Identifiability for a severely ill-posed oxygen balance model

AbstractWe are interested in recovering boundary data in a dispersive oxygen-balance model. The missing boundary condition is the flux of the biochemical oxygen demand (the amount of oxygen necessary for the oxidation of organic matter) at one extreme point. The observations are collected on the dissolved oxygen at the other extremity. This problem turns out to be severely ill-posed. We perform the mathematical analysis of it. We prove a uniqueness result owing to Pazy's theorem for parabolic boundary value problems and we prove that the compatible data set is dense

A Finite Element Method for the Boundary Data Recovery in an Oxygen-Balance Dispersion Model

International audienceThe inverse problem under investigation consists of the boundary data completion in a deoxygenation-reaeration model in stream-waters. The unidimensional transport model we deal with is based on the one introduced by Streeter and Phelps, augmented by Taylor dispersion terms. The missing boundary condition is the load or/and the flux of the biochemical oxygen demand indicator at the outfall point. The counterpart is the availability of two boundary conditions on the dissolved oxygen tracer at the same point. The major consequences of these non-standard boundary conditions is that dispersive transport equations on both oxygen tracers are strongly coupled and the resulting system becomes ill-posed. The main purpose is a finite element space-discretization of the variational problem put under a non-symmetric mixed form. Combining analytical calculations, numerical computations and theoretical justifications, we try to elucidate the characteristics related to the ill-posedness of this data completion dynamical problem and understand its mathematical structure

Reduced nitrogen and phosphorus fertilization combined with mycorrhizal inoculation enhance potato yield and soil mineral fertility

To improve sustainable agriculture it becomes crucial to minimize the use of chemical inputs and involve new practices that are more productive and healthier for humans and environment. Biofertilization with arbuscular mycorrhizal fungi (AMF), known for increasing the productivity of many crops including potatoes, could be considered as one of these sustainable practices. The purpose of this study was to determine whether mycorrhizal inoculation can minimize chemical fertilization for potato (Solanum tuberosum L.), one of the most demanding species in mineral elements. Thus, by studying the effect of chemical fertilization and mycorrhizal inoculant on potato yield and soil quality. Two doses of chemical fertilizer (NP) corresponding to 50 % and 100 % of the recommended dose has been applied in the presence or not of the mycorrhizal inoculant. The results have shown that adding arbuscular mycorrhizal fungi in combination with 50 % of the chemical fertilizer dose gives the best effects. A significant increase in yield, root colonization, and P soil content has been observed (P < 0.001). Thus mycorrhizal inoculation could reduce the application of synthetic fertilizers and thus improve yield while preserving the environment

Inverse problems of point-wise sources in dispersive transport models of contaminants : identifiability and observability

La recherche et les questions abordées dans cette thèse sont de type inverse : la reconstitution d'une source ponctuelle ou la complétion d'une donnée à la limite inconnue à l'extrémité du domaine dans les modèles paraboliques de transport de contaminants. La modélisation mathématique des problèmes de pollution des eaux fait intervenir deux traceurs, l'oxygène dissous (OD) et la demande biochimique en oxygène (DBO) qui est la quantité d'oxygène nécessaire à la biodégradation de la matière organique. En effet, au cours des procédés d'autoépuration, certaines bactéries aérobies jouent un rôle principal. Ces micro-organismes décomposent les matières organiques polluantes en utilisant l'oxygène dissous dans le milieu. Afin de compenser ces données manquantes, les champs, solutions du problème, sont observés directement ou indirectement. Les problèmes inverses qui en résultent sont quasi certainement mal-posés voire même sévèrement mal-posés pour la plupart. Dans cette thèse, nous proposons justement une analyse aussi poussée que possible sur la question de l'identifiabilité pour les deux problèmes inverses décrits ci-dessus. Nous avons démontré un résultat d'unicité pour des sources fixes dans le cas d'observations décalées. La réalité pour l'observation est nuancée et l'idéal n'est pas acquis ; des mesures directes sur la DBO sont difficiles à obtenir. En revanche collecter des données sur l'OD est possible en temps réel et avec un faible coût. La DBO est donc observée de façon indirecte, grâce au couplage dans le système de Streeter et Phelps, l'information passe de l'OD à la DBO. Pour ce problème aussi, nous avons produit un résultat d'unicité pour la reconstruction de la source ou puits ponctuel qui serait présent dans l'équation de transport sur l'OD. Nous avons ensuite examiné des questions annexes à l'identifiabilité telles que le degré d'instabilité des équations à résoudre. De ce type d'informations dépendent le comportement des méthodes numériques et des algorithmes de calcul à utiliser.The research and the questions approached on this thesis are inverse type : the reconstruction of point-wise source or the data completion problem in parabolic models of transport of contaminants. The mathematical modelling of the problems of water pollution includes two tracers, the dissolved oxygen (DO) and the biochemical demand in oxygen (BDO) which is the quantity of oxygen necessary for the biodegradation of organic matter. Indeed, during the biodegradation process, aerobic bacteria play a leading part. These micro-organisms decompose polluting organic matters by using the dissolved oxygen in the middle. To compensate these missing data, fields, solutions of the problem, are observed directly or indirectly. The resulting inverse problems are ill-posed. Their mathematical study rises big complications and their numerical treatment isn't easy. We demonstrated a uniqueness result for fixed sources in the case of moved observations. The reality for the observation is qualified and the ideal is not acquired; direct measures on the BOD are difficult to obtain. On the Other hand to collect data on the DO is possible in real time With a moderate cost. The BOD is thus observed in indirect way, thanks to the coupling in the system of Streeter and Phelps, the information passes from the DO to the BOD. For this problem, we produced a uniqueness result for the reconstruction of source. Then, we examined the degree of instability of the equation to be solved. The behaviour of numerical methods depend on this type of information

Reduced nitrogen and phosphorus fertilization combined with mycorrhizal inoculation enhance potato yield and soil mineral fertility

To improve sustainable agriculture it becomes crucial to minimize the use of chemical inputs and involve new practices that are more productive and healthier for humans and environment. Biofertilization with arbuscular mycorrhizal fungi (AMF), known for increasing the productivity of many crops including potatoes, could be considered as one of these sustainable practices. The purpose of this study was to determine whether mycorrhizal inoculation can minimize chemical fertilization for potato (Solanum tuberosum L.), one of the most demanding species in mineral elements. Thus, by studying the effect of chemical fertilization and mycorrhizal inoculant on potato yield and soil quality. Two doses of chemical fertilizer (NP) corresponding to 50 % and 100 % of the recommended dose has been applied in the presence or not of the mycorrhizal inoculant. The results have shown that adding arbuscular mycorrhizal fungi in combination with 50 % of the chemical fertilizer dose gives the best effects. A significant increase in yield, root colonization, and P soil content has been observed (P < 0.001). Thus mycorrhizal inoculation could reduce the application of synthetic fertilizers and thus improve yield while preserving the environment