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Théorie de viscosité des équations de Hamilton Jacobi du premier ordre sur certains espaces métriques
No full textThe main subject of this thesis is the study first order Hamilton Jacobi equations posed in certain classes of metric spaces. Furthermore, the Hamiltonian of these equations can potentially present some structured discontinuities.In the first part of this thesis, we study a discontinuous first order Hamilton Jacobi Bellman equation defined on a stratification of R^N. The latter is a finite and disjoint union of smooth submanifolds of R^N called the the subdomains of R^N. On each subdomain, a continuous Hamiltonian is defined on it, However the global Hamiltonian in R^N presents discontinuities once one goes from one subdomain to the other. We give an optimal control interpretation of this problem and we use nonsmooth analysis techniques to prove that the value function is the unique viscosity solution to the discontinuous Hamilton Jacobi Bellman equation in this setting. The uniqueness of the solution is guaranteed by means of a strong comparison principle valid for any lower semicontinuous supersolution and any upper semicontinuous subsolution. As far as existence of the solution is concerned, we use the dynamic programming principle verified by the value function to prove that it is a viscosity solution of the discontinuous Hamilton Jacobi equation. Moreover, we prove some stability results in the presence of perturbations on the discontinuous Hamiltonian. Finally, by virtue of the comparison principle, we prove a general convergence result of monotone numerical schemes approximating this problem.The second part of this thesis is concerned with defining a novel notion of viscosity for first order Hamilton Jacobi equations defined in proper CAT(0) spaces. A metric space is said to be a CAT(0) space if, roughly speaking, it is a geodesic space and its geodesic triangles are "thinner" than the triangles of the Euclidean plane. They can be seen as a generalization of Hilbert spaces or Hadamard manifolds. Typical examples of CAT(0) spaces include Hilbert spaces, metric trees and networks obtained by gluing a finite number of half-spaces along their common boundary. We exploit the additional structure that these spaces enjoy to study stationary and time-dependent first order Hamilton-Jacobi equation in them. In particular, we want to recover the main features of viscosity theory: the comparison principle and Perron's method}.We define the notion of viscosity using test functions that are Lipschitz and can be represented as a difference of two semiconvex function. We show that this new notion of viscosity coincides with the classical one in R^N by studying the examples of Hamilton Jacobi Bellman and Hamilton Jacobi Isaacs' equations. Furthermore, we prove existence and uniqueness of the solution of Eikonal type equations posed in networks that can result from gluing half-spaces of different Hausdorff dimension.In the third part of this thesis, we study a Mayer optimal control problem on the space of Borel probability measures over a compact Riemannian manifold M. This is motivated by certain situations where a central planner of a deterministic controlled system has only imperfect information on the initial state of the system. The lack of information here is very specific. It is described by a Borel probability measure along which the initial state is distributed. We define the new notion of viscosity in this space in a similar manner as in the previous part by taking test functions that are Lipschitz and can be written as a difference of two semiconvex functions. With this choice of test functions, we extend the notion of viscosity to Hamilton Jacobi Bellman equations in Wasserstein spaces and we establish that the value function is the unique viscosity solution of a Hamilton Jacobi Bellman equation in the Wasserstein space over M.La première partie de cette thèse est consacrée à l'étude d'une équation de Hamilton Jacobi Bellman discontinue, définie sur une stratification de R^N. Cette dernière est le résultat d'une union d'une collection finie de sous-variétés lisses et disjointes de R^N, que l'on nomme les sous-domaines. Sur chaque sous-domaine, un Hamiltonien continue y est définie. Cependant, le Hamiltonien global sur R^N présente des discontinuités lorsque l'on passe d'un sous-domaine à l'autre. On donne une interprétation commande optimale de ce problème et on utilise les techniques de l'analyse non lisse pour montrer que la fonction valeur est l'unique solution de viscosité de l'équation de Hamilton Jacobi Bellman définie dans ce chapitre. L'unicité de la solution est garantie par un principe de comparaison fort, valable pour toute sur-solution semicontinue inférieurement et toute sous-solution semicontinue supérieurement. En ce qui concerne l’existence de la solution, on utilise le principe de la programmation dynamique vérifiée par la fonction valeur pour montrer que cette dernière est une solution de viscosité du problème considéré. De plus, on prouve quelques résultats de stabilité en présence de perturbations sur le Hamiltonien discontinu. Finalement, en vertu du principe de comparaison, on montre un résultat de convergence général pour les schémas numériques monotones qui approchent ce problème.La deuxième partie de cette thèse est consacrée au développement d'une nouvelle notion de viscosité pour les équations de Hamilton Jacobi du premier ordre définies sur les espaces CAT(0) propres. Un espace métrique est dit CAT(0), s'il est un espace géodésique et si ses triangles géodésiques sont plus minces que les triangles du plan Euclidien. Les espaces CAT(0) peuvent être considérés comme une généralisation des espaces de Hilbert ou les variétés de Hadamad. Des exemples types des espaces CAT(0) sont les espaces de Hilbert, les arbres métriques et les networks obtenus en collant un nombre fini de demi-espaces selon leur frontière commune. On exploite la structure de ces espaces pour étudier les équations de Hamilton Jacobi du premier ordre stationnaires et dépendantes du temps. En particulier, le but du chapitre est de retrouver les principaux résultats de la théorie de la viscosité : le principe de comparaison et la méthode de Perron. On définit la notion de viscosité en utilisant des fonctions test qui sont Lipschitz et qui peuvent être représentées comme une différence de deux fonctions semiconvexes. On montre que cette notion de viscosité coïncide avec la notion classique développée sur R^N en étudiant quelques exemples d'équations classiques. De surcroît, on prouve l'existence et l'unicité de la solution de certaines équations du type Eikonal posées sur des networks qui peuvent résulter du collage de demi-espaces ayant différentes dimensions de Hausdorff.La troisième partie de la thèse se focalise sur l'étude d'un problème de commande optimale de Mayer sur l'espace des mesures Boréliennes de probabilité sur une variété compacte M. L'étude de ce problème est motivé par certaines situations où un planificateur central d'un système contrôlé n'a qu'une information imparfaite sur l'état initiale du système considéré. Le manque d'information est spécifique dans ce problème. Il est décrit par une mesure de probabilité Borélienne selon laquelle l'état initial est distribué. On définit la notion de viscosité sur cet espaces de la même manière que dans la deuxième partie de la thèse en considérant des fonctions test qui sont Lipschitz et qui peuvent être représentées par une différence de deux fonctions semiconvexes. Avec ce choix de fonctions test, on étend la notion de viscosité aux équations de Hamilton Jacobi Bellman définies sur l'espace de Wasserstein et on établit que la fonction valeur associée au problème de commande optimale et l'unique solution de viscosité sur l'espace de Wasserstein sur M
Development of new strategies for the synthesis of aromatic polyketides: selective C-H functionalization of naphthalenes
No full textInternational audienc
Making compounds inspired by Nature: from targeted to diversity-oriented syntheses
No full textNational audienc
Experimental characterisations and numerical simulation of fluid-structure interactions for casting control
No full textInternational audienceHollow turbojet blades are manufactured by a lost-wax casting process. After several steps aimed at manufacturing a consumable mold with inserts, the pouring or casting phase is a key step in the process, which consists of pouring a nickel-based superalloy at liquid state into the mold and then solidifying it. In order to improve the manufacturing process, chiefly the pouring step, the interaction between the molten metal and the thin walls of the casting mold are studied for several flow rates representative of the process. In order to better understand this interaction, which is difficult to observe during a casting, we use a test-calculation dialogue. Hydrodynamic and fluid-structure interaction calculations are correlated and validated using an experimental setup developed specifically for this study, in which molten metal is replaced by water based on Reynolds number similarity. A model of the experimental test bench was carried out with thecalculation code FLOW-3D®, a CFD tool dedicated to free surface movements including a fluid-structure model. The results of the flow calculations allow to find the trends of the velocity field in the fluid around the structure and its displacement with a good agreement with the experimental observations
High-Sensitivity X-ray Phase Imaging System Based on a Hartmann Wavefront Sensor
No full textInternational audienceThe Hartman wavefront sensor can be used for X-ray phase imaging with high angular resolution. The Hartmann sensor is able to retrieve both the phase and absorption from a single acquisition. The system calculates the shift in a series of apertures imaged with a detector with respect to their reference positions. In this article, the impact of the reference image on the final image quality is investigated using a laboratory setup. Deflection and absorption images of the same sample are compared using reference images acquired in air and in water. It can be easily coupled with tomographic setups to obtain 3D images of both phase and absorption. Tomographic images of a test sample are shown, where deflection images revealed details that were invisible in absorption. The findings reported in this paper can be used for the improvement of image reconstruction and for expanding the applications of X-ray phase imaging towards materials characterization and medical imaging
A robust 3D crack growth method based on the eXtended Finite Element Method and the Fast Marching Method
No full textInternational audienceIn the context of the eXtended Finite Element Method (X-FEM), the use of two level set functions allows the representation of the crack to beachieved regardless of the mesh. The initial crack geometry is represented by two distinct level set functions, and the crack propagation is simulated by anupdate of these two level set functions. In this paper, we propose a new approach, based on the Fast Marching Method (FMM), to update the level set functions. We also propose a new implementation of the FMM, designed for tetrahedral volume meshes. We then extend this method to all types of volume elements (tetrahedra, hexahedra, pentahedra, pyramids) available in a standard finite element library. The proposed approach allows one to use the same mesh to solve the mechanical problem and to update the level set functions. Non-planar quasi-static crack growth simulations are presented to demonstrate the robustness of the approach, compared to existing methods based on the integration of Hamilton-Jacobi equations or geometric approaches
Identify technologically relevant papers for 3D printing : a comparison of indicators based on scientific and patent data
No full textInternational audienceIdentify which scientific advances support technological innovation is a very dynamic area of study. Recent literature on this subject proposes two heterogene methods in order to answer this issue. Yamashita (2020) proposes the analysis of references and tracks patent citation while Ogawa and Kajikawa (2015) use another approach, they make cluster of scientific articles and extract keywords for match them with patent data. They refer to two distinct conceptions of science to technology flows. Yamashita (2020) relies on the idea that the potential contribution of a scientific article comes from the proximity between prior knowledge whereas Ogawa and Kajikawa (2015) highlight a semantic proximity that suggests a simultaneous development of science and technology. In order to better understand the mechanisms of knowledge transfer and to better define the blurring of the boundary between science and technology, this article proposes to compare the predictions of these two indicators. To do so, we focus on the case of 3D printing technologies and use original data from the LENS project which make possible to link scientific articles and patents both on the dimensions of citations and authors-inventors. We relied on the DWPI patent database in order to gather complementary patent data, including texts. Based on these data, we run regressions to compare the prediction of both indicators on the citations received from patents by articles and patent published by scientific authors. Our preliminary result shows that, in the case of 3D printing, passed citations brings to lower predicting performances than the semantic similarity approach
EAGER: Asking and Answering Questions for Automatic Reward Shaping in Language-guided RL
No full textInternational audienceReinforcement learning (RL) in long horizon and sparse reward tasks is notoriously difficult and requires a lot of training steps. A standard solution to speed up the process is to leverage additional reward signals, shaping it to better guide the learning process. In the context of language-conditioned RL, the abstraction and generalisation properties of the language input provide opportunities for more efficient ways of shaping the reward. In this paper, we leverage this idea and propose an automated reward shaping method where the agent extracts auxiliary objectives from the general language goal. These auxiliary objectives use a question generation (QG) and question answering (QA) system: they consist of questions leading the agent to try to reconstruct partial information about the global goal using its own trajectory. When it succeeds, it receives an intrinsic reward proportional to its confidence in its answer. This incentivizes the agent to generate trajectories which unambiguously explain various aspects of the general language goal. Our experimental study shows that this approach, which does not require engineer intervention to design the auxiliary objectives, improves sample efficiency by effectively directing exploration