our new characterization of the scattering poles leads to a numerical method for their computation in terms of scattering data for the corresponding interior scattering problem.

This is a joint work with F. Cakoni and D. Colton

Title: Turnpike control and deep learning (slides).

Enrique Zuazua : Friedrich-Alexander-Universität Erlangen-Nürnberg, Germany. Deusto Foundation, Bilbao, Spain. Universidad Autónoma de Madrid, Spain.

Abstract: The tunrpike principle asserts that in long time horizons optimal control strategies are nearly of a steady state nature. In this lecture we shall survey on some recent results on this topic and present some its consequences on deep supervised learning.

This lecture will be based in particular on recent joint work with C. Esteve, B. Geshkovski and D. Pighin.

Title : Is there a Golden Parachute in Sannikov's principal-agent problem ? (slides)

Nizar Touzi. Ecole Polytechnique. France.

Abstract: We provide a complete review of the continuous--time optimal contracting problem introduced by Sannikov (2008). This classical problem models the interaction between two parties by a non-zero sum Stackelberg stochastic differential game. The agent's problem is to seek for optimal effort, given the compensation scheme proposed by the principal over a random horizon. Then, given the optimal agent's response, the principal determines the best compensation scheme in terms of running payment, retirement, and lump--sum payment at retirement.

A Golden parachute is a situation where the agent ceases any efforts at some positive stopping time, and receives a payment afterwards, possibly consisting of a lump sum and/or a continuous stream of payments. We show that a Golden Parachute only exists in certain specific circumstances. More precisely, we prove that an agent with positive reservation utility is either never retired by the principal, or retired above some given threshold. In particular, different discount factors induce naturally a face-lifted utility function, which allows to reduce the whole analysis to the equal-discount factors setting. Finally, we also confirm that an agent with small reservation utility does have an informational rent, meaning that the principal optimally offers him a contract with strictly higher utility value.

Title: On complete monotonicity, Hausdorff 's moment characterization theorem and Mellin transform, with applications to infinite divisibility of probability measures (Video)

Wissem Jedidi, King Saud Uniersity, Ryadh, saudi Arabia.

Abstract: The class of completely monotone functions and Bernstein functions is important in the context of infinitely divisible probability measures of $(0,\infty)$. We shall provide for them several new characterizations via two approaches: the first one is driven algebraically via elementary preserving mappings and the second one is developed in terms of the behavior of their restriction on $N$. As a consequence, we give a complete answer to the following question: "Can we affirm that a function $f$ is completely monotone (resp. a Bernstein function) if we know that the sequence $(f (k))_k$ is completely monotone (resp. alternating)?". This approach constitutes a kind of converse to Hausdorff’s moment characterization theorem in the context of completely monotone sequences. With closely related tools, we solve an open problem raised by Harkness and Shantaram (1969) who obtained, under sufficient conditions, a limit theorem in law for sequences of nonnegative random variables build with the iterated stationary excess operator. We show that the conditions of Harkness and Shantaram are actually necessary; continuous time convergence is equivalent to discrete time convergence; and the only possible limits in distribution are mixture of exponential with log-normal distributions.

Website https://www.researchgate.net/profile/Wissem_Jedidi

Title: Issues in random networks: The Apollonian network as a case study.(Video)

Hosam Mahmoud, Department of Statistics. The George Washington University. Washington, District of Columbia, USA. Personal WEBSITE

Abstract: We briefly review a number of random networks in recent areas of interest of the speaker, and discuss the issues that arise. We present the Apollonian network as a case study. We study the distribution of the degrees of vertices as they age in the evolutionary process. Asymptotically, the (suitably-scaled) degree of a node with a fixed label has a Mittag-Leffler-like limit distribution. The degrees of nodes of later ages have different asymptotic distributions, influenced by the time of their appearance. The very late arrivals have a degenerate distribution. The result is obtained via triangular Pólya urns. Also, via the Bagchi-Pal urn, we show that the number of terminal nodes asymptotically follows a Gaussian law. We prove that the total weight of the network asymptotically follows a Gaussian law, obtained via martingale methods. Similar results carry over to the sister structure of the k-trees.

Title : Application Of The Topological Gradient To Generate Parsimonious Neural Networks. (Link to the video).

Mohamed Masmoudi. Adagos & Université Paul Sabatier. France.

Abstract : We persent a new deep learning approach based on native parsimony using the topological gradient method. This approach reduces the number of links to be identified by the learning process by several orders of magnitude and consequently, the required amount of learning data is reduced proportionally; this is a significant development when compared to methods for simplification of redundant networks.

Native parsimonious approaches outperform state-of-the-art methods when the response of the model is continuous and a high level of accuracy is expected, like long term prediction and data compression.

We will present some industrial applications.

Title: On bandwidth selection problems in nonparametric trend estimation under stationary martingale difference errors. Application to stochastic volatility error processes. (Link to the video) (Link to the slides).

Sana Louhichi. University of Grenoble. France.

Abstract: This talk is around nonparametric regression, (known also as "learning a function" in machine learning).

Recall that, the purpose of nonparametric regression is to describe and to analyse the trend between a response variable and one or more predictors. This subject was studied by several authors since 1964 and is still relevant, this is due to the fact that nonparametric regression has a lot of applications in different fields (such as in economics, medicine, biology, physics, environment, social sciences and the list is non-exhaustive).

More precisely, we are interested in the problem of smoothing parameter selection in nonparametric curve estimation under dependent errors.

We focus on kernel estimation and the case when the errors form a general stationary sequence of Martingale Difference Sequence (MDS, in short) where neither linearity assumption nor``all moments are finite" are required.

We compare the behaviors of the smoothing bandwidths obtained by minimizing either the unknown average squared error, the theoretical mean average squared error, a Mallows-type criterion adapted to the dependent case and the family of criteria known as generalized cross validation (GCV) extensions of the Mallows criterion. We prove that these three minimizers and those based on the GCV family are first-order equivalent in probability.

We give also a normal asymptotic behavior of the gap between the minimizer of the average square error and that of the Mallows-type criterion. We extend this result to the GCV family.

Finally, we apply our theoretical results to a specific case of MDS, namely to a class of stochastic volatility process.

A simulation study is conducted. To establish our theoretical results, we make use of some ingredients adapted to our case of dependent data.

Their proofs are based, in particular, on Marcinkiewicz-Zygmund type inequalities or maximal moment inequalities for MDS, weighted sums of MDS or quadratic forms for MDS that we establish using Burkholder-type moment inequalities together with some chaining arguments. Recall that chaining is a nice approach to approximate the supremum, over a non-countable set, of stochastic processes. A central limit theorem for triangular arrays of quadratic forms for MDS is also needed for the proofs of our results.

This Talk is is based on a joint paper with Karim Benhenni and Didier Girard of the LJK Laboratory. (University of Grenoble Alpes & CNRS).

Title : TREASURE, a multidisciplinary research network for water REUSE - A modeling and contrtol perspective. (Link to slides) (Link to video).

Jérôme HARMAND. Senior Scientist. website

Coordinator of the Research Euro-mediterranean network TREASURE (cf. https://www.inrae.fr/treasure).

Abstract: TREASURE - for "treatment and sustainable reuse of water in the mediterranean rim", is a international research network

dedicated to the development and the application of methods from automatic control and applied mathematics to optimze water REUSE.

With a low management load and a very flexible structure, TREASURE aims at promoting both research and education activities, the improvement

and the diffusion of knowledge on bioprocesses and microbial ecosystems – notably through co-advisoring of PhDs which is an important component of the project.

A key issue is to promote pluridisciplinary research activities in keeping close bond with applications.

The seminar will present the network, its scientific activities in terms of modeling and control of bioprocesses and microbial ecosystems for optimizing water REUSE.

Title : Controllability of the Wave Equation on a rough compact manifold.

(Link to video) (Link to slides).

Belhassen Dehmane. Faculté des Sciences de tunis. Tunisia. (website)

Abstract : The property of controllability for the wave equation has been intensively studied, mainly in a smooth framework ( smooth metric and smooth domain ). In this lecture, I will present some results on observability/control for the wave equation with rough coefficients.

More precisely, we prove that the property of exact internal or boundary controllability for a wave equation with smooth coefficients is stable with respect to Lipschitz perturbations of the metric.

We also consider the case of a C^1 metric ( the hamiltonian field is only continuous ) and we prove the propagation up to the boundary, of semi-classical measures support. The generalized Geometric Control Condition is then sufficient for exact control.

This talk comes from joint works with J. Le Rousseau ( Univ. Paris 13 ) and N. Burq ( Univ. Paris Sud ).

Title: When should one stop iterations in a domain decomposition method?

Michel Kern. Researcher at INRIA, Paris. France.

Abstract: It is customary to run domain decomposition (DD) iterations until some measure of the residual gets below a user--chosen threshold. This threshold must strike a balance between computing an accurate solution, and not using too much computing resources.

In this talk, I will present an automated choice of the threshold, based on fully computable a posteriori error estimates. The estimates allow for the separation of the part of the error due to the space and time discretizations, and that due to the DD iterations. They are based on reconstruction techniques, building a continuous pressure and conforming fluxes. The fluxes require the solution of local Neumann problems in narrow bands around the interfaces.

The method is applied to subsurface flow simulation in mixed formulation, first for a steady state model, then for a transient

situation. In the latter case the DD method is global in time and uses the Optimized Schwarz Waveform Relaxation algorithm, where the parameters in the Robin conditions can be optimized to accelerate the convergence rate, enabling the use of different time grids. An application to a simplified nuclear waste simulation is shown. An extension to a nonlinear diffusion model will be presented, with an application to capillary trapping.

This is joint work with Sarah Ali-Hassan, Elyes Ahmed, Caroline Japhet and Martin Vohralík,

Title : XLiFE++, An eXtended Library of Finite elements in C++.

(Link to the video)

Nicolas Kielbasiewicz. ENSTA Paris. France.

Abstract: XLiFE++ is an FEM-BEM C++ code developed by P.O.e.m.s. laboratory and I.R.M.A.R. laboratory, that can solve 1D / 2D / 3D, scalar / vector, transient / stationnary / harmonic problems. It provides:

• Advanced mesh tools, with refinement methods, including an advanced interface to the mesh generator Gmsh

• High order Lagrange Finite Elements (every order)

• High order edge Finite Elements (every order)

• Boundary Elements methods: Laplace, Helmholtz, Maxwell, Stokes (in progress)

• Essential conditions (periodic, quasi-periodic)

• Absorbing conditions: DtN, PML, ...

• Export to visualization tools such as Gmsh, Paraview, Matlab

• Many solvers (direct solvers, iterative solvers, eigenvalue solvers, wrappers to Arpack, UmfPack and Eigen libraries)

Title: Hölder stability of quantitative photoacoustic tomography based on partial data.

(Link to the video) (link to slides)

Faouzi triki. Grenoble Alpes University. France.

Abstract: We consider the problem of reconstructing the diffusion and absorption coefficients of the diffusion equation from internal information of the solution.

In practice, the internal information is obtained from the first step of the inverse photoacoustic tomography, and is only partially provided near the boundary

due to the high absorption property of the medium and the limitation of the equipment. Our main contribution is to prove a Hölder stability of the inverse problem

in a subregion where the internal information is reliably supplied based on the stability estimation of a Cauchy problem satisfied by the diffusion coefficient. The exponent

of the Hölder stability converges to a positive constant independent of the subregion as the subregion contracts towards the boundary. Numerical experiments demonstrates

that it is possible to locally reconstruct the diffusion and absorption coefficients for smooth and even discontinuous media.

Title: Imaging junctions of waveguides with sampling methods. (link to slides).

Laurent Bourgeois. Ecole Nationale Supérieure de Techniques Avançées. France.

Abstract: in industrial metallic structures like tubes, pipes, plates... some defects often appear in junctions, in particular in weld bead. This is why it is useful to image not only waveguides but junctions of waveguides. Sampling methods such as the Linear Sampling Method (Colton and Kirsch, 96) are very efficient to image waveguides by taking advantage of the guided modes. Sampling methods mainly rely on the fundamental solution, which has a closed-form in waveguides. But when it comes to junctions of waveguides, the fundamental solution has not a simple form any more, which is an issue from the computational point of view. In this talk, and after considering the homogeneous waveguide as a reference case, we will explain how to adapt the LSM to such situations, first to the case of an abrupt change of section, and then to a junction of two or more waveguides. Some numerical experiments will illustrate the feasibility of our approach. A more challenging problem will be presented as a perspective, that is the case of a closed waveguide partially embedded in an open waveguide. This is a joint work with Jean-François Fritsch and Arnaud Recoquillay, both from the CEA-List.

Title: Mathematical Modelling of the electrical activity of the heart from ion-channels to the body surface: Forward and Inverse problems.

(Link to the video) (link to slides)

Nejib Zemzemi. Institut de Recherches en Informatique et Automatique. France.

Abstract:

Meaningful computer based simulations of electrocardiograms (ECGs), linking models of the electrical activity of the heart to ECG signals, are a necessary step towards the development of personalized cardiac models from clinical ECG data. An ECG simulator is, in addition, a valuable tool for building a virtual data base of pathological conditions, to test and train medical devices but also to improve the knowledge on the clinical significance of some ECG signals. In this talk we will present a 3D multi-scale mathematical model based on reaction diffusion equations, named bidomain equations, representing the propagation of the electrical wave in the heart domain. These equations are coupled to a set of dynamic systems representing the electrical activity of cardiac cells. The heart model is then coupled to the Laplace equation representing the diffusion of the electrical potential in the torso. As examples of application, we will first show how this mathematical model could be used in drug cardio-toxicity assessment in safety pharmacology. Then we will show the potential of this mathematical model to improve the diagnosis of cardiac pathologies like ventricular tachycardia and fibrillation by solving various electrocardiography imaging inverse problems. A comparison between physical based and data-driven aproaches will be presented.

Title : The ants walk: finding geodesics in graphs using reinforcement learning.

Cecile Mailler: Department of Mathematical Sciences, University of Bath, Bath BA2 7AY, UK.

Abstract: How does a colony of ants find the shortest path between its nest and a source of food without any means of communication other than the pheromones each ant leave behind itself?

In this joint work with Daniel Kious (Bath) and Bruno Schapira (Marseille), we introduce a new probabilistic model for this phenomenon. In this model, the nest and the source of food are two marked nodes in a finite graph. Ants perform successive random walks from the nest to the food, and the distribution of the n-th walk depends on the trajectories of the (n-1) previous walks through some linear reinforcement mechanism.

Using stochastic approximation methods, couplings with Pólya urns, and the electric conductances method for random walks on graphs (which I will explain on some simple examples), we prove that, in this model, the ants indeed eventually find the shortest path(s) between their nest and the source food.

Title : Mathematical and numerical analysis of the PGD approximation to linear parametric elliptic PDEs. (Link to Video)

Chacón Rebollo, Tomás. Instituto de Matemáticas de la Universidad de Sevilla.

Abstract: The PGD (Proper Generalized Decomposition) is a performing technique to obtain tensorized solutions of parametric PDEs. It is increasingly used in many sectors of industrial applications and attracting the interest of a wide community of researchers. There is also an increasing interest in providing a mathematical understanding of its performances, that could lead to the design of improved methods.

In this talk we give a characterisation of the PGD solution to symmetric linear elliptic systems in terms of variational non-convex problems. This provides a theoretical framework to prove the convergence of the PGD expansion, and to compute the expansion modes by Power Iteration Method. Further, to prove the convergence of the fully discrete PGD with respect to the discretization of the PDE and that of the parameter set. We present some numerical tests to highlight the theoretical results.

We finally show how to extend the PGD to non-symmetric elliptic problems with improved numerical performances.

Title : Periodic Polya urns, the density method and asymptotics of young tableaux. (Link to Video)

Cyril Banderier. Laboratoire d'Informatique de Paris-Nord

UMR CNRS 7030. Institut Galilée - Université Paris-Nord. France.

Abstract: Pólya urns are urns where at each unit of time a ball is drawn and replaced with some other balls according to its colour. We introduce a more general model: the replacement rule depends on the colour of the drawn ball and the value of the time (modp). We extend the work of Flajolet et al. on Pólya urns: the generating function encoding the evolution of the urn is studied by methods of analytic combinatorics. We show that the initial partial differential equations lead to ordinary linear differential equations which are related to hypergeometric functions (giving the exact state of the urns at time n). When the time goes to infinity, we prove that these periodic Pólya urns have asymptotic fluctuations which are described by a product of generalized gamma distributions. With the additional help of what we call the density method (a method which offers access to enumeration and random generation of poset structures), we prove that the law of the southeast corner of a triangular Young tableau follows asymptotically a product of generalized gamma distributions. This allows us to tackle some questions related to the continuous limit of random Young tableaux and links with random surfaces..

Title : Revisited model and recent results in shell theory

Adel Blouza. Rouens University. France.

Abstract : The well known Naghdi shell model is usually considered in local covariant or contravariant basis formulation. For general shells with little regularity, which present curvature discontinuities, formulations in Cartesian basis are essential. To perform conforming FE method for such shells, relaxed variational spaces and mixed or penalized formulations are used. The drawback of this interesting approach is its high computational cost.

In this talk, we introduce a functional framework in which an hybrid formulation for Naghdi’s equations is possible and show how we can reduce the number of degrees of freedom compared to classical methods. We also present some numerical tests to highlight the theoretical results.

Finally, with the idea of free-local basis formulation, we derive a model describing the unilateral contact of a shell with an obstacle and prove the well-posedness of the resulting system of variational inequalities.

Title : Explicit solution of the 2-D scattering problem by a wedge (link to slides).

Marc Lenoir. POEMS Lab. Ecole Nationale des Techniques Avançées. Centre National de Recherche Scientifique. France.

Abstract: The solution of high-frequency scattering problems involves asymptotic expansions which are only valid when the boundary of the obstacle is smooth enough. As a consequence, the case of wedges must be the subject of a special attention.

This talk will be devoted to one of the several methods which have been devised for the solution of the scattering problem in the vicinity of a corner, namely the Sommerfeld-Malyuzhinets method. There wil be many figures and a reasonable amount of formulas.

Title: Inverse problem for the anisotropic wave equations from the Dirichlet to Neumann map. (link to video)

Mourad Bellassoued. LAMSIN, National Engineer School of Tunis. Tunis El Manar University.

Abstract : In this talk we consider the inverse problem of determining on a compact Riemannian manifold the metric tensor in the wave equation with Dirichlet data from measured Neumann sub-boundary observations. This information is enclosed in the dynamical partial Dirichlet-to-Neumann map associated with the wave equation. We prove in dimension $n\geq 2$ that

the knowledge of the partial Dirichlet-to-Neumann map for the wave equation uniquely determines the metric tensor and we establish logarithm-type stability.