The focus of this study is development of parallelised version of severely sequential and iterative numerical algorithms based on multi-threaded parallel platform such as a graphics processing unit. This requires design and development of a platform-specific numerical solution that can benefit from the parallel capabilities of the chosen platform. Graphics processing unit was chosen as a parallel platform for design and development of a numerical solution for a specific physical model in non-linear optics. This problem appears in describing ultra-short pulse propagation in bulk transparent media that has recently been subject to several theoretical and numerical studies. The mathematical model describing this phenomenon is a challenging and complex problem and its numerical modeling limited on current modern workstations. Numerical modeling of this problem requires a parallelisation of an essentially serial algorithms and elimination of numerical bottlenecks. The main challenge to overcome is parallelisation of the globally non-local mathematical model. This thesis presents a numerical solution for elimination of numerical bottleneck associated with the non-local nature of the mathematical model. The accuracy and performance of the parallel code is identified by back-to-back testing with a similar serial version.

Discrete-Event Simulation is a common and powerful tool in real world decision making, used to model systems which can not be represented analytically. Considering the likelihood of multiple objectives occurring in complex systems a framework has been designed to enable the optimization of the output from multi-objective simulation models. The SimMOp methodology intelligently combines the model with a heuristic search algorithm featuring a multi-objective optimization technique with relational database technologies. The methodology is designed to obtain solutions from a dual perspective, determining the optimal solutions, based on definitions from an optimization approach, and also to take into consideration the stochasticity that will exist within all simulation model output data.

Suppose that we observe a point pattern in an observation window W₀ C Rd. In this study we pose ourselves the following two main questions: How can one extend the pattern in a 'reasonable way' to a larger window W ) W₀? Can one predict possible gaps in the observed point pattern where points were 'expected' but somehow failed to realise. We address these questions by assuming that the point pattern is an observed part of a realisation of a non-homogeneous Poisson process in W and estimate its intensity so that to mimic a given distributional characteristic of the pattern, for instance, its sample nearest neighbour distribution. The project aims to develop prediction and extrapolation techniques for random points processes and related techniques of optimisation of functionals depending on a measure. Applications are numerous and important, including restoration of images, detection of impurities in material science, prediction of anomalies in geology, etc.

Heterogeneity in survival and recurrent event data is often due to unknown, unmeasured, or immeasurable factors. Subjects may experience heterogeneous failure times or event rates due to different levels of vulnerability to the event of interest. The more prone the subjects, the shorter the survival times and the higher the event rates. Furthermore, the presence of cured subjects who are not susceptible to the event contributes to the heterogeneity. Frailty and cure models can take into account the unexplained variation due to heterogeneity and cured fractions. This research explores the ideas of these models for failure and recurrent event data. The models are checked by simulation studies and they are applied to three data sets wherever applicable. For survival data, we investigate by simulation the results of a frailty mixture model which includes frailty and cure models. Even for a small size (e.g. 100), this model fits well to the data from either frailty or cure models. We also explore misspecification of the Cox and frailty model theoretically and by simulation when data are generated from the cure model. Although the regression parameters are underestimated under the misspecified Cox model, the frailty model fits well to simulated data with a cured fraction. Furthermore, regression parameters are underestimated under a misspecified cure model when the frailty model holds. In the case of a high rate of administrative censoring (80%), the bias is small in all misspecified models. The Aalen and Cox frailty models for failure times are compared in terms of frailty parameter estimates. Under both the Cox and Aalen frailty models, the frailty variance is underestimated. However, the frailty variance is estimated to be smaller under the Aalen frailty model because this model, as opposed to the Cox frailty model, allows for time-dependent regression parameters which can explain part of the random processes. We include a time-constant frailty term into the Aalen intensity model to construct an individual time-constant frailty model (ITCF) for recurrent event data and suggest a dynamic procedure to estimate the parameters. Estimated frailty and regression parameters are unbiased in the simulation study. Although a misspecified Aalen model ignores heterogeneity, unbiased regression parameters are obtained. However, the intensity and residuals are not estimated appropriately. Several models for clustered recurrent event data are suggested. Models can be used to estimate the correlation between subjects within the clusters and heterogeneity between them. One of the models can also consider cured fractions at the cluster and individual levels. This model can make a difference to the significant results when a cluster is event-free or the rate of event-free subjects is considerably different at various levels of a covariate. A time-dependent frailty model is also explored. This model assumes that at each time there is a frailty term with variance ξ, but there is correlation between different times. The correlation between frailties at time u and v is assumed to be p |u-v|. We use an approximation for small values of ξ to estimate the parameters. Simulation studies confirm that results are good and the bias is ignorable for both frailty and regression parameters. This model includes the ITCF model when p = 1 and a misspecified ITCF model underestimates the ξ. When p is not close to 1 (e.g. 0.8), the two models can be differentiated by composite likelihood. Three data sets are used throughout the thesis. The first (leukaemia) has single event survival times whereas the second (patient controlled analgesia) and third (Blue bay diarrhoea) have recurrent events. Clustering is present in the third dataset.

This thesis addresses data assimilation, which typically refers to the estimation of the state of a physical system given a model and observations, and its application to short-term precipitation forecasting. A general introduction to data assimilation is given, both from a deterministic and stochastic point of view. Data assimilation algorithms are reviewed, in the static case (when no dynamics are involved), then in the dynamic case. A double experiment on two non-linear models, the Lorenz 63 and the Lorenz 96 models, is run and the comparative performance of the methods is discussed in terms of quality of the assimilation, robustness in the non-linear regime and computational time.

L'objectif de cette thèse est de concevoir et d'analyser des schémas numériques performants pour la simulation d'écoulements complexes en milieux poreux. Dans un premier temps nous proposons un schéma CVFE (Control Volume Finite Element) non-linéaire pour approcher la solution de l'équation de Richards anisotrope. La mobilité d'arête est gérée à l'aide d'une procédure de décentrement. On montre d'abord que ce schéma est non-linéairement stable, qu'il admet (au moins) une solution discrète et que la saturation est bornée entre 0 et 1. Ce schéma converge sans restriction sur le maillage. Enfin, en vue de mettre en évidence l'efficacité, la stabilité et la robustesse de la méthode, nous réalisons des tests numériques dans des cas isotropes et anisotropes. Dans un second temps on étudie un schéma Volumes finis (avec décentrement des mobilités) pour un modèle d'intrusion saline. Il préserve au niveau discret les principales propriétés du problème continu: l'existence de solutions discrètes positives, la décroissance de l'énergie et le contrôle de l'entropie et sa dissipation. Nous montrons que ce schéma converge. De plus, nous illustrons numériquement le comportement du modèle. Enfin nous étudions le comportement en temps long d'un modèle d'intrusion saline. Il s'agit d'identifier les états stationnaires qui sont les minimiseurs d'une énergie convexe. On montre pour le problème continu l'existence et l'unicité des minimiseurs de l'énergie, que les minimiseurs sont des états stationnaires et que ces états stationnaires sont radiaux et uniques. Nous donnons une illustration numérique des états stationnaires et nous exhibons le taux de convergence. / This thesis is focused on the design and the analysis of efficient numerical schemes for the simulation of complex flows in porous media. First, we propose a nonlinear Control Volume Finite Element scheme (CVFE) in order to approximate the solution of Richards equation with anisotropy. This scheme is based on a suitable upwinding of the mobility which allows the negative transmissibility coefficients. We prove the nonlinear stability of the scheme, that there exists (at least) one discrete solution and that the saturation belongs to the interval [0,1]. Moreover, the convergence of the method is proved as the discretization steps tend to 0. We give some numerical experiments on isotropic and anisotropic cases illustrate the efficiency of the method. Second, we propose and analyze a finite volume scheme based on two-point flux approximation with upwind mobilities for a seawater intrusion model. The scheme preserves at the discrete level the main features of the continuous problem, namely the nonnegativity of the solutions, the decay of the energy and the control of the entropy and its dissipation. We show the convergence of this scheme. Numerical results are provided to illustrate the behavior of the model. Finally, the large time behaviour of the seawater intrusion model is studied. The goal is to identify the steady states which are the minimizers of a convex energy. We prove for the continuous problem the existence and uniqueness of the minimizers of the energy, that the minimizers are stationary states and that these stationary states are radial and unique. We give numerical illustrations of the stationary states and we exhibit the convergence rate.

Γίνετε ανασκόπηση των πιο σημαντικών προβλημάτων της Ανάλυσης Χωροθέτησης. Παρατίθονται συγκρίσεις των προβλημάτων της Ανάλυσης Χωροθέτησης με τους Αλγορίθμους Ομαδοποίησης. Ακολούθως αναγράφεται μια αριθμητική εφαρμογή μιας σύγκρισης. / This project is a review of the more important algorithms of the Locational Analysis. The main theme is the comparison of the algorithms of Location Analysis against the algorithms of Clustering.

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Ce mémoire de thèse concerne l’analyse mathématique et numérique du comportement asymptotique de certains modèles de type coagulation-fragmentation intervenant en physique ou en biologie.Dans la première partie, on considère le système d’équations de Lifshitz-Slyozov qui modélise l’immersion d’une population de macro-particules en interaction avec un bain de monomères. Ce modèle développe en temps long un comportement dépendant d’une manière très particulière de l’état initial et ses spécificités techniques en font un véritable challenge pour la simulation numérique.On introduit un nouveau schéma numérique de type volumes finis basé sur une stratégie anti-dissipative ; ce schéma parvient à capturer les profils asymptotiques attendus par la théorie et dépasse en performances les méthodes utilisées jusqu’alors. L’investigation numérique est poursuivie en prenant en compte dans le modèle des phénomènes de coalescence entremacro-particules à travers l’opérateur de Smoluchowski. La question est de déterminer par l’expérimentation numérique comment ces phénomènes influencent le comportement asymptotique. On envisage aussi une extension du modèle classique de Lifshitz-Slyozov qui prend en compte des effets spatiaux via la diffusion des monomères. On établit l’existence et l’unicité des solutions du système couplé hyperbolique-parabolique correspondant. La seconde partie de ce mémoire aborde des modèles d’agrégation fragmentation issus de la biologie. On s’intéresse en effet à des équations décrivant les phénomènes de croissance et de division pour une population de cellules caractérisée par sa densité de répartition en taille. Le comportement asymptotique de cette densité de répartition est accessible à l’expérience et peut être établi théoriquement. L’enjeu biologique consiste, à partir de données mesurées de la densité cellulaire, à estimer le taux de division cellulaire qui, lui, n’est pas expérimentalement mesurable. Ainsi, retrouver ce taux de division cellulaire fait appel à l’étude d’un problème inverse que nous abordons théoriquement et numériquement par des techniques de régularisations par quasi-reversibilité et par filtrage.La troisième partie de ce travail de thèse est consacrée à des systèmes couplés décrivant des interactions fluide-particules, avec des termes de coagulation–fragmentation, de type Becker–Döring. On étudie les propriétés de stabilité du modèle et on présente des résultats d’asymptotiques correspondant à des régimes de forte friction. / This thesis concerns the mathematical and numerical analysis of the asymptotic behavior of some coagulation-fragmentation type models arising in physics or in biology.In the first part we consider the Lifshitz-Slyozov system that models the dumping of a population of macro-particles in interaction with a bath of monomers. This model develops in long time a behavior depending in a very particular way on the initial data abd its technical specificities make a real challenge for the numerical simulation. We introduce a new numerical finite volume type scheme based on an anti-dissipative strategy; this scheme succeeds in capturing the asymptotic profiles waited by the theory and exceeds in performances the methods used before. The numerical investigation ispursued by taking into account in the model the phenomena of coalescence between macro-particles through the Smoluchowski operator. The question is to find by numerical experiment how these phenomena influence the asymptotic behavior. We also consider an extension of the classical Lifshitz-Slyozov model which takes into account the spatial effects via the diffusion of monomers. We establish the existence and the uniqueness of the solutions of the corresponding hyperbolic-parabolic coupled system.The second part of this thesis deals with approaches coagulation-fragmentation models stemming from biology. Indeed, we are interest in equations describing the phenomena of growth and division for a celles population caracterised by its size density repartition. The asymptotic behavior of this size density repartition is accessible to the experiment and can be established in theory. The biological stake consists, from measured data of the cellular density, to estimate the cellular division rate which is not experimentally measurable. So, to find this cellular division rate requires the study of an inverse problem which we approach numerically and theoretically by techniques of regularizations by quasi-reversibility and by filtering.This third part of this thesis work is devoted to coupled systems describing fluid-particles interactions with coagulation-fragmentation terms of Becker-Döring type. We study the stability properties of the model and we present some asymptotic results corresponding to the regime with strong friction force.