Séminaire Equations aux dérivées partielles
organisé par l'équipe Modélisation et contrôle
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Yvonne Alama Bronsard
Numerical approximations to nonlinear dispersive equations, from short to long times
16 janvier 2025 - 14:00Salle de conférences IRMA
The first part of this talk deals with the numerical approximation to nonlinear dispersive equations, such as the prototypical nonlinear Schrödinger equation. We introduce novel integration techniques allowing for the construction of schemes which perform well both in smooth and non-smooth settings. We obtain symmetric low-regularity schemes with very good structure preserving properties over long times. Higher order extensions will be presented, following new techniques based on decorated trees series inspired by singular stochastic PDEs via the theory of regularity structures. In the second part, we introduce a new approach for designing and analyzing schemes for some nonlinear and nonlocal integrable PDEs, including the well-known Benjamin-Ono equation. This work is heavily inspired by recent theoretical breakthroughs in the field of nonlinear integrable equations, and opens the way to numerical approximations which are far more accurate and efficient for simulating integrable PDEs, from short up to long times. -
Philippe Helluy
Schéma ALE aléatoire pour les écoulements bifluides compressibles. Application à la simulation du déferlement.
4 février 2025 - 14:00Salle de conférences IRMA
Le modèle Euler compressible bifluide ne présente pas de difficultés théoriques supplémentaires comparé au cas monofluide. Mais sa résolution numérique est notoirement plus difficile à cause du phénomène d'oscillations de pression à l'interface entre fluides. Nous présentons une approche basée sur un échantillonnage aléatoire "à la Glimm" à l'interface, qui permet de s'affranchir de ce défaut. Le schéma obtenu est applicable à des maillages non structurés, il a d'excellentes propriétés de robustesse et de convergence. Nous l'appliquons à des cas de déferlement. -
Simon Schneider
Estimatable Variation Neural Networks and their Application to Scalar Hyperbolic Conservation Laws
25 février 2025 - 14:00Salle 301
In this talk we introduce a class of neural networks for which a computationally cheap local estimate on the BV norm is available. The architecture of these networks is motivated by a linear function space we denote BMV. This space is the natural analogue to the space BPV of functions with bounded pointwise variation in one dimension. As the networks are elements of BMV, we are able to investigate the sharpness of the BV estimate. Further, we prove a universal approximation theorem in BMV and discuss practical considerations concerning the implementation.
We use these networks as ansatz functions to solve scalar hyperbolic conservations laws. Here, the big advantage of the estimate on the BV norm is that compactness in L¹ of sequences of networks can be enforced. For a loss function inspired by the finite volume method we are able to show convergence of sequences of networks under the assumption that the training error vanishes. Moreover, we show the existence of sequences of loss minimizing neural networks if the solution is an element of BMV. Several numerical test cases illustrate that it is possible to use standard techniques to minimize these loss functionals for networks with the proposed architecture. -
Elise Grosjean
Sensitivity analysis and non-intrusive two-grid reduced basis methods
4 mars 2025 - 14:00Salle de conférences IRMA
Sensitivity analysis is a crucial step in optimising the parameters of a parametric model. The objective is to determine the sensitivity of the model results to perturbations of its input parameters. In this talk, I will focus on two sensitivity analysis approaches based on differentiation (the direct and the adjoint methods). Solving a parametric problem can be less computationally expensive with the help of Reduced Basis Methods (RBM). I will present a website on RBM that I am currently developing (https://reducedbasis.github.io/), and then we will take a closer look at how to reduce computation times associated with sensitivity analysis using non-intrusive techniques inspired by the so-called two-grid method. -
Jade Le Quentrec
Theoretical and numerical study of the nonlinear Schrödinger equation with defects
11 mars 2025 - 14:00Salle de conférences IRMA
In this presentation, we present a study of the nonlinear Schrödinger equation involving a defect term, which materializes the presence of an impurity along a hypersurface.
First, we will adapt the techniques already used in the absence of a defect term to study the local or global well-posedness of our problem, as well as explosions in finite time. This will include a functional framework that takes into account the
singular term and a new viriel identity.
We will then focus on the schemes allowing us to compute numerical solutions. Special attention will be paid to the discretization of the default in finite differences. -
Giulia Sambataro
Model order reduction for parametric dynamical systems
1 avril 2025 - 14:00Salle de conférences IRMA
The numerical approximation of partial differential equations (PDEs) plays a crucial role in various fields, including engineering, mechanics and physics, for design and assessment. To accurately account for uncertainty in parameter values, we must solve the numerical model for a wide range of relevant parameters; an efficient numerical solution of this type of problem is even more challenging in a real time context.
Model order reduction (MOR) methods have the purpose to overcome the computational obstacle of numerical simulations to large-scale dynamical systems.
I will show linear and nonlinear solution manifold approximations and their validity for different applications in radioactive waste management and in contact mechanics.
Keywords: reduced basis method, Galerkin approximation, domain decomposition, supervised machine learning.