Data Assimilation for hyperbolic conservation laws. A Luenberger observer approach based on a kinetic description

Developing robust data assimilation methods for hyperbolic conservation laws is a challenging subject. Those PDEs indeed show no dissipation effects and the input of additional information in the model equations may introduce errors that propagate and create shocks. We propose a new approach based on the kinetic description of the conservation law. A kinetic equation is a first order partial differential equation in which the advection velocity is a free variable. In certain cases, it is possible to prove that the nonlinear conservation law is equivalent to a linear kinetic equation. Hence, data assimilation is carried out at the kinetic level, using a Luenberger observer also known as the nudging strategy in data assimilation. Assimilation then resumes to the handling of a BGK type equation. The advantage of this framework is that we deal with a single "linear" equation instead of a nonlinear system and it is easy to recover the macroscopic variables. The study is divided into several steps and essentially based on functional analysis techniques. First we prove the convergence of the model towards the data in case of complete observations in space and time. Second, we analyze the case of partial and noisy observations. To conclude, we validate our method with numerical results on Burgers equation and emphasize the advantages of this method with the more complex Saint-Venant system

OSAMOAL: optimized simulations by adapted models using asymptotic limits

We propose in this work to address the problem of model adaptation, dedicated to hyper- bolic models with relaxation and to their parabolic limit. The goal is to replace a hyperbolic system of balance laws (the so-called fine model) by its parabolic limit (the so-called coarse model), in delimited parts of the computational domain. Our method is based on the construction of asymptotic preserving schemes and on interfacial coupling methods between hyperbolic and parabolic models. We study in parallel the cases of the Goldstein-Taylor model and of the p-system with friction

Analytical solutions for the free surface hydrostatic Euler equations

International audienceIn this paper we propose a large set of analytical solutions (FRESH-ASSESS) for the hydrostatic incompressible Euler system in 2d and 3d. These solutions mainly concern free surface flows but partially free surface flows are also considered. These analytical solutions can be especially useful for the validation of numerical schemes

OSAMOAL: optimized simulations by adapted models using asymptotic limits

We propose in this work to address the problem of model adaptation, dedicated to hyper- bolic models with relaxation and to their parabolic limit. The goal is to replace a hyperbolic system of balance laws (the so-called fine model) by its parabolic limit (the so-called coarse model), in delimited parts of the computational domain. Our method is based on the construction of asymptotic preserving schemes and on interfacial coupling methods between hyperbolic and parabolic models. We study in parallel the cases of the Goldstein-Taylor model and of the p-system with friction

Modélisation, simulation et assimilation de données autour d'un problème de couplage hydrodynamique-biologie

This thesis is built around a biological and industrial problem: the simulation of the coupling of hydrodynamics and biology in the context of industrial microalgae culture in outdoor raceways. The numerical modeling is adressed with the use of a multilayer vertical discretization of hydrostatic Navier-Stokes equations coupled with a light sensitive Droop model. Numerically, kinetic schemes allow for the development of efficient, positivity preserving, well balanced and entropy satisfying schemes. Simulations are carried out in 2D and 3D. From a practical point of view, this model is capable of accounting for the utility of a paddlewheel and exhibits Lagrangian trajectories underwent by algae. Hence providing hints on the light history of algae in the pond, which is a key information to biologists, since it enables them to adapt their phytoplankton growth models to those particular, non natural conditions. In order to validate the models and strategies, two solutions are explored. The first one consists in providing analytical solutions to free surface Euler equations. Besides, a specific biological model is designed to permit an analytical coupling. The second one is the use of data assimilation. In order to take advantage of the kinetic description of conservation laws already used for the building of efficient schemes, an innovative data assimilation method for hyperbolic balance laws based in a Luenberger observer on the kinetic equation is developed. It provides a nice theoretical framework for scalar conservation laws, for which we study the cases of complete observations, partial observations in space, in time, and noisy observations. As far as systems are concerned, we focus on the Saint-Venant system, which is hyperbolic, nonlinear and has a topographic source term. We build an observer based only on water depths measurements. Numerical simulations are provided in the case of scalar laws and systems, in one and two dimensions, which validate the efficiency of the method.Les sujets abordés dans cette thèse s'articulent autour de la modélisation numérique du couplage entre l'hydrodynamique et la biologie pour la culture industrielle de microalgues dans des raceways. Ceci est fait au moyen d'un modèle multicouches qui disrétise verticalement les équations de Navier-Stokes hydrostatiques couplé avec un modèle de Droop photosensible pour représenter la croissance des algues, notamment la production de carbone. D'un point de vue numérique, une méthode volumes finis avec schémas cinétiques est appliquée. Elle permet d'obtenir un schéma équilibre qui préserve la positivité de la hauteur d'eau et des quantités biologiques et qui satisfait une inégalité d'énergie. Des simulations sont effectuées en 2D et en 3D, au moyen d'un code C++ développé à cet effet. Du point de vue de l'intérêt pratique de ce travail, ces simulations ont permis de mettre en évidence l'utilité de la roue à aube présente dans les raceways, mais aussi d'exhiber les trajectoires lagrangiennes réalisées par les microalgues, qui permettent de connaitre l'historique lumineux des algues, information d'une grande importance pour les biologistes car elle leur permet d'adapter leurs modèles de croissance phytoplanctoniques à ce contexte très particulier et non naturel. Afin de valider les modèles et les stratégies numériques employées, deux pistes on été explorées. La première consiste à proposer des solutions analytiques pour les équations d'Euler à surface libre, ainsi qu'un modèle biologique spécifique permettant un couplage analytique. La deuxième consiste à faire de l'assimilation de données. Afin de tirer partie de la description cinétique des lois de conservation hyperboliques, une méthode innovante basée sur la construction d'un observateur de Luenberger au niveau cinétique est développée. Elle permet d'obtenir un cadre théorique intéressant pour les lois de conservation scalaires, pour lesquelles on étudie les cas d'observations complètes, partielles en temps, en espace, et bruitées. Pour les systèmes, on se concentre particulièrement sur le système de Saint-Venant, système hyperbolique non linéaire et un observateur basé sur l'observation des hauteurs d'eau uniquement est construit. Des simulations numériques dans les cas scalaires et systèmes, en 1D et 2D sont effectuées et valident l'efficacité de la méthode

Modélisation, simulation et assimilation de données autour d'un problème de couplage hydrodynamique-biologie

This thesis is built around a biological and industrial problem: the simulation of the coupling of hydrodynamics and biology in the context of industrial microalgae culture in outdoor raceways. The numerical modeling is adressed with the use of a multilayer vertical discretization of hydrostatic Navier-Stokes equations coupled with a light sensitive Droop model. Numerically, kinetic schemes allow for the development of efficient, positivity preserving, well balanced and entropy satisfying schemes. Simulations are carried out in 2D and 3D. This model is able to account for the utility of a paddlewheel and exhibits Lagrangian trajectories of algae.Two solutions are explored to validate the models and strategies. The first one consists in providing analytical solutions to free surface Euler equations coupled with a specific biological model. The second one is the use of data assimilation. In order to take advantage of the kinetic description of conservation laws already used for the building of efficient schemes, an innovative data assimilation method for hyperbolic balance laws based in a Luenberger observer on the kinetic equation is developed. It provides a nice theoretical framework for scalar conservation laws, for which we study the cases of complete observations, partial observations in space, in time, and noisy observations. As far as systems are concerned, we focus on the Saint-Venant system, which is hyperbolic, nonlinear and has a topographic source term. We build an observer based only on water depths measurements. Numerical simulations are provided in the case of scalar laws and systems, in one and two dimensions, which validate the efficiency of the method.Cette thèse traite de la modélisation numérique du couplage hydrodynamique-biologie pour la culture industrielle de microalgues dans des raceways. On utilise un modèle multicouches qui disrétise verticalement les équations de Navier-Stokes hydrostatiques couplé avec un modèle de Droop photosensible pour représenter la croissance des algues. Une méthode volumes finis avec schémas cinétiques est appliquée. Elle permet d'obtenir un schéma équilibre qui préserve la positivité de la hauteur d'eau, des quantités biologiques et qui satisfait une inégalité d'énergie. Des simulations sont effectuées en 2D et 3D. Elles mettent en évidence l'utilité de la roue à aube présente dans les raceways, permettent de connaitre l'historique lumineux des algues.Pour valider les modèles et les stratégies numériques, on explore deux pistes. La première consiste à proposer des solutions analytiques pour les équations d'Euler à surface libre couplées avec un modèle biologique spécifique. La deuxième est l'assimilation de données. Tirant partie de la description cinétique des lois de conservation hyperboliques, on développe une méthode innovante basée sur la construction d'un observateur de Luenberger au niveau cinétique. Elle permet d'obtenir un cadre théorique intéressant pour les lois de conservation scalaires, pour lesquelles on étudie les cas d'observations complètes, partielles en temps, en espace, et bruitées. Pour les systèmes, on se concentre sur le système de Saint-Venant et on construit un observateur basé sur l'observation des hauteurs d'eau uniquement. Des simulations numériques dans les cas scalaires et sytèmes, en 1D et 2D sont effectuées et valident l'efficacité de la méthode.PARIS-BIUSJ-Mathématiques rech (751052111) / SudocSudocFranceF

On the semi-global stabilizability of the Korteweg-de Vries Equation via model predictive control

Stabilization of the nonlinear Korteweg-de Vries (KdV) equation on a bounded interval by model predictive control (MPC) is investigated. This MPC strategy does not need any terminal cost or terminal constraint to guarantee the stability. The semi-global stabilizability is the key condition. Based on this condition, the suboptimality and exponential stability of the model predictive control are investigated. Finally, numerical experiment is presented which validates the theoretical results.ERC advanced Grant 668998 (OCLOC)(VLID)278331

Numerical prediction of microalgae productivity and energy requirement during the cultivation process in raceway

International audienceno abstrac

Growth Rate Estimation of Algae in Raceway Ponds: A Novel Approach

International audienceMicroalgae mass cultivation is a promising future source of biomass for energy and food production. In order to optimize productivity of large scale plants and to make them environmentally and economically sustainable, energy requirements have to be minimized. In particular, mixing of the growth medium is a major energy input, and its effect on overall productivity should be better understood. Several dynamic models have been developed to represent the effect of a rapidly time varying light on the photosynthesis process especially for the effect of photoinhibition on growth. In order to assess the mixing effects in a complex hydrodynamic regime, we propose to reconstruct the light profile received by a single cell. A multi-layer Saint-Venant approach is used to simulate the hydrodynamics of the system. It allows for the computation of Lagrangian trajectories, and finally, when knowing the light distribution, the light pattern perceived by a cell. This pattern is then used with the dynamical model for photosynthesis. In a last step, the growth rate of the whole system is estimated as the average over a set of trajectories