In this thesis, the multi-carrier (MC) code division multiple access (CDMA) system is considered in Rayleigh fading channel. The main concern of this thesis is to devise a new direct linearly constrained constant modulus (LCCM) inverse QRD-RLS algorithm for multiple access interference (MAI) cancellation and the problem due to the mismatch of the channel estimator. In the conventional approach, two significant detectors are applied to the system for multiuser interference suppression, one is the blind adaptation algorithm and the other is adaptive linearly constrained PLIC approach. However, the mirror effect may occur when the blind adaptation algorithm is employed. It might affect the performance in terms of bit error rate (BER), although the desired signal to interference (due to other users) improvement is still acceptable. Moreover, in case that the channel coefficients could not be estimated perfectly, the mismatch problem may occur to degrade the performance of the adaptive linearly constrained PLIC approach with the LMS or RLS algorithm. To overcome the mismatch problem, the conventional approach is to use the LCCM criterion with gradient algorithm. However, the convergence rate of the gradient algorithm is too slow to be implemented in real-time wireless communication system. In this thesis, to have fast convergence rate and to circumvent the mismatch problem, the robust LCCM-IQRD algorithm is devised and applied to the MC-CDMA system in Rayleigh fading channel. The proposed robust LCCM-IQRD algorithm has shown to be more effective in terms of MAI cancellation and the mismatch due to imperfect channel estimator. The performance, in terms of BER, of the proposed algorithm is superior to that of the conventional PLIC based algorithms, the blind adaptation algorithm, and the conventional LCCM gradient algorithm.

In this thesis, the problem of multiple access interference (MAI) suppression for the multi-carrier (MC) code division multiple access (CDMA) system, based on the wavelet-based (WB) multi-carrier modulation, associated with the combining process is investigated for Rayleigh fading channel. The main concern of this thesis is to derive a new scheme, based on the linearly constrained constant modulus (LCCM) criterion with the robust inverse QR decomposition (IQRD) recursive least squares (RLS) algorithm to improve the performance of the conventional MC-CDMA system with combining process. To verify the merits of the new algorithm, the effect due to imperfect channel parameters estimation and frequency offset are investigated. We show that the proposed robust LCCM IQRD-RLS algorithm outperforms the conventional LCCM-gradient algorithm [6], in terms of output SINR, improvement percentage index (IPI), and bit error rate (BER) for MAI suppression under channel mismatch environment. Also, the performance of the WB MC-CDMA system is superior to the one with conventional MC-CDMA system. It is more robust to the channel mismatch and frequency offset. Moreover, the WB MC-CDMA system with robust LCCM IQRD-RLS algorithm does have better performance over other conventional approaches, such as the LCCM-gradient algorithm, maximum ratio combining (MRC), blind adaptation algorithm and partitioned linear interference canceller (PLIC) approach with LMS algorithm, in terms of the capability of MAI suppression and bit error rate (BER).

In an anisotropic adaptive finite element algorithm one usually needs an error estimator that yields the error size but also the stretching directions and stretching ratios of the elements of a (quasi) optimal anisotropic mesh. However the last two ingredients can not be extracted from any of the known anisotropic a posteriori error estimators. Therefore a heuristic approach is pursued here, namely, the desired information is provided by the so-called Hessian strategy. This strategy produces favourable anisotropic meshes which result in a small discretization error. The focus of this paper is on error estimation on anisotropic meshes. It is known that such error estimation is reliable and efficient only if the anisotropic mesh is aligned with the anisotropic solution. The main result here is that the Hessian strategy produces anisotropic meshes that show the required alignment with the anisotropic solution. The corresponding inequalities are proven, and the underlying heuristic assumptions are given in a stringent yet general form. Hence the analysis provides further inside into a particular aspect of anisotropic error estimation.

In an anisotropic adaptive finite element algorithm one usually needs an error estimator that yields the error size but also the stretching directions and stretching ratios of the elements of a (quasi) optimal anisotropic mesh. However the last two ingredients can not be extracted from any of the known anisotropic a posteriori error estimators. Therefore a heuristic approach is pursued here, namely, the desired information is provided by the so-called Hessian strategy. This strategy produces favourable anisotropic meshes which result in a small discretization error. The focus of this paper is on error estimation on anisotropic meshes. It is known that such error estimation is reliable and efficient only if the anisotropic mesh is aligned with the anisotropic solution. The main result here is that the Hessian strategy produces anisotropic meshes that show the required alignment with the anisotropic solution. The corresponding inequalities are proven, and the underlying heuristic assumptions are given in a stringent yet general form. Hence the analysis provides further inside into a particular aspect of anisotropic error estimation.

A singularly perturbed reaction-diffusion model problem is considered, and the choice of an appropriate norm is discussed. Particular emphasis is given to the energy norm. Certain prejudices against this norm are investigated and disproved. Moreover, an adaptive finite element algorithm is presented which exhibits an optimal error decrease in the energy norm in some simple numerical experiments. This underlines the suitability of the energy norm.

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Private data analysis on statistical data has been addressed by many recent literatures. The goal of such analysis is to measure statistical properties of a database without revealing information of individuals who participate in the database. Differential privacy is a rigorous privacy definition that protects individual information using output perturbation: a differentially private algorithm produces statistically indistinguishable outputs no matter whether the database contains a tuple corresponding to an individual or not. It is straightforward to construct differentially private algorithms for many common tasks and there are published algorithms to support various tasks under differential privacy. However methods to design error-optimal algorithms for most non-trivial tasks are still unknown. In particular, we are interested in error-optimal algorithms for sets of linear queries. A linear query is a sum of counts of tuples that satisfy a certain condition, which covers the scope of many aggregation tasks including count, sum and histogram. We present the matrix mechanism, a novel mechanism for answering sets of linear queries under differential privacy. The matrix mechanism makes a clear distinction between a set of queries submitted by users, called the query workload, and an alternative set of queries to be answered under differential privacy, called the query strategy. The answer to the query workload can then be computed using the answer to the query strategy. Given a query workload, the query strategy determines the distribution of the output noise and the power of the matrix mechanism comes from adaptively choosing a query strategy that minimizes the output noise. Our analyses also provide a theoretical measure to the quality of different strategies for a given workload. This measure is then used in accurate and approximate formulations to the optimization problem that outputs the error-optimal strategy. We present a lower bound of error to answer each workload under the matrix mechanism. The bound reveals that the hardness of a query workload is related to the spectral properties of the workload when it is represented in matrix form. In addition, we design an approximate algorithm, which generates strategies generated by our a out perform state-of-art mechanisms over (epsilon, delta)-differential privacy. Those strategies lead to more accurate data analysis while preserving a rigorous privacy guarantee. Moreover, we also combine the matrix mechanism with a novel data-dependent algorithm, which achieves differential privacy by adding noise that is adapted to the input data and to the given query workload.

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L'objectif de la thèse est l'étude mathématique et l'analyse numérique du problème non linéaire de Sobolev. Un premier chapitre est consacré à l'analyse a priori pour le problème de Sobolev où on utilise des méthodes de semi-discrétisation explicite en temps. Des estimations d'erreurs ont été obtenues assurant que les schémas numériques utilisés convergent lorsque le pas de discrétisation en temps et le pas de discrétisation en espace tendent vers zéro. Dans le second chapitre, on s'intéresse au problème de Sobolev singulièrement perturbé. En vue de la stabilité des schémas numériques, on utilise dans cette partie des méthodes numériques implicites (la méthode d'Euler et la méthode de Crank- Nicolson) pour discrétiser le problème par rapport au temps. Dans le troisième chapitre, on présente des applications et des illustrations où on utilise le logiciel "FreeFem++". Dans le dernier chapitre, on considère une équation de type Sobolev et on s'intéresse à la dérivation d'estimations d'erreur a posteriori pour la discrétisation de cette équation par la méthode des éléments finis conforme en espace et un schéma d'Euler implicite en temps. La borne supérieure est globale en espace et en temps et permet le contrôle effectif de l'erreur globale. A la fin du chapitre, on propose un algorithme adaptatif qui permet d'atteindre une précision relative fixée par l'utilisateur en raffinant les maillages adaptativement et en équilibrant les contributions en espace et en temps de l'erreur. On présente également des essais numériques. / The purpose of this work is the mathematical study and the numerical analysis of the nonlinear Sobolev problem. A first chapter is devoted to the a priori analysis for the Sobolev problem, where we use an explicit semidiscretization in time. A priori error estimates were obtained ensuring that the used numerical schemes converge when the time step discretization and the spatial step discretization tend to zero. In a second chapter, we are interested in the singularly perturbed Sobolev problem. For the stability of numerical schemes, we used in this part implicit semidiscretizations in time (the Euler method and the Crank-Nicolson method). Our estimates of Chapters 1 and 2 are confirmed in the third chapter by some numerical experiments. In the last chapter, we consider a Sobolev equation and we derive a posteriori error estimates for the discretization of this equation by a conforming finite element method in space and an implicit Euler scheme in time. The upper bound is global in space and time and allows effective control of the global error. At the end of the chapter, we propose an adaptive algorithm which ensures the control of the total error with respect to a user-defined relative precision by refining the meshes adaptively, equilibrating the time and space contributions of the error. We also present numerical experiments.