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Modélisation hybride modale-éléments finis pour le contrôle ultrasonore d'une plaque élastique. Traitement des intégrales oscillantes de la méthode HSM
No full textThis thesis focuses on the Half-Space Matching (HSM) method for solving scattering problems in an unbounded elastic plate to simulate non-destructive testing of composite plates. The HSM method is a hybrid approach that couples a finite element calculation in a box containing the defects with semi-analytical representations in four half-plates covering the plate's healthy part. Semi-analytical half-plate representations involve Green tensors, expressed with Fourier integrals and modal series. However, these expressions can be challenging to evaluate in practice (cost and accuracy), making the HSM method unusable industrially. The difficulties are first analyzed in a two-dimensional scalar (acoustic) case. Two methods are proposed for an efficient evaluation of Fourier integrals : the first one uses a far-field type approximation, and the second one is based on a deformation of the integration path in the complex plane (complexification method). These two methods are validated in the isotropic and anisotropic scalar cases, where we have the exact values of the Fourier integrals expressed using Hankel functions. They are then generalized to the three-dimensional case of the elastic plate. In this case, the representation formula is obtained by performing a Fourier transform in a direction parallel to the plate, and then, for each value of the Fourier variable ξ, a modal decomposition in the thickness. The modes involved called ξ-modes, are studied in detail and compared to classical modes (Lamb and SH in the isotropic case). In order to exploit the bi-orthogonality of the ξ modes, the half-plate formula requires the knowledge of both the displacement and the normal stress on the boundary. In the isotropic case, the analytic properties of the ξ-modes make it possible to justify and extend the complexification method, including in the presence of inverse modes. This reduces the spurious effects of modal coupling induced by the discretization of Fourier integrals. The complexification method is then used to calculate the operators involved in the HSM method, which derive from the half-plate formula. Different validations of the HSM method are thus carried out in the isotropic case. Encouraging preliminary results are also obtained for an orthotropic plate. The improvements significantly reduce the calculation time and ensure higher accuracy of the HSM method, making it possible to consider its systematic exploitation in an industrial simulation framework.Cette thèse porte sur la méthode Half-Space Matching (HSM) pour la résolution de problèmes de diffraction dans une plaque élastique non-bornée, en vue de la simulation du contrôle non-destructif de plaques composites. La méthode HSM est une approche hybride qui couple un calcul éléments finis dans une boite contenant les défauts, avec des représentations semi-analytiques dans quatre demi-plaques qui recouvrent la partie saine de la plaque. Les représentations semi-analytiques de demi-plaques font intervenir des tenseurs de Green, exprimés à l'aide d'intégrales de Fourier et de séries modales. Or ces expressions peuvent être délicates à évaluer en pratique (coût et précision), rendant la méthode HSM inexploitable industriellement. Les difficultés sont d'abord analysées dans un cas scalaire bidimensionnel (acoustique). Deux méthodes sont proposées pour une évaluation efficace des intégrales de Fourier : la première exploite une approximation de type champ lointain et la seconde repose sur une déformation du chemin d'intégration dans le plan complexe (méthode de la complexification). Ces deux méthodes sont validées dans les cas scalaires isotrope et anisotrope où l'on dispose des valeurs exactes des intégrales de Fourier exprimées à l'aide de fonctions de Hankel. Elles sont ensuite généralisées au cas tridimensionnel de la plaque élastique. Dans ce cas, la formule de représentation est obtenue en faisant une transformée de Fourier suivant une direction parallèle à la plaque, puis, pour chaque valeur de la variable de Fourier ξ, une décomposition modale dans l'épaisseur. Les modes mis en jeu, appelés ξ-modes, sont étudiés en détail et comparés aux modes classiques (Lamb et SH dans le cas isotrope). Afin d'exploiter la bi-orthogonalité des ξ-modes, la formule de demi-plaque requiert la connaissance à la fois du déplacement et de la contrainte normale sur la frontière. Dans le cas isotrope, les propriétés d'analyticité des ξ-modes permettent de justifier et d'étendre la méthode de la complexification, y compris en présence de modes inverses. Ceci réduit les effets de couplage modal parasite induits par la discrétisation des intégrales de Fourier. La méthode de la complexification est ensuite utilisée pour le calcul des opérateurs intervenant dans la méthode HSM, qui dérivent tous de la formule de demiplaque. Différentes validations de la méthode HSM sont ainsi effectuées dans le cas isotrope. Des résultats préliminaires encourageants sont également obtenus pour une plaque orthotrope. Les améliorations réalisées ont permis à la fois de réduire significativement le temps de calcul et d'assurer une plus grande précision de la méthode HSM, permettant d'envisager son exploitation systématique dans un cadre de simulation industrielle
A Symmetry-Aware Exploration of Bayesian Neural Network Posteriors
No full textInternational audienceThe distribution of modern deep neural networks (DNNs) weights - crucial for uncertainty quantification and robustness - is an eminently complex object due to its extremely high dimensionality. This paper proposes one of the first large-scale explorations of the posterior distribution of deep Bayesian Neural Networks (BNNs), expanding its study to real-world vision tasks and architectures. Specifically, we investigate the optimal approach for approximating the posterior, analyze the connection between posterior quality and uncertainty quantification, delve into the impact of modes on the posterior, and explore methods for visualizing the posterior. Moreover, we uncover weight-space symmetries as a critical aspect for understanding the posterior. To this extent, we develop an in-depth assessment of the impact of both permutation and scaling symmetries that tend to obfuscate the Bayesian posterior. While the first type of transformation is known for duplicating modes, we explore the relationship between the latter and L2 regularization, challenging previous misconceptions. Finally, to help the community improve our understanding of the Bayesian posterior, we release the \href{https://huggingface.co/datasets/torch-uncertainty/Checkpoints}{first large-scale checkpoint dataset}, including thousands of real-world models, along with our \href{https://github.com/ENSTA-U2IS-AI/torch-uncertainty}{codes}
Education numérique - Manuel Cycle 1 (3-4°) - Collection Décodage 2024
No full textInternational audienceLe manuel CODAGE d’Éducation Numérique pour le Cycle 1 a été réalisé dans le cadre du projet cantonal d’introduction d’un enseignement de l’éducation numérique dans le cursus scolaire vaudois.Il est le fruit d’une collaboration entre la Direction pédagogique de la DGEO (Direction Générale de l’Enseignement Obligatoire et de la pédagogie spécialisée) du canton de Vaud, le Centre LEARN-EPFL (École Polytechnique Fédérale de Lausanne), la HEP (Haute École Pédagogique) Vaud, avec l’expertise d’Inria (Institut français de recherche en sciences du numérique)
Pump-laser-wavelength dependence of population inversion in N 2 +
No full textInternational audienceWe investigate the pump wavelength dependence of population inversion in a nitrogen ion when nitrogen gas is irradiated by a femtosecond laser pulse in the wavelength range from ultraviolet to near-infrared. It is found that the population inversions corresponding to the transitions at 391 and 428 nm are very sensitive to the laser wavelength, with extreme values near 400 and 1000 nm. For the ultraviolet pump, the population inversions for 391 and 428 nm transitions occur at different wavelengths: a maximum inversion for the first corresponds to a minimum for the second. This is attributed to the near-resonance single-photon transition combined with the Raman process. A significantly stronger inversion is observed for the near-infrared laser, with the maximum depending on the laser intensity, ascribed to the ac Stark effect. The Floquet theory confirms this conclusion and reveals that the three-photon transition assisted by the strong laser field leads to this enhanced population inversion. This study provides the optimal conditions for air lasing based on ionic transitions of nitrogen molecules
Preface of the proceedings: ICNAAO-2022
No full textInternational audiencePreface of the proceedings: ICNAAO-202
Education numérique - Manuel Cycle 2 (7-8°) - Collection Décodage 2024
No full textInternational audienceLe manuel CODAGE d’Éducation numérique pour le cycle 2 (7ᵉ-8ᵉ) a été réalisé dans le cadre du projet cantonal d’introduction de l’Éducation numérique dans le cursus scolaire vaudois.Il est le fruit d’une collaboration entre la Direction pédagogique de la DGEO (Direction générale de l’enseignement obligatoire et de la pédagogie spécialisée) du canton de Vaud, le Centre des Sciences de l’apprentissage LEARN de l’EPFL (École polytechnique fédérale de Lausanne), la HEP Vaud (Haute école pédagogique du canton de Vaud) et l’UNIL (Université de Lausanne), avec l’expertise d’Inria (Institut français de recherche en sciences et technologies du numérique)
Viscosity solutions of centralized control problems in measure spaces
No full textInternational audienceThis work focuses on a control problem in the Wasserstein space of probability measures over Rd. Our aim is to link this control problem to a suitable Hamilton-Jacobi-Bellman (HJB) equation. We explore a notion of viscosity solution using test functions that are locally Lipschitz and locally semiconvex or semiconcave functions. This regularity allows to define a notion of viscosity and a Hamiltonian function relying on directional derivatives. Using a generalization of Ekeland's principle, we show that the corresponding HJB equation admits a comparison principle, and deduce that the value function is the unique solution in this viscosity sense. The PDE tools are developed in the general framework of Measure Differential Equations
Nouveaux modèles d'optimisation pour la construction optimale d'arbres de classification
No full textInternational audienceInterpretability is a growing concept in Machine Learning. Decision-making algorithms are more and more used in healthcare, finance or other high stakes contexts. Therefore, the need for algorithms whose decisions are understandable is of the utmost importance. Intrinsically interpretable classifiers such as decision trees are often seen as less accurate than black box models such as neural networks. For decision trees, state-of-the-art methods are recursive heuristics (e.g. CART) that may fail to find underlying characteristics in datasets. Recently, linear formulations were introduced to model the problem of the construction of the best decision tree for a given dataset. Notably, a MIO formulation, introduced by Bertsimas and al., has shown better accuracy than CART. However this model does not scale up to datasets with more than 1000 data points. Our work focuses on improvements of MIOs that speed up their resolution in order to handle larger problems. We present a quadratic formulation of the MIP devised by Bertsimas and al. as well as its linearization and another that extends a flow-formulation (from binary dataset to real-value dataset). We prove that our new formulations have stronger continuous relaxation than the MIP introduced by Bertsimas and al.. Finally, our experiments show that they have a significantly smaller resolution time than the MIP of Bertsimas and al. while maintaining or improving performances on test sets
Estimating Polyp Size From a Single Colonoscopy Image Using a Shape-From-Shading Model
No full textInternational audienceColonoscopy (CO) is the most useful procedure to estimate the polyp size as part of surveillance and therapeutic management to prevent Colorectal cancer. Studies have reported a high rate of misestimated lesions by experts, reaching a relative accuracy of 67%. This work presents a method to estimate polyp size from a raw CO frame. A shape-from-shading model trained with synthetic images estimates a depth map to reconstruct the three-dimensional (3d) colon structure. Polyp is segmented in RGB image with a U-Net, to compute diameter in pixels. This diameter is projected onto the 3d surface to compute polyp size in millimeters. The method obtained a mean absolute error of 2.16 mm and relative accuracy of 88.17% in 2 802 synthetic images, and 2.06 mm in 100 real images. In a binary classification task (≤ 10 mm or ≥ 10 mm), the method achieved macro F1 scores of 89% and 72% in the synthetic and real databases respectively.</div