Lipschitz stability estimate in the inverse Robin problem for the Stokes system

We are interested in the inverse problem of recovering a Robin coefficient defined on some non accessible part of the boundary from available data on another part of the boundary in the nonstationary Stokes system. We prove a Lipschitz stability estimate under the \textit{a priori} assumption that the Robin coefficient lives in some compact and convex subset of a finite dimensional vectorial subspace of the set of continuous functions. To do so, we use a theorem proved by L. Bourgeois which establishes Lipschitz stability estimates for a class of inverse problems in an abstract framework

Stability estimates for a Robin coefficient in the two-dimensional Stokes system

In this paper, we consider the Stokes equations and we are concerned with the inverse problem of identifying a Robin coefficient on some non accessible part of the boundary from available data on the other part of the boundary. We first study the identifiability of the Robin coefficient and then we establish a stability estimate of logarithm type thanks to a Carleman inequality due to A. L. Bukhgeim and under the assumption that the velocity of a given reference solution stays far from 0 on a part of the boundary where Robin conditions are prescribed

Lipschitz stability estimate for the Stokes system with Robin boundary conditions

In this research report, we study the inverse problem of identifying a Robin coefficient defined on some non accessible part of the boundary from measurements available on the other part of the boundary, for (u,p) solution of the Stokes system. We prove a Lipschitz stability estimate, under the a priori assumption that the Robin coefficient is piecewise constant. To do so, we use unique continuation estimates for the Stokes system proved in [BEG12] and the approach developed by E. Sincich in [Sin07] to solve a similar inverse problem for the Laplace equation

Stability estimates for the unique continuation property of the Stokes system. Application to an inverse problem

International audienceIn the first part of this paper, we prove hölderian and logarithmic stability estimates associated to the unique continuation property for the Stokes system. The proof of these results is based on local Carleman inequalities. In the second part, these estimates on the fluid velocity and on the fluid pressure are applied to solve an inverse problem: we consider the Stokes system completed with mixed Neumann and Robin boundary conditions and we want to recover the Robin coefficient (and obtain stability estimate for it) from measurements available on a part of the boundary where Neumann conditions are prescribed. For this identification parameter problem, we obtain a logarithmic stability estimate under the assumption that the velocity of a given reference solution stays far from 0 on a part of the boundary where Robin conditions are prescribed

Stability estimates for the unique continuation property of the Stokes system. Application to an inverse problem

International audienceIn the first part of this paper, we prove hölderian and logarithmic stability estimates associated to the unique continuation property for the Stokes system. The proof of these results is based on local Carleman inequalities. In the second part, these estimates on the fluid velocity and on the fluid pressure are applied to solve an inverse problem: we consider the Stokes system completed with mixed Neumann and Robin boundary conditions and we want to recover the Robin coefficient (and obtain stability estimate for it) from measurements available on a part of the boundary where Neumann conditions are prescribed. For this identification parameter problem, we obtain a logarithmic stability estimate under the assumption that the velocity of a given reference solution stays far from 0 on a part of the boundary where Robin conditions are prescribed

A quantitative central limit theorem for the effective conductance on the discrete torus

We study a random conductance problem on a $d$-dimensional discrete torus of size $L > 0$. The conductances are independent, identically distributed random variables uniformly bounded from above and below by positive constants. The effective conductance $A_L$ of the network is a random variable, depending on $L$, and the main result is a quantitative central limit theorem for this quantity as $L \to \infty$. In terms of scalings we prove that this nonlinear nonlocal function $A_L$ essentially behaves as if it were a simple spatial average of the conductances (up to logarithmic corrections). The main achievement of this contribution is the precise asymptotic description of the variance of $A_L$.Comment: 37 page

Etude de quelques problèmes inverses pour le système de Stokes. Application aux poumons.

In this work, we are interested in the resolution of some inverse problems arising from a multi-scale modeling of the airflow in the lungs. As a first step, we focus on a simplified model of the airflow in the lungs: we consider the incompressible Stokes equations with Robin boundary conditions on a part of the boundary. We want to identify the Robin coefficient defined on this non accessible part of the boundary from measurements of the velocity and the pressure available on another part of the boundary. We first prove quantification results for the unique continuation property for the Stokes system, then we establish two logarithmic stability inequalities, one valid in dimension 2 and the other one valid in any dimension. Both are based on Carleman estimates, global in the first case and local in the second one. Our stability estimates are first established for the stationary problem and the semigroup theory allows to deduce from the stationary case stability estimates for the non-stationary problem. Moreover, under the a priori assumption that the Robin coefficient is piecewise constant, we provide a Lipschitz stability estimate for the stationary problem. We conclude by coming back to the initial model which involves non-standard boundary conditions with the flux. In particular, we obtain encouraging first numerical results concerning the identification of some parameters of the model.Dans cette thèse, nous nous intéressons à la résolution de problèmes inverses provenant d'une modélisation multi-échelle de l'écoulement de l'air dans les poumons. Dans un premier temps, nous considérons une version simplifiée du modèle de l'écoulement de l'air dans les poumons : l'écoulement est modélisé par les équations de Stokes incompressibles avec des conditions aux limites de type Robin sur une partie du bord. Nous cherchons à identifier le coefficient de Robin défini sur une partie non accessible du bord à partir de mesures de la vitesse et de la pression disponibles sur une autre partie du bord. Après avoir quantifié des résultats de continuation unique pour le système de Stokes, nous établissons deux inégalités de stabilité logarithmiques, l'une valable en dimension 2 et l'autre valable en toute dimension. Toutes deux sont basées sur des inégalités de Carleman, globale dans le premier cas et locales dans le second. Les inégalités de stabilité sont d'abord montrées sur le problème stationnaire puis la théorie des semi-groupes permet de passer au problème non stationnaire. De plus, sous l'hypothèse a priori que le coefficient de Robin est constant par morceaux, nous prouvons une inégalité de stabilité Lipschitzienne pour le problème stationnaire. Nous concluons cette thèse en revenant au problème initial pour lequel nous imposons des conditions au bord non-standard faisant intervenir le flux. En particulier, nous obtenons des premiers résultats numériques encourageants concernant l'identification de certains paramètres du modèle

Etudes de quelques problèmes inverses pour le système de Stokes (Application aux poumons)

In this work, we are interested in the resolution of some inverse problems arising from a multi-scale modeling of the airflow in the lungs. As a first step, we focus on a simplified model of the airflow in the lungs: we consider the incompressible Stokes equations with Robin boundary conditions on a part of the boundary. We want to identify the Robin coefficient defined on this non accessible part of the boundary from measurements of the velocity and the pressure available on another part of the boundary. We first prove quantification results for the unique continuation property for the Stokes system, then we establish two logarithmic stability inequalities, one valid in dimension 2 and the other one valid in any dimension. Both are based on Carleman estimates, global in the first case and local in the second one. Our stability estimates are first established for the stationary problem and the semigroup theory allows to deduce from the stationary case stability estimates for the non-stationary problem. Moreover, under the a priori assumption that the Robin coefficient is piecewise constant, we provide a Lipschitz stability estimate for the stationary problem. We conclude by coming back to the initial model which involves non-standard boundary conditions with the flux. In particular, we obtain encouraging first numerical results concerning the identification of some parameters of the model.Dans cette thèse, nous nous intéressons à la résolution de problèmes inverses provenant d'une modélisation multi-échelle de l'écoulement de l'air dans les poumons. Dans un premier temps, nous considérons une version simplifiée du modèle de l'écoulement de l'air dans les poumons : l'écoulement est modélisé par les équations de Stokes incompressibles avec des conditions aux limites de type Robin sur une partie du bord. Nous cherchons à identifier le coefficient de Robin défini sur une partie non accessible du bord à partir de mesures de la vitesse et de la pression disponibles sur une autre partie du bord. Après avoir quantifié des résultats de continuation unique pour le système de Stokes, nous établissons deux inégalités de stabilité logarithmiques, l'une valable en dimension 2 et l'autre valable en toute dimension. Toutes deux sont basées sur des inégalités de Carleman, globale dans le premier cas et locales dans le second. Les inégalités de stabilité sont d'abord montrées sur le problème stationnaire puis la théorie des semi-groupes permet de passer au problème non stationnaire. De plus, sous l'hypothèse a priori que le coefficient de Robin est constant par morceaux, nous prouvons une inégalité de stabilité Lipschitzienne pour le problème stationnaire. Nous concluons cette thèse en revenant au problème initial pour lequel nous imposons des conditions au bord non-standard faisant intervenir le flux. En particulier, nous obtenons des premiers résultats numériques encourageants concernant l'identification de certains paramètres du modèle.PARIS-BIUSJ-Mathématiques rech (751052111) / SudocSudocFranceF