The representer theorem from the reproducing kernel Hilbert space theory is the origin of many kernel-based machine learning and signal modelling techniques that are popular today. Most kernel functions used in practical applications behave in a homogeneous manner across the domain of the signal of interest, and they are called stationary kernels. One open problem in the literature is the specification of a non-stationary kernel that is computationally tractable. Some recent works solve large-scale optimization problems to obtain such kernels, and they often suffer from non-identifiability issues in their optimization problem formulation. Many practical problems can benefit from using application-specific prior knowledge on the signal of interest. For example, if one can adequately encode the prior assumption that edge contours are smooth, one does not need to learn a finite-dimensional dictionary from a database of sampled image patches that each contains a circular object in order to up-convert images that contain circular edges. In the first portion of this thesis, we present a novel method for constructing non-stationary kernels that incorporates prior knowledge. A theorem is presented that ensures the result of this construction yields a symmetric and positive-definite kernel function. This construction does not require one to solve any non-identifiable optimization problems. It does require one to manually design some portions of the kernel while deferring the specification of the remaining portions to when an observation of the signal is available. In this sense, the resultant kernel is adaptive to the data observed. We give two examples of this construction technique via the grayscale image up-conversion task where we chose to incorporate the prior assumption that edge contours are smooth. Both examples use a novel local analysis algorithm that summarizes the p-most dominant directions for a given grayscale image patch. The non-stationary properties of these two types of kernels are empirically demonstrated on the Kodak image database that is popular within the image processing research community. Tensors and tensor decomposition methods are gaining popularity in the signal processing and machine learning literature, and most of the recently proposed tensor decomposition methods are based on the tensor power and alternating least-squares algorithms, which were both originally devised over a decade ago. The algebraic approach for the canonical polyadic (CP) symmetric tensor decomposition problem is an exception. This approach exploits the bijective relationship between symmetric tensors and homogeneous polynomials. The solution of a CP symmetric tensor decomposition problem is a set of p rank-one tensors, where p is fixed. In this thesis, we refer to such a set of tensors as a rank-one decomposition with cardinality p. Existing works show that the CP symmetric tensor decomposition problem is non-unique in the general case, so there is no bijective mapping between a rank-one decomposition and a symmetric tensor. However, a proposition in this thesis shows that a particular space of rank-one decompositions, SE, is isomorphic to a space of moment matrices that are called quasi-Hankel matrices in the literature. Optimization over Riemannian manifolds is an area of optimization literature that is also gaining popularity within the signal processing and machine learning community. Under some settings, one can formulate optimization problems over differentiable manifolds where each point is an equivalence class. Such manifolds are called quotient manifolds. This type of formulation can reduce or eliminate some of the sources of non-identifiability issues for certain optimization problems. An example is the learning of a basis for a subspace by formulating the solution space as a type of quotient manifold called the Grassmann manifold, while the conventional formulation is to optimize over a space of full column rank matrices. The second portion of this thesis is about the development of a general-purpose numerical optimization framework over SE. A general-purpose numerical optimizer can solve different approximations or regularized versions of the CP decomposition problem, and they can be applied to tensor-related applications that do not use a tensor decomposition formulation. The proposed optimizer uses many concepts from the Riemannian optimization literature. We present a novel formulation of SE as an embedded differentiable submanifold of the space of real-valued matrices with full column rank, and as a quotient manifold. Riemannian manifold structures and tangent space projectors are derived as well. The CP symmetric tensor decomposition problem is used to empirically demonstrate that the proposed scheme is indeed a numerical optimization framework over SE. Future investigations will concentrate on extending the proposed optimization framework to handle decompositions that correspond to non-symmetric tensors.

In this habilitation thesis a factorization theory for Toeplitz plus Hankel operators and singular integral operators with flip is established. These operators are considered with matrix-valued symbols and are thought of acting on the vector-valued analogues of the Hardy and Lebesgue spaces. A factorization theory for pure Toeplitz operators and singular integral operators without flip is known since decades and provides necessary and sufficient conditions for Fredholmness and formulas for the defect numbers. In particular, the invertibility of such operators is equivalent to the existence of a certain type of Wiener-Hopf factorization. In this thesis an analogous theory for the afore-mentioned more general classes of operators is developed. It turns out that a completely different kind of factorization is needed. This kind of factorization is studied extensively, and a corresponding Fredholm theory is established. A connection with the Hunt-Muckenhoupt-Wheeden condition is made, and several examples and applications are given as well. / In dieser Habilitationsschrift wird eine Faktorisierungstheorie für Toeplitz plus Hankel-Operatoren und singuläre Integraloperatoren mit Flip aufgestellt. Diese Operatoren werden mit matrixwertigem Symbol betrachtet und sind auf den vektorwertigen Analoga der Hardy- und Lebesgue-Räumen definiert. Eine Faktorisierungstheorie für reine Toeplitz bzw. singuläre Integraloperatoren ohne Flip ist seit Jahrzehnten bekannt. Sie liefert notwendige und hinreichende Bedingungen für die Fredholmeigenschaft und Formeln für die Defektzahlen. Insbesondere ist die Invertierbarkeit derartiger Operatoren äquivalent zur Existenz einer bestimmten Art der Wiener-Hopf-Faktorisierung. In dieser Habilitationsschrift wird eine entsprechende Theorie für die erwähnten, allgemeineren Klassen von Operatoren aufgestellt. Es stellt sich heraus, dass eine völlig andere Art der Faktorisierung benötigt wird. Diese Art der Faktorisierung wird eingehend studiert und eine entsprechende Fredholmtheorie wird entwickelt. Ein Zusammenhang mit der Hunt-Muckenhoupt-Wheeden Bedingung wird hergestellt. Mehrere Beispiele und Anwendungen werden ebenfalls angegeben.

Dans le chapitre 1, nous présentons les congruences linéaires simples et les tests de qualité des nombres pseudo-aléatoires (n.p.a.) congruentiels. L'accent est mis sur le test des treillis, le test spectral et le test sériel. Le test sériel est base sur l'estimation de la discrépance des vecteurs de n.p.a. Partant de cette estimation, on introduit une quantité appelée figure de mérite. Celle-ci nous permet de rechercher, pour m et b fixes, des multiplicateurs a tels que deux termes successifs de la suite (a,b,m,x#0) soient statistiquement indépendants. Nous débutons le chapitre 2 par l'étude des longueurs de cycle et du transitoire des suites engendrées par une congruence linéaire multidimensionnelle (c.l.m.). Ensuite, nous décrivons quelques méthodes de transformation de ces suites en suites de n.p.a. Enfin, nous faisons une discussion sur le choix des paramètres d'une c.l.m. Dans le chapitre 3, nous étudions la période d'un générateur vectoriel base sur le modèle de Daykin et une c.l.m. De période maximale, puis nous faisons un aperçu sur les principaux générateurs non linéaires de n.p.a. Le chapitre 4, réservé a la cryptographie, traite du problème du décryptage de l'ordre et du modulo d'une c.l.m

L'objet de cette thèse est d'étudier l'applicabilité de la méthode d'approximation rationnelle en norme de Hankel à des systèmes dynamiques linéaires de dimension d'état infinie. On illustre par trois exemples concrets les possibilités d'utilisation des techniques d'approximation développées ces dernières années, notamment par Curtain, Glover et Partington. Les exemples choisis représentent des phénomènes d'évolution décrits par des équations aux dérivées partielles, par rapport au temps et aux variables d'espace. Il s'agit: d'un problème de diffusion de chaleur, de type parabolique, pour lequel les techniques d'approximation s'adaptent assez directement ; de deux problèmes hyperboliques décrivant l'évolution d'une poutre en flexion et en torsion, pour lesquels une méthode originale appelée ``relaxation'' a été mise au point: préalable à l'approximation de Hankel, elle permet son application lorsque les pôes associés au système hyperbolique croissent suffisamment rapidement.

Dans cette thèse, nous visons l'amélioration de quelques algorithmes en algèbre matricielle rapide et plus spécifiquement les algorithmes rapides sur les matrices structurées en calcul formel et numérique. Nous nous intéressons en particulier aux matrices de Hankel et de Toeplitz. Nous introduisons un nouvel algorithme de diagonalisation par blocs approchée de matrices réelles de Hankel. Nous décrivons la relation naturelle entre l'algorithme d'Euclide et notre factorisation par blocs approchée pour les matrices de Hankel associées à deux polynômes, ainsi que pour les matrices de Bézout associées aux mêmes polynômes. Enfin, dans le cas complexe, nous présentons un algorithme révisé de notre diagonalisation par blocs approchée des matrices de Hankel, en calculant la suite des restes et la suite des quotients apparues au cours de l'exécution de l'algorithme d'Euclide.

Cette thèse présente une contribution à l'amélioration de certains résultats concernant les algorithmes en Algèbre linéaire et plus particulièrement les algorithmes sur les matrices structurées. Nous présentons un nouvel algorithme de diagonalisation par blocs des matrices de Hankel, particulièrement efficace. Dans le cas où la matrice de Hankel correspond à une suite récurrente linéaire, nous retrouvons ainsi l'algorithme de Berlekamp-Massey, mais dans une version simplifiée (plus facile à expliquer et à programmer) et accélérée par des troncatures. En outre notre version permet une gestion dynamique des données. Notre diagonalisation par blocs, qui s'applique sur un corps arbitraire, nous permet de donner une démonstration purement algébrique et simple d'un délicat théorème de Frobenius pour la signature d'une forme de Hankel réelle. Nous donnons également une étude approfondie de l'algorithme d'Euclide signé et de ses versions matricielles pour les matrices de Hankel et de Bezout associées à un couple de polynômes. Nous expliquons les rapports existants entre différents algorithmes connus dans la littérature.

Acoustic plane wave intearctions with a rough seabed with a continuously varying density and sound speed in a fluid-like sediment layer overlying an elastic basement is considered in this thesis. The acoustic properties in the sediment layer possess an exponential type of variation in density and one of the three classes of sound speed profiles, which are constant, k^2-linear, or inverse-square variations. Analytical solutions for the Helmholtz equation in the sediment layer, combined with a formulation based upon boundary perturbation theory, facilitate numerical implementation for the solution of coherent field. The coherent reflection coefficients corresponding to the aformentioned density and sound speed profiles for various frequencies, roughness parameters, basement stiffness, are numerically generated and analyzed. Physical interpretations are provided for various results. This simple model characterizes three important features of an realistic sea floor, including seabed roughness, sediment inhomogenieties, and basement shear property,%Two dimensions is considered in the seafloor environment and the random roughness is belong to one dimension space.% , therefore, provides a canonical model for the study of seabed acoustics. The variation of the acoustic properties takes such a form that it is not only geologically realistic, but also renders analytical solutions for the Helmholtz equation, thus facilitating the formulation of the problem. The computational algorithm for the spatial spectrum of the scattered field due to random seabed has been developed based upon a boundary perturbation method. %About scattering field, only one time reflection from the sediment is taked account of, because the higher numerical order is, the lower scattering energy exist.% The results have shown that, while the coherent field mainly depends upon the gross structure of the rough seabed represented by the RMS roughness, the scattered field heavily depends upon the details of the roughness structure specialized by the roughness power spectrum and the spatial correlation length of the rough surface. The dependence of the spatial spectrum on the sediment stratification is also carefully examined.

Consider the D-optimal designs for the dth-degree polynomial regression model with a bounded and positive weight function on a compact interval. As the degree of the model goes to infinity, we show that the D-optimal design converges weakly to the arcsin distribution. If the weight function is equal to 1, we derive the formulae of the values of the D-criterion for five classes of designs including (i) uniform density design; (ii) arcsin density design; (iii) J_{1/2,1/2} density design; (iv) arcsin support design and (v) uniform support design. The comparison of D-efficiencies among these designs are investigated; besides, the asymptotic expansions and limits of their D-efficiencies are also given. It shows that the D-efficiency of the arcsin support design is the highest among the first four designs.

This work is concerned with the applicability of several approximation methods (finite section method, Galerkin and collocation methods with maximum defect splines for uniform and non uniform meshes) to operators belonging to the closed subalgebra generated by operators of multiplication bz piecewise continuous functions and convolution operators also with piecewise continuous generating function.

A practical procedure based on implicit time integration methods applied to the differential Lyapunov equations arising in the square root balanced truncation method is presented. The application of high order time integrators results in indefinite right-hand sides of the algebraic Lyapunov equations that have to be solved within every time step. Therefore, classical methods exploiting the inherent low-rank structure often observed for practical applications end up in complex data and arithmetic. Avoiding the additional effort treating complex quantities, a symmetric indefinite factorization of both the right-hand side and the solution of the differential Lyapunov equations is applied.