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In this thesis, we develop some new models and efficient algorithms for image denoising. The total variation model of Rudin, Osher, and Fatemi(ROF) for image denoising is considered to be one of the most successful deterministic denoising models. It exploits the non-smooth total variation (TV) semi-norm to preserve discontinuities and to keep the edges of smooth regions sharp. Despite its simple form, the TV semi-norm results in a strongly nonlinear Euler-Lagrange equation and poses computational challenge in solving the model efficiently. Moreover, this model produces so-called staircase effect. In this thesis, we propose several new algorithms and models to solve these problems. We study the discretized ROF model and propose a new algorithm which does not involve partial differential equations. Convergence of the algorithm is analyzed. Numerical results show that this algorithm is efficient and stable. We then introduce a denoising model which utilizes high-order difference to approximate piece-wise smooth functions. This model eliminates undesirable staircases, and improves both visual quality and signal-to-noise ratio. Our algorithm is generalized to solve the high-order models. A relaxation technique is proposed for the iteration scheme, aiming to accelerate our solution process. Finally, we propose a method combining total variation and wavelet packets to improve performance on texture-rich images. The ROF model is utilized to eliminate noise, and a wavelet packet transform is used to enhance textures. The numerical results show that the combinational method exploits the advantages of both total variation and wavelet packets. / Mathematics

In this thesis, we consider second-order fully nonlinear partial differential equations of elliptic type. Our aim is to develop computational methods using convergent difference schemes for stationary Hamilton-Jacobi equations with Dirichlet and Neumann type boundary conditions in arbitrary two-dimensional domains. First, we introduce the notion of viscosity solutions in both continuous and discontinuous frameworks. Next, we review Barles-Souganidis approach using monotone, consistent, and stable schemes. In particular, we show that these schemes converge locally uniformly to the unique viscosity solution of the first-order Hamilton-Jacobi equations under mild assumptions. To solve the scheme numerically, we use Euler map with some initial guess. This iterative method gives the viscosity solution as a limit. Moreover, we illustrate our numerical approach in several two-dimensional examples.

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Cette thèse est un recueil de constructions et de résultats variés autour de problèmes de résonances et de stabilités. Premièrement, on s'intéresse à la conception et à l'analyse de méthodes numériques pour des problèmes académiques tels que le problème de Dirichlet sur un segment ou l'équation de transport associée à une rotation du plan. Ensuite, on étend l'analyse linéaire classique des équations de Vlasov-Poisson autour d'états d'équilibre homogènes pour décrire des phénomènes multidimensionnels et non linéaires. Enfin, une large partie est consacrée à l'étude d'équations de Schrödinger non linéaires en dimension 1. D'une part, on étudie l'impact d'une semi-discrétisation naturelle sur les ondes solitaires progressives et la croissance des normes de Sobolev. D'autre part, on développe une nouvelle famille de formes normales permettant de décrire la dynamique des petites solutions régulières pendant des temps très longs. / This manuscript deals with many problems about resonance and stability. First, we design and analyse numerical methods for academic problems like the Dirichlet problem on a segment line or the transport equation associated with a two dimensional rotation. Then, we extend the classical linear analysis of Vlasov-Poisson equations near homogeneous equilibria to describe nonlinear and multidimensional phenomena. Finally, a large part of this thesis is devoted to nonlinear Schrödinger equations in dimension 1. On the one hand, we study the impact of a natural semi-discretisation on the solitary traveling waves and on the growth of the high order Sobolev norms. On the other hand, we develop a new family of normal forms to describe the dynamic of small and smooth solutions for very long times.

The modelling of the processes of advection and dispersion-diffusion is the most crucial factor in solute transport simulations. It is generally appreciated that the first order upwind difference scheme gives rise to excessive numerical diffusion, whereas the conventional second order central difference scheme exhibits severe oscillations for advection dominated transport, especially in regions of high solute gradients or discontinuities. Higher order schemes have therefore become increasingly used for improved accuracy and for reducing grid scale oscillations. Two such schemes are the QUICK (Quadratic Upwind Interpolation for Convective Kinematics) and TOASOD (Third Order Advection Second Order Diffusion) schemes, which are similar in formulation but different in accuracy, with the two schemes being second and third order accurate in space respectively for finite difference models. These two schemes can be written in various finite difference forms for transient solute transport models, with the different representations having different numerical properties and computational efficiencies. Although these two schemes are advectively (or convectively) stable, it has been shown that the originally proposed explicit QUICK and TOASOD schemes become numerically unstable for the case of pure advection. The stability constraints have been established for each scheme representation based upon the von Neumann stability analysis. All the derived schemes have been tested for various initial solute distributions and for a number of continuous discharge cases, with both constant and time varying velocity fields. The 1-D QUICKEST (QUICK with Estimated Streaming Term) scheme is third order accurate both in time and space. It has been shown analytically and numerically that a previously derived quasi 2-D explicit QUICKEST scheme, with a reduced accuracy in time, is unstable for the case of pure advection. The modified 2-D explicit QUICKEST, ADI-TOASOD and ADI-QUICK schemes have been developed herein and proved to be numerically stable, with the bility sta- region of each derived 2-D scheme having also been established. All these derived 2-D schemesh ave been tested in a 2-D domain for various initial solute distributions with both uniform and rotational flow fields. They were further tested for a number of 2-D continuous discharge cases, with the corresponding exact solutions having also been derived herein. All the numerical tests in both the 1-D and 2-D cases were compared with the corresponding exact solutions and the results obtained using various other difference schemes, with the higher order schemes generally producing more accurate predictions, except for the characteristic based schemes which failed to conserve mass for the 2-D rotational flow tests. The ADI-TOASOD scheme has also been applied to two water quality studies in the U. K., simulating nitrate and faecal coliform distributions respectively, with the results showing a marked improvement in comparison with the results obtained by the second order central difference scheme. Details are also given of a refined numerical representation of flooding and drying of tidal flood plains for hydrodynamic modelling, with the results showing considerable improvements in comparison with a number of existing models and in good agreement with the field measured data in a natural harbour study.

The modelling of the processes of advection and dispersion-diffusion is the most crucial factor in solute transport simulations. It is generally appreciated that the first order upwind difference scheme gives rise to excessive numerical diffusion, whereas the conventional second order central difference scheme exhibits severe oscillations for advection dominated transport, especially in regions of high solute gradients or discontinuities. Higher order schemes have therefore become increasingly used for improved accuracy and for reducing grid scale oscillations. Two such schemes are the QUICK (Quadratic Upwind Interpolation for Convective Kinematics) and TOASOD (Third Order Advection Second Order Diffusion) schemes, which are similar in formulation but different in accuracy, with the two schemes being second and third order accurate in space respectively for finite difference models. These two schemes can be written in various finite difference forms for transient solute transport models, with the different representations having different numerical properties and computational efficiencies. Although these two schemes are advectively (or convectively) stable, it has been shown that the originally proposed explicit QUICK and TOASOD schemes become numerically unstable for the case of pure advection. The stability constraints have been established for each scheme representation based upon the von Neumann stability analysis. All the derived schemes have been tested for various initial solute distributions and for a number of continuous discharge cases, with both constant and time varying velocity fields. The 1-D QUICKEST (QUICK with Estimated Streaming Term) scheme is third order accurate both in time and space. It has been shown analytically and numerically that a previously derived quasi 2-D explicit QUICKEST scheme, with a reduced accuracy in time, is unstable for the case of pure advection. The modified 2-D explicit QUICKEST, ADI-TOASOD and ADI-QUICK schemes have been developed herein and proved to be numerically stable, with the bility sta- region of each derived 2-D scheme having also been established. All these derived 2-D schemesh ave been tested in a 2-D domain for various initial solute distributions with both uniform and rotational flow fields. They were further tested for a number of 2-D continuous discharge cases, with the corresponding exact solutions having also been derived herein. All the numerical tests in both the 1-D and 2-D cases were compared with the corresponding exact solutions and the results obtained using various other difference schemes, with the higher order schemes generally producing more accurate predictions, except for the characteristic based schemes which failed to conserve mass for the 2-D rotational flow tests. The ADI-TOASOD scheme has also been applied to two water quality studies in the U. K., simulating nitrate and faecal coliform distributions respectively, with the results showing a marked improvement in comparison with the results obtained by the second order central difference scheme. Details are also given of a refined numerical representation of flooding and drying of tidal flood plains for hydrodynamic modelling, with the results showing considerable improvements in comparison with a number of existing models and in good agreement with the field measured data in a natural harbour study.

First we consider implicit finite difference schemes on uniform grids in time and space for second order linear stochastic partial differential equations of parabolic type. Under sufficient regularity conditions, we prove the existence of an appropriate asymptotic expansion in powers of the the spatial mesh and hence we apply Richardson's method to accelerate the convergence with respect to the spatial approximation to an arbitrarily high order. Then we extend these results to equations where the parabolicity condition is allowed to degenerate. Finally, we consider implicit finite difference approximations for deterministic linear second order partial differential equations of parabolic type and give sufficient conditions under which the approximations in space and time can be simultaneously accelerated to an arbitrarily high order.

Les méthodes particulaires sont des méthodes numériques adaptées à la résolution d'équations de conservation. Leur principe consiste à introduire des particules ``numériques'' conservant localement l'inconnue sur un petit volume, puis à les transporter le long de leur trajectoire. Lorsqu'un terme source est présent dans les équations, l'évolution de la solution le long des caractéristiques est prise en compte par une intéraction entre les particules. Ces méthodes possèdent de bonnes propriétés de conservation et ne sont pas soumises aux conditions habituelles de CFL qui peuvent être contraignantes pour les méthodes Eulériennes. Cependant, une contrainte de recouvrement entre les particules doit être satisfaite pour vérifier des propriétés de convergence de la méthode. Pour satisfaire cette condition de recouvrement, un remaillage périodique des particules est souvent utilisé. Elle consiste à recréer régulièrement de nouvelles particules uniformément réparties, à partir de celles ayant été advectées à l'itération précédente. Quand cette étape de remaillage est effectuée à chaque pas de temps, l'analyse numérique de ces méthodes particulaires remaillées nécessite d'être reconsidérée, ce qui représente l'objectif de ces travaux de thèse. Pour mener à bien cette analyse, nous nous basons sur une analogie entre méthodes particulaires avec remaillage et schémas de grille. Nous montrons que pour des grands pas de temps les schémas numériques obtenus souffrent d'une perte de précision. Nous proposons des méthodes de correction, assurant la consistance des schémas en tout point de grille, le pas de temps étant contraint par une condition sur le gradient du champ de vitesse. Cette méthode est construite en dimension un. Des techniques de limitation sont aussi introduites de manière à remailler les particules sans créer d'oscillations en présence de fortes variations de la solution. Enfin, ces méthodes sont généralisées aux dimensions plus grandes que un en s'inspirant du principe de splitting d'opérateurs. Les applications numériques présentées dans cette thèse concernent la résolution de l'équation de transport sous forme conservative en dimension un à trois, dans des régimes linéaires ou non-linéaires. / Particle methods are numerical methods designed to solve advection dominated conservation equations. Their principle is to introduce ``numerical'' particles that concentrate the unknown locally on a small volume, and to transport them along their trajectories. These methods have good conservation properties and are not subject to the usual CFL conditions that can be binding for the Eulerian methods. However, an overlap condition must be satisfied between the particles to ensure convergence properties of the method. To satisfy this condition, a periodic remeshing of the particles is often used. New particles uniformly distributed are created on a regular mesh. When this remeshing step is performed at every time step, numerical analysis of particle methods needs to be revisited. This is the purpose of this thesis. To carry out this analysis, we rely on an analogy between remeshed particle methods and grid schemes. We show that for large time step the numerical schemes have a loss of accuracy. We propose correction methods wich ensure consistency at any grid point, provided the time step satisfies a condition based on the gradient of the velocity field. Limitation techniques are also introduced to remesh particles without creating any oscillations in the presence of strong variations of the solution. Finally, these methods are generalized to dimensions greater than one. Numerical example on various transport equations are given to illustrate the benefit of the proposed algorithms.

Many partial differential equations (PDEs) admit conserved quantities like mass or energy. Those quantities are often essential to establish well-posed results. When approximating a PDE by a finite-difference scheme, it is natural to ask whether related discretized quantities remain conserved under the scheme. Such conservation may establish the stability of the numerical scheme. We present an algorithm for checking the preservation of a polynomial quantity under a polynomial finite-difference scheme. In our algorithm, schemes can be explicit or implicit, have higher-order time and space derivatives, and an arbitrary number of variables. Additionally, we present an algorithm for, given a scheme, finding conserved quantities. We illustrate our algorithm by studying several finite-difference schemes.