No / In this paper we discuss techniques for 3D data modelling and processing where the data are usually provided as point clouds which arise from 3D scanning devices. The particular approaches we adopt in modelling 3D data involves the use of Partial Differential Equations (PDEs). In particular we show how the continuous and discrete versions of elliptic PDEs can be used for data modelling. We show that using PDEs it is intuitively possible to model data corresponding to complex scenes. Furthermore, we show that data can be stored in compact format in the form of PDE boundary conditions. In order to demonstrate the methodology we utlise several examples of practical nature.

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This paper discusses how a boundary-based surface fitting approach can be utilised to smoothly reconstruct a given human face where the scan data corresponding to the face is provided. In particular, the paper discusses how a solution to the Biharmonic equation can be used to set up the corresponding boundary value problem. We show how a compact explicit solution method can be utilised for efficiently solving the chosen Biharmonic equation. Thus, given the raw scan data of a 3D face, we extract a series of profile curves from the data which can then be utilised as boundary conditions to solve the Biharmonic equation. The resulting solution provides us a continuous single surface patch describing the original face.

We present an efficient method for reconstructing complex geometry using an elliptic Partial Differential Equation (PDE) formulation. The integral part of this work is the use of three-dimensional curves within the physical space which act as boundary conditions to solve the PDE. The chosen PDE is solved explicitly for a given general set of curves representing the original shape and thus making the method very efficient. In order to improve the quality of results for shape representation we utilize an automatic parameterization scheme on the chosen curves. With this formulation we discuss our methodology for shape representation using a series of practical examples.

We study a parameter identification problem for the steady state diffusion equations. In this thesis, we transform this identification problem into a minimization problem by considering an appropriate cost functional and propose a finite element method for the identification of the parameter for the linear and nonlinear partial differential equation. The cost functional involves the classical output least square term, a term approximating the derivative of the piezometric head 𝑢(𝑥), an equation error term plus some regularization terms, which happen to be a norm or a semi-norm of the variables in the cost functional in an appropriate Sobolev space. The existence and uniqueness of the minimizer for the cost functional is proved. Error estimates in a weighted 𝐻⁻¹-norm, 𝐿²-norm and 𝐿¹-norm for the numerical solution are derived. Numerical examples will be given to show features of this numerical method. / Ph. D.

The manuscript presents my research on hyperbolic Partial Differential Equations (PDE), especially on conservation laws. My works began with this thought in my mind: ''Existence and uniqueness of solutions is not the end but merely the beginning of a theory of differential equations. The really interesting questions concern the behavior of solutions.'' (P.D. Lax, The formation and decay of shock waves 1974). To study or highlight some behaviors, I started by working on geometric optics expansions (WKB) for hyperbolic PDEs. For conservation laws, existence of solutions is still a problem (for large data, $L^\infty$ data), so I early learned method of characteristics, Riemann problem, $BV$ spaces, Glimm and Godunov schemes, \ldots In this report I emphasize my last works since 2006 when I became assistant professor. I use geometric optics method to investigate a conjecture of Lions-Perthame-Tadmor on the maximal smoothing effect for scalar multidimensional conservation laws. With Christian Bourdarias and Marguerite Gisclon from the LAMA (Laboratoire de \\ Mathématiques de l'Université de Savoie), we have obtained the first mathematical results on a $2\times2$ system of conservation laws arising in gas chromatography. Of course, I tried to put high oscillations in this system. We have obtained a propagation result exhibiting a stratified structure of the velocity, and we have shown that a blow up occurs when there are too high oscillations on the hyperbolic boundary. I finish this subject with some works on kinetic équations. In particular, a kinetic formulation of the gas chromatography system, some averaging lemmas for Vlasov equation, and a recent model of a continuous rating system with large interactions are discussed. Bernard Rousselet (Laboratoire JAD Université de Nice Sophia-Antipolis) introduced me to some periodic solutions related to crak problems and the so called nonlinear normal modes (NNM). Then I became a member of the European GDR: ''Wave Propagation in Complex Media for Quantitative and non Destructive Evaluation.'' In 2008, I started a collaboration with Bruno Lombard, LMA (Laboratoire de Mécanique et d'Acoustique, Marseille). We details mathematical results and challenges we have identified for a linear elasticity model with nonlinear interfaces. It leads to consider original neutral delay differential systems.

CoordenaÃÃo de AperfeiÃoamento de Pessoal de NÃvel Superior / Conselho Nacional de Desenvolvimento CientÃfico e TecnolÃgico / Neste presente trabalho, faremos o estudo de importantes propriedades geomÃtricas e analÃticas de soluÃÃes de equaÃÃes diferenciais parciais elÃpticas totalmente nÃo-lineares do tipo: singulares e degeneradas. O estudo de processos de combustÃo que se degeneram ao longo do conjunto de anulamento da densidade de um gÃs, um caso particular de problemas do tipo "quenching", apresentam em sua modelagem equaÃÃes singulares que estÃo descritas neste trabalho. Nesta primeira parte iremos obter propriedades de uma soluÃÃo minimal, que vÃo desde o controle completo Ãtimo, atà a obtenÃÃo de estimativas de Hausdorff da fronteira livre singular. Por fim, iremos obter a regularidade Ãtima de soluÃÃes de equaÃÃes em que suas propriedades de difusÃo(elipticidade) se deterioram na ordem de uma potÃncia do seu gradiente ao longo do conjunto em que tal taxa de variaÃÃo se anula. / In this work we study important geometric and analytic properties to solutions of fully nonlinear elliptic partial differential equations, both singular and degenerate types. The study of combustion processes that degenerate along the null-set of the density of a gas, a particular case of quenching problems, present in their modeling, equations described in this work. In this first part we obtain properties of a minimal solution, since the complete optimal control until the Hausdorff estimates of the singular free boundary. Ultimately, we obtain the optimal regularity to equation solutions where their diffusion property (elipticity) deterorate in a power of their gradient along the set where such rate of variation nullifies.

In this thesis, I discuss two main subjects coming from biology and I propose two models that mimic the behaviours of the biological networks studied. The first part of the thesis deals with intracellular transport of molecules. Proteins, RNA and, generally, any kind of cargo molecules move freely in the cytoplasm: intracellular transport as a consequence of Brownian motion is classically modelled as a diffusion process. Some specific proteins, like the tumour suppressor p53, use microtubules to facilitate their way towards the nucleus. Microtubules are a dense network of filaments that point towards the cell centre. Motor proteins bind to these filaments and move along, bearing a cargo bound to them. I propose a simplified bi-dimensional model of nucleocytoplasmic transport taking into account the kinetic processes linked to microtubule transport. Unlike in other models we know, I represented the position of a single MT filament. This model is given by a system of partial differential equations which are cast in different dimensions and connected by suitable exchange rules. A numerical scheme is introduced and several scenarios are presented and discussed to answer to the question of which proteins benefit from microtubule transport, depending on their diffusion coefficients. In the second part of the thesis, I design and analyse a physiologically based model representing the accumulation of protein p53 in the nucleus after triggering of the sentinel protein ATM by DNA damage. The p53 protein plays an essential role in the physiological maintenance of healthy tissue integrity in multicellular organisms (regulation of cell cycle arrest, repair pathways and apoptosis). Firstly, I developed a compartmental ODE model to represent the temporal dynamics of the protein. Since the p53 protein is known for its oscillatory behaviour, I performed a numerical bifurcation study to verify the existence, in the model, of stable periodic solutions. Next, I have expanded the model by the addition of a spatial variable and analysed the spatio-temporal dynamics of p53. After checking the existence of oscillations in the spatial setting, I have analysed the robustness of the system under spatial variations (diffusion and permeability coefficients, cell shape and size).