Ma thèse porte sur l'étude des couches limites et de perturbations singulières (\textit{i.e.} des problèmes caractérisés par la présence d'un petit paramètre qui tend vers zéro) dans des conditions plus délicates que d'habitude, à savoir lorsque la solution limite n'est pas régulière. Je considère ainsi deux classes de problèmes réguliers associes à un laplacien et à un bilaplacien, et un problème non linéaire dérivé du problème de Plateau (surfaces minimas), pour lequels la fonction limite possède une singularité (discontinuité simple pour les premiers problèmes, dérivée normale infinie sur certaines parties de la frontière pour le second).\\ La première partie de cette thèse est consacrée à l'étude de deux modèles linéaires singuliers associés à des perturbations singulières pour des EDPs ayant une fonction source singulière. Ce type d'équations fait l'objet de plusieurs applications, par exemple les problèmes de flambement en élasticité, les tourbillons singuliers en mécanique des fluides, le problème de la charge critique pour une poutre ou une plaque élastoplastique, le problème du contrôle automatique de la trajectoire d'un mobile et le problème du bord arrière pour l'écoulement autour d'une aile. De manière classique, la présence d'un petit paramètre dans des équations aux dérivées partielles entraîne, dans certains cas, l'apparition d'une couche limite classique près du bord du domaine pour la solution dite régularisée. Cependant, si on considère en plus une fonction source discontinue (voire une distribution), on constate que de nouvelles couches limites apparaissent à l'intérieur du domaine; l'étude de celles-ci constitue le principal but de cette première partie. Dans la deuxième partie, on s'intéresse à l'étude du problème des surfaces minimales sur une couronne. Pour certaines classes de données au bord, ce problème n'admet pas de solution et sa solution faible dite ``généralisée'' admet une dérivée infinie. On introduit alors une méthode de régularisation elliptique qui entraîne une couche limite près du bord. Le résultat fondamental de cette partie consiste à donner explicitement une approximation pour cette solution régularisée.

The capillary surface $u(x,y)$ near a cusp region satisfies the boundary value problem: \begin{eqnarray} \nabla \cdot \frac{\nabla u}{\sqrt{1+\left|\nabla u \right|^2}}&=&\kappa u \qquad \textrm{in }\left\{(x,y): 0<x,f_2(x)<y<f_1(x)\right\}\,, \label{0.1}\\ \nu \cdot \frac{\nabla u}{\sqrt{1+\left|\nabla u \right|^2}}&=& \cos \gamma_1 \qquad \textrm{on } y=f_1(x)\,,\\ \nu \cdot \frac{\nabla u}{\sqrt{1+\left|\nabla u \right|^2}}&=& \cos \gamma_2 \qquad \textrm{on } y=f_2(x)\,, \label{0.3} \end{eqnarray} where $\lim_{x\rightarrow 0}f_1(x),f_2(x)=0$, $\lim_{x\rightarrow 0}f'_1(x),f'_2(x)=0$. It is shown that the capillary surface is unbounded at the cusp and satisfies $u(x,y)=O\left(\frac{1}{f_1(x)-f_2(x)}\right)$, even for types of cusp not investigated previously (e.g. exponential cusps). By using a tangent cylinder coordinate system, we show that the exact solution $v(x,y)$ of the boundary value problem: \begin{eqnarray} \nabla \cdot \frac{\nabla v}{\left|\nabla v \right|}&=&\kappa v \qquad \textrm{in }\left\{(x,y): 0<x,f_2(x)<y<f_1(x)\right\}\,,\\ \nu \cdot \frac{\nabla v}{\left|\nabla v \right|}&=& \cos \gamma_1 \qquad \textrm{on } y=f_1(x)\,,\\ \nu \cdot \frac{\nabla v}{\left|\nabla v \right|}&=& \cos \gamma_2 \qquad \textrm{on } y=f_2(x)\,, \end{eqnarray} exhibits sixth order asymptotic accuracy to the capillary equations~\eqref{0.1}$-$\eqref{0.3} near a circular cusp. Finally, we show that the solution is bounded and can be defined to be continuous at a symmetric cusp ($f_1(x)=-f_2(x)$) with the supplementary contact angles ($\gamma_2=\pi-\gamma_1$). Also it is shown that the solution surface is of the order $O\left(f_1(x)\right)$, and moreover, the formal asymptotic series for a symmetric circular cusp region is derived.

The capillary surface $u(x,y)$ near a cusp region satisfies the boundary value problem: \begin{eqnarray} \nabla \cdot \frac{\nabla u}{\sqrt{1+\left|\nabla u \right|^2}}&=&\kappa u \qquad \textrm{in }\left\{(x,y): 0<x,f_2(x)<y<f_1(x)\right\}\,, \label{0.1}\\ \nu \cdot \frac{\nabla u}{\sqrt{1+\left|\nabla u \right|^2}}&=& \cos \gamma_1 \qquad \textrm{on } y=f_1(x)\,,\\ \nu \cdot \frac{\nabla u}{\sqrt{1+\left|\nabla u \right|^2}}&=& \cos \gamma_2 \qquad \textrm{on } y=f_2(x)\,, \label{0.3} \end{eqnarray} where $\lim_{x\rightarrow 0}f_1(x),f_2(x)=0$, $\lim_{x\rightarrow 0}f'_1(x),f'_2(x)=0$. It is shown that the capillary surface is unbounded at the cusp and satisfies $u(x,y)=O\left(\frac{1}{f_1(x)-f_2(x)}\right)$, even for types of cusp not investigated previously (e.g. exponential cusps). By using a tangent cylinder coordinate system, we show that the exact solution $v(x,y)$ of the boundary value problem: \begin{eqnarray} \nabla \cdot \frac{\nabla v}{\left|\nabla v \right|}&=&\kappa v \qquad \textrm{in }\left\{(x,y): 0<x,f_2(x)<y<f_1(x)\right\}\,,\\ \nu \cdot \frac{\nabla v}{\left|\nabla v \right|}&=& \cos \gamma_1 \qquad \textrm{on } y=f_1(x)\,,\\ \nu \cdot \frac{\nabla v}{\left|\nabla v \right|}&=& \cos \gamma_2 \qquad \textrm{on } y=f_2(x)\,, \end{eqnarray} exhibits sixth order asymptotic accuracy to the capillary equations~\eqref{0.1}$-$\eqref{0.3} near a circular cusp. Finally, we show that the solution is bounded and can be defined to be continuous at a symmetric cusp ($f_1(x)=-f_2(x)$) with the supplementary contact angles ($\gamma_2=\pi-\gamma_1$). Also it is shown that the solution surface is of the order $O\left(f_1(x)\right)$, and moreover, the formal asymptotic series for a symmetric circular cusp region is derived.

Consider a cognitive overlaid network (CON) that has two tiers with different priorities: a primary tier vs. a secondary tier, which is an emerging network scenario with the advancement of cognitive radio (CR) technologies. The primary tier consists of randomly distributed primary radios (PRs) of density n, which have an absolute priority to access the spectrum. The secondary tier consists of randomly distributed CRs of density m = n^y with y greater than or equal to 1, which can only access the spectrum opportunistically to limit the interference to PRs. In this dissertation, the fundamental limits of such a network are investigated in terms of the asymptotic throughput and packet delay performance when m and n approaches infinity. The following two types of CONs are considered: 1) selfish CONs, in which neither the primary tier nor the secondary tier is willing to route the packets for the other, and 2) supportive CONs, in which the secondary tier is willing to route the packets for the primary tier while the primary tier does not. It is shown that in selfish CONs, both tiers can achieve the same throughput and delay scaling laws as a stand-alone network. In supportive CONs, the throughput and delay scaling laws of the primary tier could be significantly improved with the aid of the secondary tier, while the secondary tier can still achieve the same throughput and delay scaling laws as a stand-alone network. Finally, the throughput and packet delay of a CON with a small number of nodes are investigated. Specifically, we investigate the power and rate control schemes for multiple CR links in the same neighborhood, which operate over multiple channels (frequency bands) in the presence of PRs with a delay constraint imposed on data transmission. By further considering practical limitations in spectrum sensing, an efficient algorithm is proposed to maximize the average sum-rate of the CR links over a finite time horizon under the constraints on the CR-to-PR interference and the average transmit power for each CR link. In the proposed algorithm, the PR occupancy of each channel is modeled as a discrete-time Markov chain (DTMC). Based on such a model, a novel power and rate control strategy based on dynamic programming (DP) is derived, which is a function of the spectrum sensing output, the instantaneous channel gains for the CR links, and the remaining power budget for the CR transmitter. Simulation results show that the proposed algorithm leads to a significant performance improvement over heuristic algorithms.

The transient analysis of large-scale systems is often difficult even when the systems belong to the simplest M/M/n type of queues. To address analytical difficulties, previous studies have been conducted under various asymptotic regimes by suitably accelerating parameters, thereby establishing some useful mathematical frameworks and giving insights into important characteristics and intuitions. However, some studies show significant limitations when used to approximate real service systems: (i) they are more relevant to steady-state analysis; (ii) they emphasize proofs of convergence results rather than numerical methods to obtain system performance; and (iii) they provide only one set of limit processes regardless of actual system size. Attempting to overcome the drawbacks of previous studies, this dissertation studies the transient analysis of large-scale service systems with time-dependent parameters. The research goal is to develop a methodology that provides accurate approximations based on a technique called uniform acceleration, utilizing the theory of strong approximations. We first investigate and discuss the possible inaccuracy of limit processes obtained from employing the technique. As a solution, we propose adjusted fluid and diffusion limits that are specifically designed to approximate large, finite-sized systems. We find that the adjusted limits significantly improve the quality of approximations and hold asymptotic exactness as well. Several numerical results provide evidence of the effectiveness of the adjusted limits. We study both a call center which is a canonical example of large-scale service systems and an emerging peer-based Internet multimedia service network known as P2P. Based on our findings, we introduce a possible extension to systems which show non-Markovian behavior that is unaddressed by the uniform acceleration technique. We incorporate the denseness of phase-type distributions into the derivation of limit processes. The proposed method offers great potential to accurately approximate performance measures of non-Markovian systems with less computational burden.

Machine learning is emerging as a powerful tool to data science and is being applied in almost all subjects. In many applications, the number of features is com- parable to the number of samples, and both grow large. This setting is usually named the high-dimensional regime. In this regime, new challenges arise when it comes to the application of machine learning. In this work, we conduct a high-dimensional performance analysis of some popular classification and regression techniques. In a first part, discriminant analysis classifiers are considered. A major challenge towards the use of these classifiers in practice is that they depend on the inverse of covariance matrices that need to be estimated from training data. Several estimators for the inverse of the covariance matrices can be used. The most common ones are estimators based on the regularization approach. In this thesis, we propose new estimators that are shown to yield better performance. The main principle of our proposed approach is the design of an optimized inverse covariance matrix estimator based on the assumption that the covariance matrix is a low-rank perturbation of a scaled identity matrix. We show that not only the proposed classifiers are easier to implement but also, outperform the classical regularization-based discriminant analysis classifiers. In a second part, we carry out a high-dimensional statistical analysis of linear support vector regression. Under some plausible assumptions on the statistical dis- tribution of the data, we characterize the feasibility condition for the hard support vector regression and, when feasible, derive an asymptotic approximation for its risk. Similarly, we study the test risk for the soft support vector regression as a function of its parameters. The analysis is then extended to the case of kernel support vector regression under generalized linear models assumption. Based on our analysis, we illustrate that adding more samples may be harmful to the test performance of these regression algorithms, while it is always beneficial when the parameters are optimally selected. Our results pave the way to understand the effect of the underlying hyper- parameters and provide insights on how to optimally choose the kernel function.

<p>In this thesis, analytical analysis and design techniques for wireless communications with diversity are studied. The impact of impairments such as correlated fading is analyzed using statistical models. Countermeasures designed to overcome, or even exploit, such effects are proposed and examined. In particular two applications are considered, satellite broadcast to vehicular terminals and communication using transmitters and receivers equipped with multiple antennas.</p><p>Mobile satellite broadcast systems offer the possibility of high data rate services with reliability and ubiquitous coverage. The design of system architectures providing such services requires complex trade-offs involving technical, economical, and regulatory aspects. A satisfactory availability can be ensured using space, terrestrial, and time diversity techniques. The amount of applied diversity affects the spectral efficiency and system performance. Also, dedicated satellite and terrestrial networks represent significant investments and regulatory limitations may further complicate system design.</p><p>The work presented in this thesis provides insights to the technical</p><p>aspects of the trade-offs above. This is done by deriving an efficient method for estimating what resources in terms of spectrum and delay are required for a broadcast service to reach a satisfactory number of end users using a well designed system. The results are based on statistical models of the mobile satellite channel for which efficient analytical design and error rate estimation methods are derived. We also provide insight to the achievable spectral efficiency using different transmitter and receiver configurations.</p><p>Multiple-element antenna communication is a promising technology for future high speed wireless infrastructures. By adding a spatial dimension, radio resources in terms of transmission power and spectrum can be used more efficiently. Much of the design and analysis work has focused on cases where the transmitter either has access to perfect channel state information or it is blind and the spatial channels are uncorrelated.</p><p>Herein, systems where the fading of the spatial channels is correlated and/or the transmitter has access to partial channel state information are considered. While maintaining perfect channel knowledge at the transmitter may prove difficult, updating parameters that change on a slower time scale could be realistic. Here we formulate analysis and design techniques based on statistical models of the multichannel propagation. Fundamental properties of the multi-element antenna channel and limitations given by information theory are investigated under an asymptotic assumption on the number of antennas on either side of the system. For example, limiting normal distributions are derived for the squared singular values of the channel matrix and the mutual information. We also propose and examine a practical scheme capable of exploiting partial channel state information.</p><p>In both applications outlined above, by using statistical models of the channel characteristics in the system design, performance can be improved. The main contribution of this thesis is the development of efficient techniques for estimating the system performance in different scenarios. Such techniques are vital to obtain insights to the impact of different impairments and how countermeasures against these should be designed.</p>

The Manakov system appears in the physics of optical fibers, as well as in quantum mechanics, as multi-component versions of the Nonlinear Schr\&quot;odinger and the Gross-Pitaevskii equations.Although the Manakov system is completely integrable its solutions are far from being explicit in most cases. However, the Inverse Scattering Transform (IST) can be exploited to obtain asymptotic information about solutions.This thesis will describe the IST of the Manakov system, and its asymptotic behavior at short times. I will compare the focusing and defocusing behavior, numerically and analytically, for squared barrier initial potentials. Finally, I will show that the continuous spectrum gives the dominant contribution at short-times.

Fretting fatigue is a type of contact fatigue which causes premature failure in a number of engineering assemblies subjected to vibration or other forms of cyclic loading. It is concerned with the nucleation of cracks due to oscillatory micro slip between contacting bodies. Therefore, a detailed knowledge of the interface conditions and the means of quantifying crack nucleation are very important, and will be the ultimate goal of this thesis. The analysis of an incomplete contact (Herzian contact) is considered first followed by various complete contacts. Fretting fatigue tests employing a Hertzian contact are analysed accurately by introducing several modifications needed to the classical formulation. With the total state of stress in a strip established, the crack tip stress intensity factor for a crack growing inward from the trailing edge of the contact is determined by the distributed dislocation technique. The results are then correlated with local solutions for the contact stress field which enable an estimate of the crack nucleation life, and hence a characteristic material property quantifying initiation, to be found. The interfacial contact pressure distribution beneath a complete sliding contact between elastically similar components, in the presence of friction, has been studied in detail, with particular reference to contacts whose edge angles are 60 degree, 90 degree and 120 degree. The possible types of behaviour at the edge of contacts, namely power order singularity, power order bounded and square root bounded, are discussed. A full understanding of the behaviour requires a detailed study of a characteristic equation, and this shows the kinds of pressure distribution to be anticipated, which can vary very markedly. The transition from power order behaviour to local separation and bounded behaviour is examined, and an appropriate asymptotic form developed. The problem of trapezium shaped punches pressed into a frictional, elastically similar half-plane, and subject to sequential normal and shear loading, under partial slip, is studied. Detailed considerations have again been given to the specific cases of 60 degree, 90 degree and 120 degree punches, and maps have been developed showing the initial mix of stick, slip and separation regions, together with the steady state response when the shearing force is cycled. Conditions for full stick are established.

<div>Non-stationary point processes are often used to model systems whose rates vary over time. Estimating underlying rate functions is important for input to a discrete-event simulation along with various statistical analyses. We study nonparametric estimators to the marked point process, the infinite-server queueing model, and the transitory queueing model. We conduct statistical inference for these estimators by establishing a number of asymptotic results.</div><div><br></div><div>For the marked point process, we consider estimating the offered load to the system over time. With direct observations of the offered load sampled at fixed intervals, we establish asymptotic consistency, rates of convergence, and asymptotic covariance through a Functional Strong Law of Large Numbers, a Functional Central Limit Theorem, and a Law of Iterated Logarithm. We also show that there exists an asymptotically optimal interval width as the sample size approaches infinity.</div><div><br></div><div>The infinite-server queueing model is central in many stochastic models. Specifically, the mean number of busy servers can be used as an estimator for the total load faced to a multi-server system with time-varying arrivals and in many other applications. Through an omniscient estimator based on observing both the arrival times and service requirements for n samples of an infinite-server queue, we show asymptotic consistency and rate of convergence. Then, we establish the asymptotics for a nonparametric estimator based on observations of the busy servers at fixed intervals.</div><div><br></div><div>The transitory queueing model is crucial when studying a transitory system, which arises when the time horizon or population is finite. We assume we observe arrival counts at fixed intervals. We first consider a natural estimator which applies an underlying nonhomogeneous Poisson process. Although the estimator is asymptotically unbiased, we see that a correction term is required to retrieve an accurate asymptotic covariance. Next, we consider a nonparametric estimator that exploits the maximum likelihood estimator of a multinomial distribution to see that this estimator converges appropriately to a Brownian Bridge.</div>