- Nos aurons le plaisir d'assister aux présentations suivantes (cliquez sur les images pdf pour les télécharger) :
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Mathieu COLLOWALD : Reconstruction of n-dimensional shapes from directional moments. Application to computed tomography.
The shape-from-moments problem is a well-known problem used in many domains like medicine or signal processing. It consists from measurements to recover a discrete approximation of the shape : a polytope. This problem was studied since many years but only in the 2D-case [1]. Recently and with the help of a new formula [2] which links n-dimensional polytopes and directional moments, we develop an algorithm to recover the vertices of the sought polytope, which is more efficient and more robust than a first one given in [3]. During the talk I will present the algorithm itself with some examples to illustrate it. As this method used simple mathematical tools and can be used in many domains, I hope being as interesting as the other activities proposed during the Colloque doctorants.
[1] Golub G., Milanfar P., Varah J., A stable numerical method for inverting shape from moments, 1999
[2] Beck M., Robins S., Computing the continuous discretely, 2007
[3] Gravin N., Lasserre J., Pasechnik D., Robins S., The inverse moment for convex polytopes, 2012 Keywords : shape-from-moments problem, inversion problem, symbolic-numeric computation -
Brice EICHWALD : Modified exponential integrator for nonlinear waves.
Efficient time stepping algorithms are crucial for accurate long time simulations of nonlinear waves. In particular, adaptive time stepping combined with an integrating factor [1] are known to be very effective. We propose a simple modification of the existing technique to improve the efficiency.
The trick consists in subtracting a certain-order polynomial to a PDE. Then, like for the integrating factor, a change of variables is performed to remove the linear part. But, here, we hope to remove something more to make the PDE less stiff to numerical resolution.
The polynomial is chosen as a low-order Taylor expansion around the initial time t0 of the solution. In order to calculate the different derivatives, we use a dense output which gives a ‘cheap’ possibility to approximate the derivatives of the solution at any time [2].
The modified integrating factor being applied, a classical time-stepping method can be used to solve the remaining equation. We focus on various Runge–Kutta schemes with a varying step size.
This method is similar to that of [3], the main differences being on the implementation and on the calculation of the derivatives. We take advantage of embedded methods and use an evolved adaptive step control. We do not need to calculate new functions and loose time of calculation only by using already estimated values during the temporal evolution.
Numerical tests show that the actual efficiency of the method varies along cases. For example, we verified that steeper waves profiles give rise to better behaviour of the method. For fully nonlinear water wave simulations, we can save up to 30% of total time steps with a Dormand-Prince Runge–Kutta scheme. For Serre’s equations, we save up to 95% with the Bogacki–Shampine scheme.
[1] J. D. Lawson. Generalized Runge-Kutta Processes for Stable Systems with Large Lipschitz Constants, SIAM Journal on Numerical Analysis, 4, 3,372-380, (1967).
[2] Ernst Hairer and Syvert Paul Nørsett and Gerhard Wanner. Solving Ordinary Differential Equations I: Nonstiff Problems, Springer Series in Computational Mathematics, (2000).
[3] S. Krogstad. Generalized integrating factor methods for stiff PDEs, Journal of Computational Physics, 203, 1, 72-88, (2005). -
Camille POIGNARD : On the Synchronization and Desynchronization of dynamical systems.
This talk is an overview of the questions dealt with in my thesis, for which I will briefly present basic tools on Dynamical Systems and Chaos theory. The second part will be short.
Given a dynamical system with a very stable (regular) behavior, we can wonder if it is possible to destabilize this stability, that is to say to induce Chaos in this system. This problem has obvious concrete motivations in systemic biology, for a natural way to destroy an embarrassing organism is to destabilize its metabolism. In this context, we'll explain how to prove that a particular Gene Regulatory Network represented by a system with four equations can behave in a chaotic way, for a good choice of the parameters.
From the reverse side, we can take interest in the Synchronization problem, which consists in making the trajectories of n dynamical systems behave identically. In this context we'll explain Global synchronization results for a Cantor set of systems organized hierarchically, that generalize the well-understood finite case. We'll finish by looking at the synchronization problem in case of an infinity of broken links appearing inside the infinite structure. -
Mathieu SART : Estimation par tests robustes.
Lorsque l'on observe n variables aléatoires indépendantes de loi P, une méthode traditionnelle pour estimer P est celle du maximum de vraisemblance. Malheureusement, cette méthode n'est pas générale et il existe de nombreux exemples dans lesquels cet estimateur est connu pour ne pas fonctionner. En outre, l'estimateur du maximum de vraisemblance n'est pas robuste, ce qui peut rendre son utilisation en pratique délicate. Une méthode alternative est d'utiliser des tests robustes pour estimer P comme cela fut proposé par Lucien Lecam il y a 45 ans. Cette stratégie a connu un regain d'intérêt dans les années 2000 lorsque Birgé à montré qu'elle permet d'obtenir des résultats très généraux dans divers cadres statistiques. Le but de l'exposer est de présenter ce domaine.