Séminaire de mathématiques appliquées (archives)

Afin de développer de nouveaux schémas numériques préservant l’asymptotique pour des équations cinétiques en régimes hyperbolique ou diffusif, il est intéressant d’appliquer la technique d’intégration projective introduite par Gear et Kevrekidis pour des grands systèmes différentiels apparaissant en chimie. Après avoir introduit l’intégration projective pour les équations cinétiques, j’exposerai notamment une méthode pour la montée en ordre de ces schémas.

Au cours de cet exposé, je présenterai une propriété d'hypercoercivité non-linéaire en mécanique des fluides. Je montrerai que sous cet objet mathématique se cache la présence de deux vitesses et d'une fraction de mélange $\kappa$. Cela permettra notamment de démontrer le caractère bien posé globalement en temps pour des systèmes de type faible nombre de Mach ou pour Navier-Stokes compressible avec viscosité dégénérée. Je discuterai également de plusieurs extensions de ces résultats. Cet exposé est le fruit de collaborations avec B. Desjardins, V. Giovangigli et E. Zatorska.

We consider the regression model\begin{equation} Y_{i}=g(x_i)+\varepsilon _{i},\,\,\,\,i=0,1,2...,n, \end{equation}where the regression function derivative has a jump point at an unknown position $\theta .$ We propose a nonparametric Kernel-based estimator of the jump location $\theta .$ Assume that $\sup_{\left| i-j\right| \geq k}\left| Cov\left( \varepsilon _{i},\varepsilon _{j}\right) \right|\leq Ck^{-\rho }$ for $0<\rho \leq 1.$ Under very general conditions, we prove the $(nh)^{\frac{-\rho}{2}}$ convergence rate of the estimator, where $h$ is the window of the kernel. This includes short-range dependent as well as long-range dependent and even non-stationary errors. Finally, we gives conditions on the windows $h$ to obtain the best rate of convergence. The obtained rate is known to be optimal for i.i.d. errors as well as for LRD errors

Cette présentation porte sur les équations gyro-cinétiques et traite un développement rigoureux des limites de l'équation de Vlasov avec différents opérateurs de collision dans un champ magnétique fort, ainsi que du développement de méthodes numériques basées sur une décomposition micro-macro de la fonction de distribution des particules.

We present a discretization strategy for compressible fluids systems for unstrutctured grids based a Lagrange-Remap approach that does not involve any moving mesh. We present the stability properties of this solver and present a natural semi-implicit extension of the method that allows to remain stable under a CFL condition involving only the material velocity. We present another modification of the solver that allows to provide an accurate and stable solver for simulation involving low-Mach regions in the flow. This work is a collaboration with Christophe CHALONS and Mathieu GIRARDIN.

Un certain nombre d'équations aux dérivées partielles stochastiques très singulières présentent des problèmes dans leur définition même. C'est le cas, entre autres, de l'équation de KPZ, mais aussi de l'équation de quantisation stochastique en dimension 3. Les méthodes classiques d'analyse ne permettent pas de définir cette équation, et il s'avère que pour lui donner un sens, il est nécessaire de soustraire un constante infinie et de considérer formellement un nouveau problème. La théorie des distributions paracontrolées, qui combine des idées de la théorie des chemins rugueux avec la décomposition de Paley-Littlewood et le paraproduit, est un bon cadre pour donner un sens à cette renormalisation, et résoudre (localement) cette équation.

Dans une première partie nous introduirons donc la notion de distributions paracontrolées, et dans une deuxième partie, nous montrerons comment cette théorie peut s'appliquer à l'équation de quantisation stochastique en dimension 3.

Chemotaxis is the phenomenon in which cells direct their motion according to a chemical present in their environment. Since experimental observations have shown that the motion of bacteria (e.g. Escherichia Coli) is due to the alternation of 'runs and tumbles', mathematical modelling thanks to a kinetic description has been proposed. The starting point of the study is the so-called Othmer-Dunbar-Alt model governing the dynamics of the distribution function of cells. From this system, macroscopic model can be derived after rescaling. When the taxis is small compared to the unbiased movement of cells, the scaling must be of diffusive type.

We investigate the coupling of the Multi-dimensional Optimal Order Detection (MOOD) method and the Arbitrary high order DERivatives (ADER) approach in order to design a new high order accurate, robust and computationally efficient Finite Volume (FV) scheme dedicated to solve nonlinear systems of hyperbolic conservation laws on unstructured triangular and tetrahedral meshes in two and three space dimensions, respectively.

The Multi-dimensional Optimal Order Detection (MOOD) method for 2D and 3D geometries has been introduced in a recent series of papers [1, 2, 3] for mixed unstructured meshes. It is an arbitrary high-order accurate Finite Volume scheme in space, using polynomial reconstructions with a posteriori detection and polynomial degree decrementing processes to deal with shock waves and other discontinuities.

In the following work, the time discretization is performed with an elegant and efficient one-step ADER procedure [4, 5]. Doing so, we retain the good properties of the MOOD scheme, that is to say the optimal high-order of accuracy is reached on smooth solutions, while spurious oscillations near singularities are prevented. The ADER technique permits not only to reduce the cost of the overall scheme as shown on a set of numerical tests in 2D and 3D, but it also increases the stability of the overall scheme.
A systematic comparison between classical unstructured ADER-WENO schemes and the new ADER-MOOD approach has been carried out for high-order schemes in space and time in terms of cost, robustness, accuracy and efficiency.

A large suite of classical numerical test problems has been solved on unstructured meshes for three challenging multi-dimensional systems of conservation laws: the Euler equations of compressible gas dynamics, the classical equations of ideal magneto-Hydrodynamics (MHD) and finally the relativistic MHD equations (RMHD), which constitutes a particularly challenging nonlinear system of hyperbolic partial differential equation.

If time permits we will present the extension of the a posteriori treatment to construct a subcell limiter for Discontinuous Galerkin methods of high accuracy (polynomial degree 9).

References:
[1] S. Clain, S. Diot, and R. Loubère, A high-order finite volume method for systems of conservation laws-multi-dimensional optimal order detection (MOOD). Journal of Computational Physics, 230(10):4028 – 4050, 2011.
[2] S. Diot, S. Clain, and R. Loubère, Improved detection criteria for the multi-dimensional optimal order detection (MOOD) on unstructured meshes with very high-order polynomials. Computers and Fluids, 64:43 – 63, 2012.
[3] S. Diot, R. Loubère, and S. Clain, The MOOD method in the three-dimensional case: Very-high-order finite volume method for hyperbolic systems. International Journal of Numerical Methods in Fluids, 73:362–392, 2013.
[4] M. Dumbser, Arbitrary high order PNPM schemes on unstructured meshes for the compressible Navier- Stokes equations. Computers and Fluids, 39:60–76, 2010.
[5] M. Dumbser, M. Castro, C. Parés, and E.F. Toro, ADER schemes on unstructured meshes for non-conservative hyperbolic systems: Applications to geophysical flows. Computers and Fluids, 38:1731 – 1748, 2009.

In this presentation, we focus on a theoretical analysis and the use of statistical and optimization methods in the context of sparse linear regressions in a high-dimensional setting. The first part of this work is dedicated to the study of statistical learning methods, more precisely penalized methods and greedy algorithms. The second part concerns the application of these methods for gene regulatory networks inference. Gene regulatory networks are powerful tools to represent and analyse complex biological systems, and enable the modelling of functional relationships between elements of these systems. We thus propose to develop optimization methods to estimate relationships in such networks.