Notions of optimal transport theory and how to implement them on a computer

This article gives an introduction to optimal transport, a mathematical theory that makes it possible to measure distances between functions (or distances between more general objects), to interpolate between objects or to enforce mass/volume conservation in certain computational physics simulations. Optimal transport is a rich scientific domain, with active research communities, both on its theoretical aspects and on more applicative considerations, such as geometry processing and machine learning. This article aims at explaining the main principles behind the theory of optimal transport, introduce the different involved notions, and more importantly, how they relate, to let the reader grasp an intuition of the elegant theory that structures them. Then we will consider a specific setting, called semi-discrete, where a continuous function is transported to a discrete sum of Dirac masses. Studying this specific setting naturally leads to an efficient computational algorithm, that uses classical notions of computational geometry, such as a generalization of Voronoi diagrams called Laguerre diagrams.Comment: 32 pages, 17 figure

On the uniqueness and stability of an inverse problem in photo-acoustic tomography

International audienceThis article deals with the uniqueness and stability of the solution of a problem of optimal control related to the photo-acoustic tomog-raphy process. We prove stability results of the optimal solution with respect to the source and to the observation data and we compute the corresponding derivatives

On the reconstruction of obstacles and of rigid bodies immersed in a viscous incompressible fluid

International audienceWe consider the geometrical inverse problem consisting in recovering an unknown obstacle in a viscous incompressible fluid by measurement of the Cauchy force on a part of the exterior boundary. We deal with the case where the fluid equations are the non stationary Stokes system and using the enclosure method, we can recover the convex hull of the obstacle and the distance from a point to the obstacle. With the same method, we can obtain the same result in the case of a linear fluid--structure system composed by a rigid body and a viscous incompressible fluid. We also tackle the corresponding nonlinear systems: the Navier--Stokes system and a fluid--structure system with free boundary. Using complex spherical waves, we obtain some information on the distance from a point to the obstacle

Problèmes d’interaction entre un Fluide Newtonien Incompressible et une Structure

Tesis en cotutellaThis thesis deals with two different fluid--structure interactionproblems in the three dimensional case: in the first problem, wemake a theoretical analysis of a problem of interaction between adeformable structure and an incompressible Newtonian fluid(Chapter 2); in the second problem, we consider ageometrical inverse problem associated to a fluid--rigid bodysystem (Chapter 3).For the first problem, we prove a result of existence anduniqueness of strong solutions by using, for the elasticstructure, an approximation of the equations of linear elasticityby a finite-dimensional system.In the second problem, we prove the well-posedness of thecorresponding system and we show an identifiability result: theform of a convex body and its initial position are identified bythe measurement, at a positive time, of the Cauchy force of thefluid on an open part of the exterior boundary. Moreover, astability result for this system is tackled.En esta tesis se abordan dos problemas diferentes de interacciónfluido--estructura en el caso tridimensional: en el primero deellos realizamos un estudio teórico de un problema de interacciónentre una estructura deformable y un fluido Newtonianoincompresible (Capítulo 2), y en el segundo problema,consideramos un problema inverso geométrico asociado a un sistemafluido--cuerpo rígido (Capítulo 3).Para el primer problema probamos un resultado de existencia yunicidad de soluciones fuertes considerando para la estructuraelástica una aproximación finito-dimensional de la ecuación deelasticidad lineal.En el segundo problema, demostramos el buen planteamiento delcorrespondiente sistema fluido--estructura rígida y probamos unresultado de identificabilidad: la forma de un cuerpo convexo y suposición inicial son identificadas, vía la medición, en algúntiempo positivo, del tensor de Cauchy del fluido sobre unsubconjunto abierto de la frontera exterior. También un resultadode estabilidad es estudiado para este problema.Cette thèse porte sur deux problèmes différents d'interactionfluide--structure dans le cas tridimensionnel: dans le premierproblème, on effectue une étude théorique d'un problèmed'interaction entre une structure déformable et un fluideNewtonien incompressible (Chapitre 2); dans le deuxièmeproblème, on considère un problème inverse géométrique associé àun système fluide--corps rigide (Chapitre 3). Pour le premier problème nous démontrons unrésultat d'existence et d'unicité des solutions fortes, enutilisant, pour la structure élastique, une approximation deséquations de l'élasticité linéaire par un système dedimension finie.Dans le deuxième problème, nous démontrons le caractère bièn-posédu système associé et nous montrons un résultatd'identifiabilité: la forme d'un corps convexe et sa positioninitiale sont identifiées par la mesure, en un temps positif, dutenseur de Cauchy du fluide sur une partie ouverte de la frontièreextérieure. De plus, un résultat de stabilité pour ce problème estabordé

Problemas de interaccion entre un fluido newtoniano incompresible y una estructura

Cette thèse porte sur deux problèmes différents d'interaction fluide-structure dans le cas tridimensionnel: dans le premier problème, on effectue une étude théorique d'un problème d'interaction entre une structure déformable et un fluide Newtonien incompressible (Chapitre 2); dans le deuxième problème, on considère un problème inverse géométrique associé à un système fluide-corps rigide (Chapitre 3). Pour le premier problème nous démontrons un résultat d'existence et d'unicité des solutions fortes, en utilisant, pour la structure élastique, une approximation des équations de l'élasticité linéaire par un système de dimension finie. Dans le deuxième problème, nous démontrons le caractère bien-posé du système associé et nous montrons un résultat d'identifiabilité: la forme d'un corps convexe et sa position initiale sont identifiées par la mesure, en un temps positif, du tenseur de Cauchy du fluide sur une partie ouverte de la frontière extérieure. De plus, un résultat de stabilité pour ce problème est abordé.This thesis deals with two different fluid-structure interaction problems in the three dimensional case: in the first problem, we make a theoretical analysis of a problem of interaction between a deformable structure and an incompressible Newtonian fluid (Chapter 2); in the second problem, we consider a geometrical inverse problem associated to a fluid-rigid body system (Chapter 3). For the first problem, we prove a result of existence and uniqueness of strong solutions by using, for the elastic structure, an approximation of the equations of linear elasticity by a finite-dimensional system. In the second problem, we prove the well-posedness of the corresponding system and we show an identifiability result: the form of a convex body and its initial position are identified by the measurement, at a positive time, of the Cauchy force of the fluid on an open part of the exterior boundary. Moreover, a stability result for this system is tackled

A MINIMAX FORMULA FOR THE BEST NATURAL C([0, 1])-APPROXIMANT BY NONDECREASING FUNCTIONS

International audienceLet f be a function in C([0, 1]). We denote by fp the best approxi-mant to f in Lp([0, 1]) by nondecreasing functions. It is well known that the limit f * := limp→∞ fp exists and f * is a best approximant to f in C([0, 1]) by nondecreasing functions. In this paper we show an explicit formula for the function f * and we prove some additional minimization properties of f *

Simultaneous optimal controls for non-stationary Stokes systems

International audienceThis paper deal with optimal control problems for a non-stationary Stokes system. We study a simultaneous distributed-boundary optimal control problem with distributed observation. We prove the existence and uniqueness of a simultaneous optimal control and we give the first order optimality condition for this problem. We also consider a distributed optimal control problem and a boundary optimal control problem and we obtain estimations between the simultaneous optimal control and the optimal controls of these last ones. Finally, some regularity results are presented

REGULARITY RESULTS FOR A CLASS OF HYPERBOLIC EQUATIONS WITH VMO COEFFICIENTS

In this note we show a regularity result for an hyperbolic system with discontinuous coefficients. More precisely, we deal with coefficients in the function space VMO and we prove the existence and uniqueness of a solution $u \in L^{\infty}(0,T;H^2(\Omega))$ with also suitable regularity for $\frac{\partial u }{\partial t}$,$\frac{\partial^2 u }{\partial t^2}$ and $\frac{\partial^3 u }{\partial t^3}

SIMULTANEOUS OPTIMAL CONTROLS FOR UNSTEADY STOKES SYSTEMS

International audienceThis paper deal with optimal control problems for an unsteady Stokes system. We consider a simultaneous distributed-boundary optimal control problem with distributed observation. We prove the existence and uniqueness of an optimal control and we give the first order optimality condition for this problem. We also consider a distributed optimal control problem and a boundary optimal control problem and we obtain estimations between the simultaneous optimal control and the optimal controls of these last problems