Diss. (sammanfattning) Uppsala : Univ., 2004. / Härtill 3 uppsatser.

Cette thèse porte sur l’étude de nouveaux modèles de jeux à champ moyen. On étudie dans un premier temps des modèles d’arrêt optimal et de contrôle impulsionnel en l’absence de bruit commun. On construit pour ces modèles une notion de solution adaptée pour laquelle on prouve des résultats d’existence et d’unicité sous des hypothèses naturelles. Ensuite, on s’intéresse à plusieurs propriétés des jeux à champ moyen. On étudie la limite de ces modèles vers des modèles d’évolution pures lorsque l’anticipation des joueurs tend vers 0. On montre l’unicité des équilibres pour des systèmes fortement couples (couples par les stratégies) sous certaines hypothèses. On prouve ensuite certains résultats de régularités sur une ”master equation” qui modélise un jeu à champ moyen avec bruit commun dans un espace d’états discret. Par la suite on présente une généralisation de l’algorithme standard d’Uzawa et on l’applique à la résolution numérique de certains modèles de jeux à champ moyen, notamment d’arrêt optimal ou de contrôle impulsionnel. Enfin on présente un cas concret de jeu à champ moyen qui provient de problèmes faisant intervenir un grand nombre d’appareils connectés dans les télécommunications. / This thesis is concerned with new models of mean field games. First, we study models of optimal stopping and impulse control in the case when there is no common noise. We build an appropriate notion of solutions for those models. We prove the existence and the uniqueness of such solutions under natural assumptions. Then, we are interested with several properties of mean field games. We study the limit of such models when the anticipation of the players vanishes. We show that uniqueness holds for strongly coupled mean field games (coupled via strategies) under certain assumptions. We then prove some regularity results for the master equation in a discrete state space case with common noise. We continue by giving a generalization of Uzawa’s algorithm and we apply it to solve numerically some mean field games, especially optimal stopping and impulse control problems. The last chapter presents an application of mean field games. This application originates from problems in telecommunications which involve a huge number of connected devices.

Retracer l’histoire d’un bassin est un préalable essentiel à toute recherche d’hydrocarbures. Pour cela, on a recours à un modèle stratigraphique, qui simule l'évolution des bassins sédimentaires sur de grandes échelles de temps (millions d'années) et d'espace (centaines de kilomètres). Le logiciel Dionisos, développé à IFPEN depuis 1992 et très apprécié par les compagnies pétrolières, permet d’effectuer ce type de calculs en prenant en compte deux grands processus physiques : (1) le transport gravitaire des sédiments dû à l’inclinaison du sol ; (2) l’écoulement de l’eau provenant des fleuves et des précipitations. Le transport gravitaire est décrit par une équation de diffusion dans laquelle le flux de sédiments dépend de la pente du sol. Initialement, cette dépendance est linéaire. Pour mieux s’approcher des observations réelles, on souhaite la rendre non-linéaire par l’intermédiaire d’un p-Laplacien. Ce changement nécessite la conception d’une nouvelle méthode de résolution numérique, qui doit offrir non seulement une grande rapidité d’exécution, mais aussi des garanties de robustesse et de précision des résultats. De plus, elle doit être compatible avec une contrainte sur le taux d’érosion présente dans le modèle. L’ajout de l’écoulement de l’eau est aussi une sophistication récente du modèle physique de Dionisos. Il se traduit par l’introduction d’une nouvelle équation aux dérivées partielles, couplée à celle du transport. Là encore, il est important d’élaborer une stratégie de résolution numérique innovante, en ce sens qu’elle doit être à la fois performante et bien adaptée au fort couplage de ces deux phénomènes. L'objectif de cette thèse est de moderniser le cœur numérique de Dionisos afin de traiter plus adéquatement les processus physiques ci-dessus. On cherche notamment à élaborer un schéma implicite par rapport à toutes les inconnues qui étend et améliore le schéma actuel. Les méthodologies retenues serviront de base à la prochaine génération du calculateur. / An essential prerequisite to finding hydrocarbons is to trace back the history of a basin. To this end, geologists resort to a stratigraphic model, which simulates the evolution of sedimentary basins over large time scales (million years) and space (hundreds of kilometers). The Dionisos software, developed by IFPEN since 1992 and highly praised by oil companies, makes this type of calculation possible by accounting for two main physical processes: (1) the sediment transport due to gravity; (2) the flow of water from rivers and rains. The gravity transport is described by a diffusion equation in which the sediment flow depends on the slope of the ground. Initially, this dependence is linear. To better match experimental observations, we wish to make it nonlinear by means of a p-Laplacian. This upgrade requires to design a dedicated numerical method which should not only run fast but also provide guarantees of robustness and accuracy. In addition, it must be compatible with a constraint on the erosion rate in the present model. The water flow due to rivers and rains is also a recent enhancement brought to the physical model of Dionisos. This is achieved by introducing a new partial differential equation, coupled with that of sediment transport. Again, it is capital to work out an innovative numerical strategy, in the sense that it must be both efficient and well suited to the strong coupling of these two phenomena. The objective of this thesis is to rejuvenate the numerical schemes that lie at the heart of Dionisos in order to deal more adequately with the physical processes above. In particular, we look for an implicit scheme with respect to all the unknowns that extends and improves the current scheme. The methodologies investigated in this work will serve as a basis for the next generation of stratigraphic modelling softwares.

This dissertation deals with the use of Shewhart control charts, modified to have variable sampling intervals, to simultaneously monitor a set of parameters. Fixed sampling interval control charts are modified to utilize sampling intervals that vary depending on what is being observed from the data. Two problems are emphasized, namely, the simultaneous monitoring of the mean and the variance and the simultaneous monitoring of several means. For each problem, two basic strategies are investigated. One strategy uses separate control charts for each parameter. A second strategy uses a single statistic which combines the information in the entire sample and is sensitive to shifts in any of the parameters. Several variations on these two basic strategies are studied. Numerical studies investigate the optimal number of sampling intervals and the length of the sampling intervals to be used. Each procedure is compared to corresponding fixed interval procedures in terms of time and the number of samples taken to signal. The effect of correlation on multiple means charts is studied through simulation. For both problems, it is seen that the variable sampling interval approach is substantially more efficient than fixed interval procedures, no matter which strategy is used. / Ph. D.

The main emphasis of this thesis is to study a posteriori error analysis of discontinuous Galerkin (DG) methods for the elliptic variational inequalities. The DG methods have become very pop-ular in the last two decades due to its nature of handling complex geometries, allowing irregular meshes with hanging nodes and different degrees of polynomial approximation on different ele-ments. Moreover they are high order accurate and stable methods. Adaptive algorithms refine the mesh locally in the region where the solution exhibits irregular behaviour and a posteriori error estimates are the main ingredients to steer the adaptive mesh refinement. The solution of linear elliptic problem exhibits singularities due to change in boundary con-ditions, irregularity of coefficients and reentrant corners in the domain. Apart from this, the solu-tion of variational inequality exhibits additional irregular behaviour due to occurrence of the free boundary (the part of the domain which is a priori unknown and must be found as a component of the solution). In the lack of full elliptic regularity of the solution, uniform refinement is inefficient and it does not yield optimal convergence rate. But adaptive refinement, which is based on the residuals ( or a posteriori error estimator) of the problem, enhance the efficiency by refining the mesh locally and provides the optimal convergence. In this thesis, we derive a posteriori error estimates of the DG methods for the elliptic variational inequalities of the first kind and the second kind. This thesis contains seven chapters including an introductory chapter and a concluding chap-ter. In the introductory chapter, we review some fundamental preliminary results which will be used in the subsequent analysis. In Chapter 2, a posteriori error estimates for a class of DG meth-ods have been derived for the second order elliptic obstacle problem, which is a prototype for elliptic variational inequalities of the first kind. The analysis of Chapter 2 is carried out for the general obstacle function therefore the error estimator obtained therein involves the min/max func-tion and hence the computation of the error estimator becomes a bit complicated. With a mild assumption on the trace of the obstacle, we have derived a significantly simple and easily com-putable error estimator in Chapter 3. Numerical experiments illustrates that this error estimator indeed behaves better than the error estimator derived in Chapter 2. In Chapter 4, we have carried out a posteriori analysis of DG methods for the Signorini problem which arises from the study of the frictionless contact problems. A nonlinear smoothing map from the DG finite element space to conforming finite element space has been constructed and used extensively, in the analysis of Chapter 2, Chapter 3 and Chapter 4. Also, a common property shared by all DG methods allows us to carry out the analysis in unified setting. In Chapter 5, we study the C0 interior penalty method for the plate frictional contact problem, which is a fourth order variational inequality of the second kind. In this chapter, we have also established the medius analysis along with a posteriori analy-sis. Numerical results have been presented at the end of every chapter to illustrate the theoretical results derived in respective chapters. We discuss the possible extension and future proposal of the work presented in the Chapter 6. In the last chapter, we have documented the FEM codes used in the numerical experiments.

This work aims to investigate some applications of the conjugate duality for scalar and vector optimization problems to the construction of gap functions for variational inequalities and equilibrium problems. The basic idea of the approach is to reformulate variational inequalities and equilibrium problems into optimization problems depending on a fixed variable, which allows us to apply duality results from optimization problems. Based on some perturbations, first we consider the conjugate duality for scalar optimization. As applications, duality investigations for the convex partially separable optimization problem are discussed. Afterwards, we concentrate our attention on some applications of conjugate duality for convex optimization problems in finite and infinite-dimensional spaces to the construction of a gap function for variational inequalities and equilibrium problems. To verify the properties in the definition of a gap function weak and strong duality are used. The remainder of this thesis deals with the extension of this approach to vector variational inequalities and vector equilibrium problems. By using the perturbation functions in analogy to the scalar case, different dual problems for vector optimization and duality assertions for these problems are derived. This study allows us to propose some set-valued gap functions for the vector variational inequality. Finally, by applying the Fenchel duality on the basis of weak orderings, some variational principles for vector equilibrium problems are investigated.

Dans une première partie, nous étudions la stabilité des solutions d'une inégalité variationnelle de la forme cône normal perturbé par une fonction. Pour ce faire, nous généralisons la méthode de S. Robinson, basée sur le degré topologique, aux espaces de Hilbert et à une classe de multi-applications non nécessairement convexes, appelées multi-applications prox-régulières.  Dans une deuxième partie, nous étudions des problèmes de contrôle optimal liés à la modélisation de problèmes de bio-procédés, et l'on s'intéresse à des contraintes périodiques sur l'état. Ainsi, nous étendons les résultats d'existence de solutions périodiques des EDOs de Kolmogorov au cadre du contrôle en rajoutant un paramètre contrôlé à ces équations. Ceci nous permet d'étudier par la suite un problème de commande optimale d'un chemostat sous forçage périodique, et d'en déduire la synthèse optimale pour ce problème. / In the first part, we study stability of solutions of a variational inequality of the form normal cone perturbed by a mapping. To do so, we generalize the method introduced by S. Robinson, based on the topological degree, to the general Hilbert setting on the class of non-necessarily convex set-valued mapping, called prox-regular set-valued mapping. In the second part, we study optimal control problems connected to the modelization of bio-processes and we consider periodic constraints on the state variable. We first extend the existence result of periodic solutions of Kolmogorov ODEs to the setting of control by adding a controlled parameter to those ODEs. This allows us to study an optimal control problem modeling a chemostat under a periodic forcing for which we give the optimal synthesis.

[pt] O mercado brasileiro de derivados de petróleo está sendo aberto para competição este ano, saindo de um ambiente de preços regulados pelo Governo Federal para um ambiente onde os preços são estabelecidos pelas leis de oferta e demanda. Neste contexto, existe a preocupação de como serão estes preços, e seus impactos sobre os consumidores e sobre os produtores locais. Esta Tese propõe alguns modelos matemáticos para estimar preços, níveis de produção, níveis de consumo (demanda), e importação e exportação de derivados de petróleo nas diversas regiões do mercado brasileiro. O fornecimento de derivados de petróleo não é considerado um mercado competitivo, e sim oligopolizado, principalmente no curto prazo, devido à capacidade instalada de refinarias e aos altos custos envolvidos na construção de novas refinarias. Estes modelos são multi- produto, considerando um fato importante na produção de derivados que é a impossibilidade de produzir apenas um derivado. Assim, existem restrições onde a oferta de um derivado é relacionada a oferta dos outros. O primeiro modelo considera um mercado de oligopólio fechado, com um número fixo de firmas. Tal modelo é formulado como um problema de equilíbrio a Nash. Um segundo modelo é apresentado expandindo o primeiro para o caso em que existem preços teto de demanda definidos politicamente. O terceiro modelo relaxa a suposição do mercado fechado, com número fixo de firmas, e considera a possibilidade de competição de novas firmas no mercado. Um quarto modelo é discutido, onde assume-se que existe uma firma líder no mercado, que consegue definir sua estratégia antes das demais firmas, semelhante ao problema econômico de Stackelberg. Todos os modelos foram formulados como problemas de inequações variacionais, sendo que o último modelo é ainda um problema de programação binível. Algoritmos de solução são propostos para os três primeiros modelos. Simulações sobre o mercado brasileiro de derivados são apresentadas. / [en] The Brazilian petroleum refined products market is being opened to competition this year, leaving an environment of regulated prices to another one where the prices are defined by the supply demand interactions. Considering this new scenario, there is a concern about how high the prices will be, and about their impact on the consumers and on the local producers. This thesis proposes some mathematical models to predict prices, production, consumption, and import and export levels of petroleum-refined products in all the sub-regions of the Brazilian market. Instead of a competitive market, the supply of refined products is considered an oligopoly market, especially in the short term, given the already installed refining capacity and the high costs involved in building new refineries. These models are multi-products, and they consider an important characteristic of the production of refined products, the impossibility of producing only one refined product. Hence, constraints where the production of one refined product is related to the production of the others are considered. The first model considers a closed oligopoly market, with a fixed number of firms. This problem is formulated as a Nash equilibrium problem. A second model is presented generalizing the first one to consider the possibility of ceiling demand prices politically defined. The third model relaxes the assumption of a fixed number of firms in the first model, and considers the possibility of competition by new entrants. A fourth model is discussed, where it is assumed that there is a leader firm in the market, which can define its strategy before the other firms, similar to the economic problem of Stackelberg. All the models are formulated as variational inequalities problems, and the last model is also a bi-level programming problem. Solution algorithms for the three first models are proposed. Some analyses of the Brazilian petroleum refined- products market are presented.

In this paper we combine ideas from cutting plane and interior point methods in order to solve variational inequality problems efficiently. In particular, we introduce a general framework that incorporates nonlinear as well as linear "smarter" cuts. These cuts utilize second order information on the problem through the use of a gap function. We establish convergence as well as complexity results for this framework. Moreover, in order to devise more practical methods, we consider an affine scaling method as it applies to symmetric, monotone variationalinequality problems and demonstrate its convergence. Finally, in order to further improve the computational efficiency of the methods in this paper, we combine the cutting plane approach with the affine scaling approach.

Incremental Sheet Forming (ISF) is a progressive metal forming process, where the deformation occurs locally around the point of contact between a tool and the metal sheet. The final work-piece is formed cumulatively by the movements of the tool, which is usually attached to a CNC milling machine. The ISF process is dieless in nature and capable of producing different parts of geometries with a universal tool. The tooling cost of ISF can be as low as 5–10% compared to the conventional sheet metal forming processes. On the laboratory scale, the accuracy of the parts created by ISF is between ±1.5 mm and ±3mm. However, in order for ISF to be competitive with a stamping process, an accuracy of below ±1.0 mm and more realistically below ±0.2 mm would be needed. In this work, we first studied the ISF deformation process by a simplified phenomenal linear model and employed a predictive controller to obtain an optimised tool trajectory in the sense of minimising the geometrical deviations between the targeted shape and the shape made by the ISF process. The algorithm is implemented at a rig in Cambridge University and the experimental results demonstrate the ability of the model predictive controller (MPC) strategy. We can achieve the deviation errors around ±0.2 mm for a number of simple geometrical shapes with our controller. The limitations of the underlying linear model for a highly nonlinear problem lead us to study the ISF process by a physics based model. We use the elastoplastic constitutive relation to model the material law and the contact mechanics with Signorini’s type of boundary conditions to model the process, resulting in an infinite dimensional system described by a partial differential equation. We further developed the computational method to solve the proposed mathematical model by using an augmented Lagrangian method in function space and discretising by finite element method. The preliminary results demonstrate the possibility of using this model for optimal controller design.