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1
Wigestrand, Jan. "Inequalities in Hilbert Spaces." Thesis, Norwegian University of Science and Technology, Department of Mathematical Sciences, 2008. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-9673.
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<p>The main result in this thesis is a new generalization of Selberg's inequality in Hilbert spaces with a proof. In Chapter 1 we define Hilbert spaces and give a proof of the Cauchy-Schwarz inequality and the Bessel inequality. As an example of application of the Cauchy-Schwarz inequality and the Bessel inequality, we give an estimate for the dimension of an eigenspace of an integral operator. Next we give a proof of Selberg's inequality including the equality conditions following [Furuta]. In Chapter 2 we give selected facts on positive semidefinite matrices with proofs or references. Then we use this theory for positive semidefinite matrices to study inequalities. First we give a proof of a generalized Bessel inequality following [Akhiezer,Glazman], then we use the same technique to give a new proof of Selberg's inequality. We conclude with a new generalization of Selberg's inequality with a proof. In the last section of Chapter 2 we show how the matrix approach developed in Chapter 2.1 and Chapter 2.2 can be used to obtain optimal frame bounds. We introduce a new notation for frame bounds.</p>
2
Ameur, Yacin. "Interpolation of Hilbert spaces." Doctoral thesis, Uppsala universitet, Matematiska institutionen, 2002. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-1753.
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(i) We prove that intermediate Banach spaces A, B with respect to arbitrary Hilbert couples H, K are exact interpolation iff they are exact K-monotonic, i.e. the condition f0∊A and the inequality K(t,g0;K)≤K(t,f0;H), t>0 imply g0∊B and ||g0||B≤||f0||A (K is Peetre's K-functional). It is well-known that this property is implied by the following: for each ρ>1 there exists an operator T : H→K such that Tf0=g0, and K(t,Tf;K)≤ρK(t,f;H), f∊H0+H1, t>0.Verifying the latter property, it suffices to consider the "diagonal" case where H=K is finite-dimensional. In this case, we construct the relevant operators by a method which allows us to explicitly calculate them. In the strongest form of the theorem, it is shown that the statement remains valid when substituting ρ=1. (ii) A new proof is given to a theorem of W. F. Donoghue which characterizes certain classes of functions whose domain of definition are finite sets, and which are subject to certain matrix inequalities. The result generalizes the classical Löwner theorem on monotone matrix functions, and also yields some information with respect to the finer study of monotone functions of finite order. (iii) It is shown that with respect to a positive concave function ψ there exists a function h, positive and regular on ℝ+ and admitting of analytic continuation to the upper half-plane and having positive imaginary part there, such that h≤ψ≤ 2h. This fact is closely related to a theorem of Foiaş, Ong and Rosenthal, which states that regardless of the choice of a concave function ψ, and a weight λ, the weighted l2-space l2(ψ(λ)) is c-interpolation with respect to the couple (l2,l2(λ)), where we have c≤√2 for the best c. It turns out that c=√2 is best possible in this theorem; a fact which is implicit in the work of G. Sparr. (iv) We give a new proof and new interpretation (based on the work (ii) above) of Donoghue's interpolation theorem; for an intermediate Hilbert space H* to be exact interpolation with respect to a regular Hilbert couple H it is necessary and sufficient that the norm in H* be representable in the form ||f||*= (∫[0,∞] (1+t-1)K2(t,f;H)2dρ(t))1/2 with some positive Radon measure ρ on the compactified half-line [0,∞]. (v) The theorem of W. F. Donoghue (item (ii) above) is extended to interpolation of tensor products. Our result is related to A. Korányi's work on monotone matrix functions of several variables.
3
Ameur, Yacin. "Interpolation of Hilbert spaces /." Uppsala : Matematiska institutionen, Univ. [distributör], 2001. http://publications.uu.se/theses/91-506-1531-9/.
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Panayotov, Ivo. "Conjugate gradient in Hilbert spaces." Thesis, McGill University, 2004. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=82402.
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In this thesis, we examine the Conjugate Gradient algorithm for solving self-adjoint positive definite linear systems in Cn . We generalize the algorithm by proving a convergence result of Conjugate Gradient for self-adjoint positive definite operators in an arbitrary Hilbert space H. Then, we use the Maple software for symbolic manipulation to implement a general version of Conjugate Gradient and to demonstrate, by examples, that the algorithm can be used directly to solve problems in Hilbert spaces other than Cn .
5
Bahmani, Fatemeh. "Ternary structures in Hilbert spaces." Thesis, Queen Mary, University of London, 2011. http://qmro.qmul.ac.uk/xmlui/handle/123456789/697.
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Ternary structures in Hilbert spaces arose in the study of in nite dimensional manifolds in di erential geometry. In this thesis, we develop a structure theory of Hilbert ternary algebras and Jordan Hilbert triples which are Hilbert spaces equipped with a ternary product. We obtain several new results on the classi - cation of these structures. Some results have been published in [2]. A Hilbert ternary algebra is a real Hilbert space (V; h ; i) equipped with a ternary product [ ; ; ] satisfying h[a; b; x]; yi = hx; [b; a; y]i for a; b; x and y in V . A Jordan Hilbert triple is a real Hilbert space in which the ternary product f ; ; g is a Jordan triple product. It is called a JH-triple if the identity hfa; b; xg; xi = hx; fb; a; xgi holds in V . JH-triples correspond to a class of Lie algebras which play an important role in symmetric Riemannian manifolds. We begin by proving new structure results on ideals, centralizers and derivations of Hilbert ternary algebras. We characterize primitive tripotents in Hilbert ternary algebras and use them as coordinates to classify abelian Hilbert ternary algebras. We show that they are direct sums of simple ones, and each simple abelian Hilbert ternary algebra is ternary isomorphic to the algebra C2(H;K) of Hilbert-Schmidt operators between real, complex or quaternion Hilbert spaces H and K. Further, we describe completely the ternary isomorphisms and ternary antiisomorphisms between abelian Hilbert ternary algebras. We show that each ternary isomorphism between simple algebras C2(H;K) and C2(H0;K0) is of the form (x) = Jxj where j : H0 ! H and J : K ! K0 are isometries. A ternary anti-isomorphism is of the form (x) = Jx j where j : H0 ! K and J : H ! K0 are isometries. The structures of Hilbert ternary algebras and JH-triples are closely related. Indeed, we show that each JH-triple (V; f ; ; g) admits a decomposi- 6 tion V = Vs L V ? s where (Vs; f ; ; g) is a Hilbert ternary algebra which is usually nonabelian and unless V = Vs, the orthogonal complement V ? s is always a nonabelian Hilbert ternary algebra in the derived ternary product [a; b; c]d = fa; b; cg fb; a; cg. Hence JH-triples provide important examples of nonabelian Hilbert ternary algebras. We determine exactly when Vs and V ? s are Jordan triple ideals of V . We show, in each dimension at least 2, there is a JH-triple (V; f ; ; g) for which V 6= Vs, equivalently, (V; f ; ; g) is not a Hilbert ternary algebra. 7
6
Das, Tushar. "Kleinian Groups in Hilbert Spaces." Thesis, University of North Texas, 2012. https://digital.library.unt.edu/ark:/67531/metadc149579/.
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The theory of discrete groups acting on finite dimensional Euclidean open balls by hyperbolic isometries was borne around the end of 19th century within the works of Fuchs, Klein and Poincaré. We develop the theory of discrete groups acting by hyperbolic isometries on the open unit ball of an infinite dimensional separable Hilbert space. We present our investigations on the geometry of limit sets at the sphere at infinity with an attempt to highlight the differences between the finite and infinite dimensional theories. We discuss the existence of fixed points of isometries and the classification of isometries. Various notions of discreteness that were equivalent in finite dimensions, no longer turn out to be in our setting. In this regard, the robust notion of strong discreteness is introduced and we study limit sets for properly discontinuous actions. We go on to prove a generalization of the Bishop-Jones formula for strongly discrete groups, equating the Hausdorff dimension of the radial limit set with the Poincaré exponent of the group. We end with a short discussion on conformal measures and their relation with Hausdorff and packing measures on the limit set.
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Harris, Terri Joan Mrs. "HILBERT SPACES AND FOURIER SERIES." CSUSB ScholarWorks, 2015. https://scholarworks.lib.csusb.edu/etd/244.
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I give an overview of the basic theory of Hilbert spaces necessary to understand the convergence of the Fourier series for square integrable functions. I state the necessary theorems and definitions to understand the formulations of the problem in a Hilbert space framework, and then I give some applications of the theory along the way.
8
Dieuleveut, Aymeric. "Stochastic approximation in Hilbert spaces." Thesis, Paris Sciences et Lettres (ComUE), 2017. http://www.theses.fr/2017PSLEE059/document.
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Le but de l’apprentissage supervisé est d’inférer des relations entre un phénomène que l’on souhaite prédire et des variables « explicatives ». À cette fin, on dispose d’observations de multiples réalisations du phénomène, à partir desquelles on propose une règle de prédiction. L’émergence récente de sources de données à très grande échelle, tant par le nombre d’observations effectuées (en analyse d’image, par exemple) que par le grand nombre de variables explicatives (en génétique), a fait émerger deux difficultés : d’une part, il devient difficile d’éviter l’écueil du sur-apprentissage lorsque le nombre de variables explicatives est très supérieur au nombre d’observations; d’autre part, l’aspect algorithmique devient déterminant, car la seule résolution d’un système linéaire dans les espaces en jeupeut devenir une difficulté majeure. Des algorithmes issus des méthodes d’approximation stochastique proposent uneréponse simultanée à ces deux difficultés : l’utilisation d’une méthode stochastique réduit drastiquement le coût algorithmique, sans dégrader la qualité de la règle de prédiction proposée, en évitant naturellement le sur-apprentissage. En particulier, le cœur de cette thèse portera sur les méthodes de gradient stochastique. Les très populaires méthodes paramétriques proposent comme prédictions des fonctions linéaires d’un ensemble choisi de variables explicatives. Cependant, ces méthodes aboutissent souvent à une approximation imprécise de la structure statistique sous-jacente. Dans le cadre non-paramétrique, qui est un des thèmes centraux de cette thèse, la restriction aux prédicteurs linéaires est levée. La classe de fonctions dans laquelle le prédicteur est construit dépend elle-même des observations. En pratique, les méthodes non-paramétriques sont cruciales pour diverses applications, en particulier pour l’analyse de données non vectorielles, qui peuvent être associées à un vecteur dans un espace fonctionnel via l’utilisation d’un noyau défini positif. Cela autorise l’utilisation d’algorithmes associés à des données vectorielles, mais exige une compréhension de ces algorithmes dans l’espace non-paramétrique associé : l’espace à noyau reproduisant. Par ailleurs, l’analyse de l’estimation non-paramétrique fournit également un éclairage révélateur sur le cadre paramétrique, lorsque le nombre de prédicteurs surpasse largement le nombre d’observations. La première contribution de cette thèse consiste en une analyse détaillée de l’approximation stochastique dans le cadre non-paramétrique, en particulier dans le cadre des espaces à noyaux reproduisants. Cette analyse permet d’obtenir des taux de convergence optimaux pour l’algorithme de descente de gradient stochastique moyennée. L’analyse proposée s’applique à de nombreux cadres, et une attention particulière est portée à l’utilisation d’hypothèses minimales, ainsi qu’à l’étude des cadres où le nombre d’observations est connu à l’avance, ou peut évoluer. La seconde contribution est de proposer un algorithme, basé sur un principe d’accélération, qui converge à une vitesse optimale, tant du point de vue de l’optimisation que du point de vue statistique. Cela permet, dans le cadre non-paramétrique, d’améliorer la convergence jusqu’au taux optimal, dans certains régimes pour lesquels le premier algorithme analysé restait sous-optimal. Enfin, la troisième contribution de la thèse consiste en l’extension du cadre étudié au delà de la perte des moindres carrés : l’algorithme de descente de gradient stochastiqueest analysé comme une chaine de Markov. Cette approche résulte en une interprétation intuitive, et souligne les différences entre le cadre quadratique et le cadre général. Une méthode simple permettant d’améliorer substantiellement la convergence est également proposée<br>The goal of supervised machine learning is to infer relationships between a phenomenon one seeks to predict and “explanatory” variables. To that end, multiple occurrences of the phenomenon are observed, from which a prediction rule is constructed. The last two decades have witnessed the apparition of very large data-sets, both in terms of the number of observations (e.g., in image analysis) and in terms of the number of explanatory variables (e.g., in genetics). This has raised two challenges: first, avoiding the pitfall of over-fitting, especially when the number of explanatory variables is much higher than the number of observations; and second, dealing with the computational constraints, such as when the mere resolution of a linear system becomes a difficulty of its own. Algorithms that take their roots in stochastic approximation methods tackle both of these difficulties simultaneously: these stochastic methods dramatically reduce the computational cost, without degrading the quality of the proposed prediction rule, and they can naturally avoid over-fitting. As a consequence, the core of this thesis will be the study of stochastic gradient methods. The popular parametric methods give predictors which are linear functions of a set ofexplanatory variables. However, they often result in an imprecise approximation of the underlying statistical structure. In the non-parametric setting, which is paramount in this thesis, this restriction is lifted. The class of functions from which the predictor is proposed depends on the observations. In practice, these methods have multiple purposes, and are essential for learning with non-vectorial data, which can be mapped onto a vector in a functional space using a positive definite kernel. This allows to use algorithms designed for vectorial data, but requires the analysis to be made in the non-parametric associated space: the reproducing kernel Hilbert space. Moreover, the analysis of non-parametric regression also sheds some light on the parametric setting when the number of predictors is much larger than the number of observations. The first contribution of this thesis is to provide a detailed analysis of stochastic approximation in the non-parametric setting, precisely in reproducing kernel Hilbert spaces. This analysis proves optimal convergence rates for the averaged stochastic gradient descent algorithm. As we take special care in using minimal assumptions, it applies to numerous situations, and covers both the settings in which the number of observations is known a priori, and situations in which the learning algorithm works in an on-line fashion. The second contribution is an algorithm based on acceleration, which converges at optimal speed, both from the optimization point of view and from the statistical one. In the non-parametric setting, this can improve the convergence rate up to optimality, even inparticular regimes for which the first algorithm remains sub-optimal. Finally, the third contribution of the thesis consists in an extension of the framework beyond the least-square loss. The stochastic gradient descent algorithm is analyzed as a Markov chain. This point of view leads to an intuitive and insightful interpretation, that outlines the differences between the quadratic setting and the more general setting. A simple method resulting in provable improvements in the convergence is then proposed
9
Boralugoda, Sanath Kumara. "Prox-regular functions in Hilbert spaces." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 1998. http://www.collectionscanada.ca/obj/s4/f2/dsk2/tape15/PQDD_0006/NQ34740.pdf.
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Lapinski, Felicia. "Hilbert spaces and the Spectral theorem." Thesis, Uppsala universitet, Analys och sannolikhetsteori, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:uu:diva-454412.
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Tipton, James Edward. "Reproducing Kernel Hilbert spaces and complex dynamics." Diss., University of Iowa, 2016. https://ir.uiowa.edu/etd/2284.
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Both complex dynamics and the theory of reproducing kernel Hilbert spaces have found widespread application over the last few decades. Although complex dynamics started over a century ago, the gravity of it's importance was only recently realized due to B.B. Mandelbrot's work in the 1980's. B.B. Mandelbrot demonstrated to the world that fractals, which are chaotic patterns containing a high degree of self-similarity, often times serve as better models to nature than conventional smooth models. The theory of reproducing kernel Hilbert spaces also having started over a century ago, didn't pick up until N. Aronszajn's classic was written in 1950. Since then, the theory has found widespread application to fields including machine learning, quantum mechanics, and harmonic analysis.
In the paper, Infinite Product Representations of Kernel Functions and Iterated Function Systems, the authors, D. Alpay, P. Jorgensen, I. Lewkowicz, and I. Martiziano, show how a kernel function can be constructed on an attracting set of an iterated function system. Furthermore, they show that when certain conditions are met, one can construct an orthonormal basis of the associated Hilbert space via certain pull-back and multiplier operators.
In this thesis we take for our iterated function system, the family of iterates of a given rational map. Thus we investigate for which rational maps their kernel construction holds as well as their orthornormal basis construction. We are able to show that the kernel construction applies to any rational map conjugate to a polynomial with an attracting fixed point at 0. Within such rational maps, we are able to find a family of polynomials for which the orthonormal basis construction holds. It is then natural to ask how the orthonormal basis changes as the polynomial within a given family varies. We are able to determine for certain families of polynomials, that the dynamics of the corresponding orthonormal basis is well behaved. Finally, we conclude with some possible avenues of future investigation.
12
EUSEBI, Anita. "Quantum Cryptography in d-dimensional Hilbert spaces." Doctoral thesis, Università degli Studi di Camerino, 2011. http://hdl.handle.net/11581/401811.
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The topic of this thesis is Quantum Cryptography. Based on the laws of Quantum Mechanics, this allows two parties to share a secure key, by using quantum states to carry classical information. Namely, the Quantum Key Distribution (QKD) process, completed with the classical algorithm One Time Pad, leads to an unconditionally secure cryptosystem. Most QKD protocols, like the famous BB84, are realized on unidirectional quantum channels and have probabilistic character. Recently, a new protocol, named LM05, has been proposed, which works on bidirectional quantum channels and in a deterministic way. This thesis is mainly concerned with protocols extending LM05 from binary to d-ary alphabets (dmajor2), by using multi-level quantum systems in d-dimensional Hilbert spaces. The construction of such protocols relies on the notion of Mutually Unbiased Bases (MUB). But the total number of MUB in a Hilbert space of dimension d is known only if d is a prime power. Accordingly, the new protocols are realized under this assumption. As a preliminary step, an explicit expression for MUB encompassing powers of both even and odd primes is discussed. The first proposed extension, called EM09, uses shift operators on MUB to encode information and the usual quantum measurement to realize the control procedure. This guarantees maximal security for dimension d=3 against a powerful individual attack. The second extension, named EM11, is characterized by an innovative control strategy based on a suitable unitary transformation rather than on quantum measurement. Such protocol only works for d an odd prime power, due to the particular choice of the control operator, which is proved to be the only possible with the required properties. The EM11 protocol leads to a relevant improvement. In fact, the security against the same attack is much better than in the EM09 and it increases in terms of the dimension d. Some partial results are also obtained about the possible extensions of the probabilistic protocol called SARG04 to higher dimensions. Finally, it is considered the use of the EM09 and EM11 protocols in the Quantum Direct Communication, where the meaningful message is transmitted without any encryption, as allowed by their deterministic character. In this context, the asymptotic security turns out to be optimal in dimension d=2 for EM09 and in dimension d=3 for EM11.
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Kiteu, Marco M. "Orbits of operators on Hilbert space and some classes of Banach spaces." Kent State University / OhioLINK, 2012. http://rave.ohiolink.edu/etdc/view?acc_num=kent1341850621.
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Sprungk, Björn. "Numerical Methods for Bayesian Inference in Hilbert Spaces." Doctoral thesis, Universitätsbibliothek Chemnitz, 2018. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-226748.
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Bayesian inference occurs when prior knowledge about uncertain parameters in mathematical models is merged with new observational data related to the model outcome. In this thesis we focus on models given by partial differential equations where the uncertain parameters are coefficient functions belonging to infinite dimensional function spaces. The result of the Bayesian inference is then a well-defined posterior probability measure on a function space describing the updated knowledge about the uncertain coefficient. For decision making and post-processing it is often required to sample or integrate wit resprect to the posterior measure. This calls for sampling or numerical methods which are suitable for infinite dimensional spaces. In this work we focus on Kalman filter techniques based on ensembles or polynomial chaos expansions as well as Markov chain Monte Carlo methods. We analyze the Kalman filters by proving convergence and discussing their applicability in the context of Bayesian inference. Moreover, we develop and study an improved dimension-independent Metropolis-Hastings algorithm. Here, we show geometric ergodicity of the new method by a spectral gap approach using a novel comparison result for spectral gaps. Besides that, we observe and further analyze the robustness of the proposed algorithm with respect to decreasing observational noise. This robustness is another desirable property of numerical methods for Bayesian inference. The work concludes with the application of the discussed methods to a real-world groundwater flow problem illustrating, in particular, the Bayesian approach for uncertainty quantification in practice<br>Bayessche Inferenz besteht daraus, vorhandenes a-priori Wissen über unsichere Parameter in mathematischen Modellen mit neuen Beobachtungen messbarer Modellgrößen zusammenzuführen. In dieser Dissertation beschäftigen wir uns mit Modellen, die durch partielle Differentialgleichungen beschrieben sind. Die unbekannten Parameter sind dabei Koeffizientenfunktionen, die aus einem unendlich dimensionalen Funktionenraum kommen. Das Resultat der Bayesschen Inferenz ist dann eine wohldefinierte a-posteriori Wahrscheinlichkeitsverteilung auf diesem Funktionenraum, welche das aktualisierte Wissen über den unsicheren Koeffizienten beschreibt. Für Entscheidungsverfahren oder Postprocessing ist es oft notwendig die a-posteriori Verteilung zu simulieren oder bzgl. dieser zu integrieren. Dies verlangt nach numerischen Verfahren, welche sich zur Simulation in unendlich dimensionalen Räumen eignen. In dieser Arbeit betrachten wir Kalmanfiltertechniken, die auf Ensembles oder polynomiellen Chaosentwicklungen basieren, sowie Markowketten-Monte-Carlo-Methoden. Wir analysieren die erwähnte Kalmanfilter, indem wir deren Konvergenz zeigen und ihre Anwendbarkeit im Kontext Bayesscher Inferenz diskutieren. Weiterhin entwickeln und studieren wir einen verbesserten dimensionsunabhängigen Metropolis-Hastings-Algorithmus. Hierbei weisen wir geometrische Ergodizität mit Hilfe eines neuen Resultates zum Vergleich der Spektrallücken von Markowketten nach. Zusätzlich beobachten und analysieren wir die Robustheit der neuen Methode bzgl. eines fallenden Beobachtungsfehlers. Diese Robustheit ist eine weitere wünschenswerte Eigenschaft numerischer Methoden für Bayessche Inferenz. Den Abschluss der Arbeit bildet die Anwendung der diskutierten Methoden auf ein reales Grundwasserproblem, was insbesondere den Bayesschen Zugang zur Unsicherheitsquantifizierung in der Praxis illustriert
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VIEIRA, PAULO CESAR MARQUES. "STABILITY FOR DISCRETE LINEAR SYSTEMS IN HILBERT SPACES." PONTIFÍCIA UNIVERSIDADE CATÓLICA DO RIO DE JANEIRO, 1988. http://www.maxwell.vrac.puc-rio.br/Busca_etds.php?strSecao=resultado&nrSeq=8426@1.
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COORDENAÇÃO DE APERFEIÇOAMENTO DO PESSOAL DE ENSINO SUPERIOR<br>Este trabalho aborda o problema da estabilidade de
sistemas lineares, invariantes no tempo, a tempo discreto,
com o espaço de estado sendo um espaço de Hilbert complexo
e separável de dimensão infinita. São investigadas
condições necessárias e/ou suficientes para quatro
conceitos diferentes de estabilidade: estabilidade
assintótica uniforme e estabilidade assintótica forte,
estabilidade assintótica fraca e estabilidade limitada.
Identifica-se e analisa-se as conexões entre os problemas
de estabilidade e dois problemas em aberto da teoria de
operadores em espaços de Hilbert: o problema do subespaço
invariante e o problemas da similaridade e contração.
Diversos resultados, oriundos de tentativas de solução
para os dois problemas acima, ou motivados por aquelas
tentativas, são utilizadas para fornecer caracterizações
adicionais (principalmente caracterizações espectrais)
para os quatro conceitos de estabilidade em questão.<br>This work deals with the stability problem for time-
invariant discrete linear systems evolving in a separable
infinite-dimensional Hilbert space. Necessary and/or
sufficient conditions for uniform, strong and weak
asymptotic stability, as well as to bounded stability
problems to two open problems in operator theory, namely,
the invariant subspace and the similarity to contractions,
are identified and analysed in detail. Several results
from the many attempts, of solving the above mentioned
open problems, or motivated by those attempts, are used to
supply additional characterizations (mainly spectral
characterization) for the four stabilty concepts under
consideration.
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Sprungk, Björn. "Numerical Methods for Bayesian Inference in Hilbert Spaces." Doctoral thesis, Technische Universität Chemnitz, 2017. https://monarch.qucosa.de/id/qucosa%3A20754.
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Bayesian inference occurs when prior knowledge about uncertain parameters in mathematical models is merged with new observational data related to the model outcome. In this thesis we focus on models given by partial differential equations where the uncertain parameters are coefficient functions belonging to infinite dimensional function spaces. The result of the Bayesian inference is then a well-defined posterior probability measure on a function space describing the updated knowledge about the uncertain coefficient.
For decision making and post-processing it is often required to sample or integrate wit resprect to the posterior measure. This calls for sampling or numerical methods which are suitable for infinite dimensional spaces. In this work we focus on Kalman filter techniques based on ensembles or polynomial chaos expansions as well as Markov chain Monte Carlo methods.
We analyze the Kalman filters by proving convergence and discussing their applicability in the context of Bayesian inference. Moreover, we develop and study an improved dimension-independent Metropolis-Hastings algorithm. Here, we show geometric ergodicity of the new method by a spectral gap approach using a novel comparison result for spectral gaps. Besides that, we observe and further analyze the robustness of the proposed algorithm with respect to decreasing observational noise. This robustness is another desirable property of numerical methods for Bayesian inference.
The work concludes with the application of the discussed methods to a real-world groundwater flow problem illustrating, in particular, the Bayesian approach for uncertainty quantification in practice.<br>Bayessche Inferenz besteht daraus, vorhandenes a-priori Wissen über unsichere Parameter in mathematischen Modellen mit neuen Beobachtungen messbarer Modellgrößen zusammenzuführen. In dieser Dissertation beschäftigen wir uns mit Modellen, die durch partielle Differentialgleichungen beschrieben sind. Die unbekannten Parameter sind dabei Koeffizientenfunktionen, die aus einem unendlich dimensionalen Funktionenraum kommen. Das Resultat der Bayesschen Inferenz ist dann eine wohldefinierte a-posteriori Wahrscheinlichkeitsverteilung auf diesem Funktionenraum, welche das aktualisierte Wissen über den unsicheren Koeffizienten beschreibt.
Für Entscheidungsverfahren oder Postprocessing ist es oft notwendig die a-posteriori Verteilung zu simulieren oder bzgl. dieser zu integrieren. Dies verlangt nach numerischen Verfahren, welche sich zur Simulation in unendlich dimensionalen Räumen eignen. In dieser Arbeit betrachten wir Kalmanfiltertechniken, die auf Ensembles oder polynomiellen Chaosentwicklungen basieren, sowie Markowketten-Monte-Carlo-Methoden.
Wir analysieren die erwähnte Kalmanfilter, indem wir deren Konvergenz zeigen und ihre Anwendbarkeit im Kontext Bayesscher Inferenz diskutieren. Weiterhin entwickeln und studieren wir einen verbesserten dimensionsunabhängigen Metropolis-Hastings-Algorithmus. Hierbei weisen wir geometrische Ergodizität mit Hilfe eines neuen Resultates zum Vergleich der Spektrallücken von Markowketten nach. Zusätzlich beobachten und analysieren wir die Robustheit der neuen Methode bzgl. eines fallenden Beobachtungsfehlers. Diese Robustheit ist eine weitere wünschenswerte Eigenschaft numerischer Methoden für Bayessche Inferenz.
Den Abschluss der Arbeit bildet die Anwendung der diskutierten Methoden auf ein reales Grundwasserproblem, was insbesondere den Bayesschen Zugang zur Unsicherheitsquantifizierung in der Praxis illustriert.
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Giulini, Ilaria. "Generalization bounds for random samples in Hilbert spaces." Thesis, Paris, Ecole normale supérieure, 2015. http://www.theses.fr/2015ENSU0026/document.
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Ce travail de thèse porte sur l'obtention de bornes de généralisation pour des échantillons statistiques à valeur dans des espaces de Hilbert définis par des noyaux reproduisants. L'approche consiste à obtenir des bornes non asymptotiques indépendantes de la dimension dans des espaces de dimension finie, en utilisant des inégalités PAC-Bayesiennes liées à une perturbation Gaussienne du paramètre et à les étendre ensuite aux espaces de Hilbert séparables. On se pose dans un premier temps la question de l'estimation de l'opérateur de Gram à partir d'un échantillon i. i. d. par un estimateur robuste et on propose des bornes uniformes, sous des hypothèses faibles de moments. Ces résultats permettent de caractériser l'analyse en composantes principales indépendamment de la dimension et d'en proposer des variantes robustes. On propose ensuite un nouvel algorithme de clustering spectral. Au lieu de ne garder que la projection sur les premiers vecteurs propres, on calcule une itérée du Laplacian normalisé. Cette itération, justifiée par l'analyse du clustering en termes de chaînes de Markov, opère comme une version régularisée de la projection sur les premiers vecteurs propres et permet d'obtenir un algorithme dans lequel le nombre de clusters est déterminé automatiquement. On présente des bornes non asymptotiques concernant la convergence de cet algorithme, lorsque les points à classer forment un échantillon i. i. d. d'une loi à support compact dans un espace de Hilbert. Ces bornes sont déduites des bornes obtenues pour l'estimation d'un opérateur de Gram dans un espace de Hilbert. On termine par un aperçu de l'intérêt du clustering spectral dans le cadre de l'analyse d'images<br>This thesis focuses on obtaining generalization bounds for random samples in reproducing kernel Hilbert spaces. The approach consists in first obtaining non-asymptotic dimension-free bounds in finite-dimensional spaces using some PAC-Bayesian inequalities related to Gaussian perturbations and then in generalizing the results in a separable Hilbert space. We first investigate the question of estimating the Gram operator by a robust estimator from an i. i. d. sample and we present uniform bounds that hold under weak moment assumptions. These results allow us to qualify principal component analysis independently of the dimension of the ambient space and to propose stable versions of it. In the last part of the thesis we present a new algorithm for spectral clustering. It consists in replacing the projection on the eigenvectors associated with the largest eigenvalues of the Laplacian matrix by a power of the normalized Laplacian. This iteration, justified by the analysis of clustering in terms of Markov chains, performs a smooth truncation. We prove nonasymptotic bounds for the convergence of our spectral clustering algorithm applied to a random sample of points in a Hilbert space that are deduced from the bounds for the Gram operator in a Hilbert space. Experiments are done in the context of image analysis
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Sorensen, Julian Karl. "White noise analysis and stochastic evolution equations." Title page, contents and abstract only, 2001. http://web4.library.adelaide.edu.au/theses/09PH/09phs713.pdf.
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Agora, Elona. "Boundedness of the Hilbert Transform on Weighted Lorentz Spaces." Doctoral thesis, Universitat de Barcelona, 2012. http://hdl.handle.net/10803/108930.
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The main goal of this thesis is to characterize the weak-type (resp. strong-type) boundedness of the Hilbert transform H on weighted Lorentz spaces Λpu(w). The characterization is given in terms of some geometric conditions on the weights u and w and the weak-type (resp. strong-type) boundedness of the Hardy-Littlewood maximal operator on the same spaces. Our results extend and unify simultaneously the theory of the boundedness of H on weighted Lebesgue spaces Lp(u) and Muckenhoupt weights Ap, and the theory on classical Lorentz spaces Λp(w) and Ariño-Muckenhoupt weights Bp.<br>Títol: Acotaciò de l'operador de Hilbert sobre espais de Lorentz amb pesos
Resum: L'objectiu principal d'aquesta tesi es caracteritzar l'acotació de l'operador de Hilbert sobre els espais de Lorentz amb pesos Λpu(w). També estudiem la versió dèbil.
La caracterització es dona en terminis de condicions geomètriques sobre els pesos u i w, i l'acotació de l'operador maximal de Hardy-Littlewood sobre els mateixos espais.
Els nostres resultats unifiquen dues teories conegudes i aparentment no relacionades entre elles, que tracten l'acotació de l'operador de Hilbert sobre els espais de Lebegue amb pesos Lp(u) per una banda i els espais de Lorentz clàssics Λp(w) per altre banda.
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ru, neretin@main mccme rssi. "Groups of Vassalomorphisms and Hilbert Spaces Associated with Trees." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi1047.ps.
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Struble, Dale William. "Wavelets on manifolds and multiscale reproducing kernel Hilbert spaces." Related electronic resource:, 2007. http://proquest.umi.com/pqdweb?did=1407687581&sid=1&Fmt=2&clientId=3739&RQT=309&VName=PQD.
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Saraivanov, Michael S. "Quantum Circuit Synthesis using Group Decomposition and Hilbert Spaces." Thesis, Portland State University, 2013. http://pqdtopen.proquest.com/#viewpdf?dispub=1542568.
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<p> The exponential nature of Moore's law has inadvertently created huge data storage complexes that are scattered around the world. Data elements are continuously being searched, processed, erased, combined and transferred to other storage units without much regard to power consumption. The need for faster searches and power efficient data processing is becoming a fundamental requirement. Quantum computing may offer an elegant solution to speed and power through the utilization of the natural laws of quantum mechanics. Therefore, minimal cost quantum circuit implementation methodologies are greatly desired. </p><p> This thesis explores the decomposition of group functions and the Walsh spectrum for implementing quantum canonical cascades with minimal cost. Three different methodologies, using group decomposition, are presented and generalized to take advantage of different quantum computing hardware, such as ion traps and quantum dots. Quantum square root of swap gates and fixed angle rotation gates comprise the first two methodologies. The third and final methodology provides further quantum cost reduction by more efficiently utilizing Hilbert spaces through variable angle rotation gates. The thesis then extends the methodology to realize a robust quantum circuit synthesis tool for single and multi-output quantum logic functions.</p>
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Linder, Kevin A. (Kevin Andrew). "Spectral multiplicity theory in nonseparable Hilbert spaces : a survey." Thesis, McGill University, 1991. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=60478.
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Spectral multiplicity theory solves the problem of unitary equivalence of normal operate on a Hilbert space ${ cal H}$ by associating with each normal operator N a multiplicity function, such that two operators are unitarily equivalent if and only if their multiplicity functions are equal. This problem was first solved in the classical case in which ${ cal H}$ is separable by Hellinger in 1907, and in the general case in which ${ cal H}$ is nonseparable by Wecken in 1939. This thesis develops the later versions of multiplicity theory in the nonseparable case given by Halmos and Brown, and gives the simplification of Brown's version to the classical theory. Then the versions of Halmos and Brown are shown directly to be equivalent. Also, the multiplicity function of Brown is expressed in terms of the multiplicity function of Halmos.
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McKain, David. "Transference and the Hilbert transform on Banach function spaces." Thesis, University of Edinburgh, 2000. http://hdl.handle.net/1842/12628.
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The thesis begins with a summary of the classical theory of Banach function spaces, including the notions of saturation and associate norms along with the various well-known ideas of completeness. We then go on to establish some rather more "practical" results. In particular we look at the problem of establishing when certain subspaces, such as the simple or continuous functions, are dense in a Banach function space <i>L<sub>p</sub>. </i>We shall see that our intention fails us slightly when considering continuous functions, requiring us to approach that idea of saturation from a slightly different angle. This interplay between topology, measure and norm is studied in more depth when we look at function norms over locally compact abelian groups, and results will be illuminated by reviewing well-known functions spaces such as Lorentz spaces and weighted <i>L<sup>p</sup></i> spaces. The chapter finishes with the idea of vector-valued function spaces. In the second chapter we motivate and develop the idea of mixed (or iterated) norms, as introduced for <i>L<sup>p</sup></i> spaces by Benedek and Panzone, before going on to identify dense subspaces and some other elementary results. We shall see that there are certain interesting measurability problems to address here which are not evident when considering <i>L<sup>p</sup></i> spaces. One rather technical highlight of this measure theory will be to make rigorous the canonical identification between most mixed norm spaces and vector-valued Banach function spaces. Motivated by a trivial application of Fubini's theorem which allows us to interchange two <i>L<sup>p</sup> </i>norms, i.e. ||||<i>f(x, y)</i>||<i><sub>Lp(dy)</sub></i>||<i><sub>Lp(dx)</sub></i> = ||||<i>f(x, y)</i>||<i><sub>Lp(dx)</sub></i>||<i><sub>Lp(dy)</sub></i>, we then consider when interchanging two general mixed norms is bounded. Although there are some positive results we shall see that this idea fails in many cases. In particular we shall show that two iterated Lorentz <i>L<sup>pq</sup></i> norms can be interchanged if and only if <i>p = q.</i> In chapter three we study how the classical transference theorem of Coifman and Weiss can be generalised from <i>L<sup>p</sup></i> spaces to arbitrary rearrangement invariant spaces.
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Quiggin, Peter Philip. "Generalisations of Pick's theorem to reproducing Kernel Hilbert spaces." Thesis, Lancaster University, 1994. http://eprints.lancs.ac.uk/61962/.
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Pick's theorem states that there exists a function in H1, which is bounded by 1 and takes given values at given points, if and only if a certain matrix is positive. H1 is the space of multipliers of H2 and this theorem has a natural generalisation when H1 is replaced by the space of multipliers of a general reproducing kernel Hilbert space H(K) (where K is the reproducing kernel). J. Agler showed that this generalised theorem is true when H(K) is a certain Sobolev space or the Dirichlet space. This thesis widens Agler's approach to cover reproducing kernel Hilbert spaces in general and derives sucient (and usable) conditions on the kernel K, for the generalised Pick's theorem to be true for H(K). These conditions are then used to prove Pick's theorem for certain weighted Hardy and Sobolev spaces and for a functional Hilbert space introduced by Saitoh. The reproducing kernel approach is then used to derived results for several related problems. These include the uniqueness of the optimal interpolating multiplier, the case of operator-valued functions and a proof of the Adamyan-Arov-Kren theorem.
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Marx, Gregory. "The Complete Pick Property and Reproducing Kernel Hilbert Spaces." Thesis, Virginia Tech, 2014. http://hdl.handle.net/10919/24783.
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We present two approaches towards a characterization of the complete Pick property. We first discuss the lurking isometry method used in a paper by J.A. Ball, T.T. Trent, and V. Vinnikov. They show that a nondegenerate, positive kernel has the complete Pick property if $1/k$ has one positive square. We also look at the one-point extension approach developed by P. Quiggin which leads to a sufficient and necessary condition for a positive kernel to have the complete Pick property. We conclude by connecting the two characterizations of the complete Pick property.<br>Master of Science
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Kaldas, Hany Kamel Halim. "Relativistic Gamow vectors : state vectors for unstable particles /." Full text (PDF) from UMI/Dissertation Abstracts International, 2000. http://wwwlib.umi.com/cr/utexas/fullcit?p3004300.
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Boulton, Lyonell. "Topics in the spectral theory of non adjoint operators." Thesis, King's College London (University of London), 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.272412.
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Hansmann, Marcel. "On the discrete spectrum of linear operators in Hilbert spaces." Clausthal-Zellerfeld Universitätsbibliothek Clausthal, 2010. http://d-nb.info/1001898664/34.
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Giménez, Febrer Pere Joan. "Matrix completion with prior information in reproducing kernel Hilbert spaces." Doctoral thesis, Universitat Politècnica de Catalunya, 2021. http://hdl.handle.net/10803/671718.
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In matrix completion, the objective is to recover an unknown matrix from a small subset of observed entries. Most successful methods for recovering the unknown entries are based on the assumption that the unknown full matrix has low rank. By having low rank, each of its entries are obtained as a function of a small number of coefficients which can be accurately estimated provided that there are enough available observations. Hence, in low-rank matrix completion the estimate is given by the matrix of minimum rank that fits the observed entries.
Besides low rankness, the unknown matrix might exhibit other structural properties which can be leveraged in the recovery process. In a smooth matrix, it can be expected that entries that are close in index distance will have similar values. Similarly, groups of rows or columns can be known to contain similarly valued entries according to certain relational structures. This relational information is conveyed through different means such as covariance matrices or graphs, with the inconvenient that these cannot be derived from the data matrix itself since it is incomplete. Hence, any knowledge on how the matrix entries are related among them must be derived from prior information.
This thesis deals with matrix completion with prior information, and presents an outlook that generalizes to many situations. In the first part, the columns of the unknown matrix are cast as graph signals with a graph known beforehand. In this, the adjacency matrix of the graph is used to calculate an initial point for a proximal gradient algorithm in order to reduce the iterations needed to converge to a solution. Then, under the assumption that the graph signals are smooth, the graph Laplacian is incorporated into the problem formulation with the aim to enforce smoothness on the solution. This results in an effective denoising of the observed matrix and reduced error, which is shown through theoretical analysis of the proximal gradient coupled with Laplacian regularization, and numerical tests.
The second part of the thesis introduces a framework to exploit prior information through reproducing kernel Hilbert spaces. Since a kernel measures similarity between two points in an input set, it enables the encoding of any prior information such as feature vectors, dictionaries or connectivity on a graph. By associating each column and row of the unknown matrix with an item in a set, and defining a pair of kernels measuring similarity between columns or rows, the missing entries can be extrapolated by means of the kernel functions. A method based on kernel regression is presented, with two additional variants aimed at reducing computational cost, and online implementation. These methods prove to be competitive with existing techniques, especially when the number of observations is very small.
Furthermore, mean-square error and generalization error analyses are carried out, shedding light on the factors impacting algorithm performance. For the generalization error analysis, the focus is on the transductive case, which measures the ability of an algorithm to transfer knowledge from a set of labelled inputs to an unlabelled set. Here, bounds are derived for the proposed and existing algorithms by means of the transductive Rademacher complexity, and numerical tests confirming the theoretical findings are presented.
Finally, the thesis explores the question of how to choose the observed entries of a matrix in order to minimize the recovery error of the full matrix. A passive sampling approach is presented, which entails that no labelled inputs are needed to design the sampling distribution; only the input set and kernel functions are required. The approach is based on building the best Nyström approximation to the kernel matrix by sampling the columns according to their leverage scores, a metric that arises naturally in the theoretical analysis to find an optimal sampling distribution.<br>A matrix completion, l'objectiu és recuperar una matriu a partir d'un subconjunt d'entrades observables. Els mètodes més eficaços es basen en la idea que la matriu desconeguda és de baix rang. Al ser de baix rang, les seves entrades són funció d'uns pocs coeficients que poden ser estimats sempre que hi hagi suficients observacions. Així, a matrix completion la solució s'obté com la matriu de mínim rang que millor s'ajusta a les entrades visibles. A més de baix rang, la matriu desconeguda pot tenir altres propietats estructurals que poden ser aprofitades en el procés de recuperació. En una matriu suau, pot esperar-se que les entrades en posicions pròximes tinguin valor similar. Igualment, grups de columnes o files poden saber-se similars. Aquesta informació relacional es proporciona a través de diversos mitjans com ara matrius de covariància o grafs, amb l'inconvenient que aquests no poden ser derivats a partir de la matriu de dades ja que està incompleta. Aquesta tesi tracta sobre matrix completion amb informació prèvia, i presenta metodologies que poden aplicar-se a diverses situacions. En la primera part, les columnes de la matriu desconeguda s'identifiquen com a senyals en un graf conegut prèviament. Llavors, la matriu d'adjacència del graf s'usa per calcular un punt inicial per a un algorisme de gradient pròxim amb la finalitat de reduir les iteracions necessàries per arribar a la solució. Després, suposant que els senyals són suaus, la matriu laplaciana del graf s'incorpora en la formulació del problema amb tal forçar suavitat en la solució. Això resulta en una reducció de soroll en la matriu observada i menor error, la qual cosa es demostra a través d'anàlisi teòrica i simulacions numèriques. La segona part de la tesi introdueix eines per a aprofitar informació prèvia mitjançant reproducing kernel Hilbert spaces. Atès que un kernel mesura la similitud entre dos punts en un espai, permet codificar qualsevol tipus d'informació tal com vectors de característiques, diccionaris o grafs. En associar cada columna i fila de la matriu desconeguda amb un element en un set, i definir un parell de kernels que mesuren similitud entre columnes o files, les entrades desconegudes poden ser extrapolades mitjançant les funcions de kernel. Es presenta un mètode basat en regressió amb kernels, amb dues variants addicionals que redueixen el cost computacional. Els mètodes proposats es mostren competitius amb tècniques existents, especialment quan el nombre d'observacions és molt baix. A més, es detalla una anàlisi de l'error quadràtic mitjà i l'error de generalització. Per a l'error de generalització, s'adopta el context transductiu, el qual mesura la capacitat d'un algorisme de transferir informació d'un set de mostres etiquetades a un set no etiquetat. Després, es deriven cotes d'error per als algorismes proposats i existents fent ús de la complexitat de Rademacher, i es presenten proves numèriques que confirmen els resultats teòrics. Finalment, la tesi explora la qüestió de com triar les entrades observables de la matriu per a minimitzar l'error de recuperació de la matriu completa. Una estratègia de mostrejat passiva és proposada, la qual implica que no és necessari conèixer cap etiqueta per a dissenyar la distribució de mostreig. Només les funcions de kernel són necessàries. El mètode es basa en construir la millor aproximació de Nyström a la matriu de kernel mostrejant les columnes segons la seva leverage score, una mètrica que apareix de manera natural durant l'anàlisi teòric.
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Sinha, Suddhasattwa. "Coherent control of dipolar coupled spins in large Hilbert spaces." Thesis, Massachusetts Institute of Technology, 2006. http://hdl.handle.net/1721.1/41278.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Nuclear Science and Engineering, 2006.<br>Includes bibliographical references (p. 101-106).<br>Controlling the dynamics of a dipolar-coupled spin system is critical to the development of solid-state spin-based quantum information processors. Such control remains challenging, as every spin is coupled to a large number of surrounding spins. In this thesis, we primarily focus on developing coherent control techniques for such large spin systems. We start by experimentally simulating spin squeezing using a liquid-state NMR quantum information processor. We demonstrate that the precision of quantum control obtained using strongly modulating pulses was sufficient to reproduce the theoretically expected behavior of the spin observables and the associated entanglement measures among the underlying qubits. We then investigate coherent control in a more complex solid-state spin system consisting of an ensemble of spin pairs. Using pulse amplitude modulation techniques, we decouple the weaker interactions between different pairs and extend the coherence lifetimes within the two-spin system. This is achieved without decoupling the stronger interaction between the two spins within a pair. We thus demonstrated that it is possible to restrict the evolution of a dipolar coupled spin network to a much smaller subspace of the system Hilbert space which allows us to significantly extend the phase coherence times for selected states. Finally, we demonstrate the sensitivity of highly correlated multiple-quantum states to the presence of rare spin defects in a solid-state spin system.<br>(cont.) We design two multiple-pulse control sequences - one that suspends all spin interactions in the system including that of the defect spins, while the other selectively allows the defect spins to interact only with the abundant spins. By measuring the effective relaxation time of the rare spins, we demonstrate the efficiency of the two control sequences. Furthermore we observe that for small spin cluster sizes, the sensitivity of the highly correlated spin states to the spin defects depends on the coherence order of these correlated spin states. But beyond a certain cluster size, one observes a saturation effect as the higher coherence orders are no longer increasingly sensitive to the defect spin dynamics.<br>by Suddhasattwa Sinha.<br>Ph.D.
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Thompson, Kinney. "Frames for Hilbert spaces and an application to signal processing." VCU Scholars Compass, 2012. http://scholarscompass.vcu.edu/etd/2735.
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The goal of this paper will be to study how frame theory is applied within the field of signal processing. A frame is a redundant (i.e. not linearly independent) coordinate system for a vector space that satisfies a certain Parseval-type norm inequality. Frames provide a means for transmitting data and, when a certain about of loss is anticipated, their redundancy allows for better signal reconstruction. We will start with the basics of frame theory, give examples of frames and an application that illustrates how this redundancy can be exploited to achieve better signal reconstruction. We also include an introduction to the theory of frames in infinite dimensional Hilbert spaces as well as an interesting example.
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Zandler, Andersson Nils. "Boundedness of a Class of Hilbert Operators on Modulation Spaces." Thesis, Linnéuniversitetet, Institutionen för matematik (MA), 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:lnu:diva-84932.
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In this work we take interest in frames and modulation spaces. On the basis of their properties, we show how frame expansions can be used to prove the boundedness of a particular class of Hilbert operators on modulation spaces taking advantage of the special category of piece-wise polynomial functions known as B-splines.
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Khalil, Asma Mohammed. "Structure of scalar-type operators on Lp spaces and well-bounded operators on Hilbert spaces." Thesis, University of Edinburgh, 2002. http://hdl.handle.net/1842/10983.
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It is known that every scalar-type spectral operator on a Hilbert space <i>H</i> is similar to a multiplication operator on some <i>L</i><sup>2</sup><i> </i>space. The purpose of the main theorem in Chapter 2 of this thesis is to show that every scalar-type spectral operator on an <i>L</i><sup>1</sup> space whose spectral measure has finite multiplicity is similar to a multiplication operator on the same <i>L</i><sup>1</sup> space. Then we prove a similar result for scalar-type spectral operators on <i>L<sup>p</sup> </i>(Ω, S<sub>Ω</sub>, <i>m</i>), <i>p </i>≠<i> </i> 2, 1 < <i>p</i> < ∞, with spectral measure <i>E</i>(<sup>.</sup>) of finite uniform multiplicity provided an extra condition is satisfied. Also, we give conditions that make a scalar-type spectral operator on <i>L</i><sup>2</sup>(Ω, S<sub>Ω</sub>, <i>m</i>) similar to a multiplication operator on the same <i>L</i><sup>2</sup>(Ω, S<sub>Ω</sub>, <i>m</i>). In 1954, Dunford proved that a bounded operator <i>T</i> on a Banach space <i>X</i> is spectral if and only if it has the canonical decomposition <i>T = S</i> +<i>Q</i>, where <i>S</i> is a scalar-type operator and <i>Q</i> is a quasinilpotent operator which commutes with <i>S</i>. In Chapter 3, we prove that any well-bounded operator <i>T</i> on a Hilbert space <i>H</i> has the form <i>T = A + Q</i>, where <i>A</i> is a self-adjoint operator and <i>Q</i> is a quasinilpotent operator such that <i>AQ - QA</i> is quasinilpotent. Then we prove that a trigonometrically well-bounded operator <i>T </i>on <i>H</i> can be decomposed as <i>T = U</i>(<i>Q + I</i>) where <i>U</i> is a unitary operator and <i>Q</i> is quasinilpotent such that <i>UQ = QU </i>is also quasinilpotent. In Chapter 4 we prove that an AC-operator with discrete spectrum on <i>H</i> can be decomposed as a sum of a normal operator <i>N</i> and a quasinilpotent <i>Q</i> such that <i>NQ - QN </i>is quasinilpotent. However, the converse of each of the last three theorems is not true in general. In the final chapter we introduce a new class of operators on <i>L</i><sup>2</sup>([<i>a,b</i>]) which is larger than the class of well-bounded operators on <i>L</i><sup>2</sup>([<i>a,b</i>]) and we call them operators with an <i>AC</i><sub>2</sub>-functional calculus. Then we give an example of an operator with an <i>AC</i><sub>2</sub>-functional calculus on <i>L</i><sup>2</sup>([0,1]) which can be decomposed as a sum of a self-adjoint operator and a quasinilpotent. We also discuss the possibility of decomposing every operator <i>T</i> with an <i>AC</i><sub>2</sub>-functional calculus on <i>L</i><sup>2</sup>([<i>a,b</i>]) into the sum of a self-adjoint operator <i>A</i> and a quasinilpotent operator <i>Q</i> such that <i>AQ - QA </i>is quasinilpotent.
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Becker, Tanja. "Moduli spaces of (G,h)-constellations." Nantes, 2011. http://www.theses.fr/2011NANT2073.
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Nous construisons l'espace de modules M(X) des (G; h)constellations stables sur X pour un groupe réductif G qui agit sur un schéma ane X sur C et pour une fonction de Hilbert h: IrrG ! N0. Cet espace de modules est une généralisation commune du schéma de Hilbert invariant d'après Alexeev et Brion [AB05] et de l'espace de modules des Gconstellations stables pour un groupe ni G introduit par Craw et Ishii [CI04]. Notre construction d'un morphisme M(X) ! X//G fait de cet espace de modules un candidat pour une résolution des singularités du quotient X//G. De plus, nous déterminons le schéma de Hilbert invariant de la bre en zéro de l'application moment d'une action de Sl2 sur (C2)6. C'est un des premiers exemples d'un schéma de Hilbert invariant avec multiplicités. Ceci nous amène à décrire une façon générale de procéder pour eectuer de tels calculs. En outre, nous démontrons que notre schéma de Hilbert invariant est lisse et connexe : Cet exemple est donc une résolution des singularités de la réduction symplectique de l'action<br>Given a reductive group G acting on an a#ne scheme X over C and a Hilbert function h: IrrG ! N0, we construct the moduli space M#(X) of ##stable (G; h)#constellations on X, which is a common generalisation of the invariant Hilbert scheme after Alexeev and Brion [AB05] and the moduli space of ##stable G#constellations for #nite groups G introduced by Craw and Ishii [CI04]. Our construction of a morphism M#(X) ! X//G makes this moduli space a candidate for a resolution of singularities of the quotient X//G. Furthermore, we determine the invariant Hilbert scheme of the zero #bre of the moment map of an action of Sl2 on (C2)#6 as one of the #rst examples of invariant Hilbert schemes with multiplicities. While doing this, we present a general procedure for the realisation of such calculations. We also consider questions of smoothness and connectedness and thereby show that our Hilbert scheme gives a resolution of singularities of the symplectic reduction of the action
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Van, Tuyl Adam Leonard. "Sets of points in multi-projective spaces and their Hilbert function." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2001. http://www.collectionscanada.ca/obj/s4/f2/dsk3/ftp05/NQ63465.pdf.
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Wickramasekara, Sujeewa. "Differentiable representations of finite dimensional lie groups in rigged Hilbert spaces /." Digital version accessible at:, 1999. http://wwwlib.umi.com/cr/utexas/main.
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Wickramasekara, Sujeewa, and sujeewa@physics utexas edu. "Symmetry Representations in the Rigged Hilbert Space Formulation of." ESI preprints, 2001. ftp://ftp.esi.ac.at/pub/Preprints/esi993.ps.
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Allen, Cristian Gerardo. "A Classification of the Homogeneity of Countable Products of Subsets of Real Numbers." Thesis, University of North Texas, 2017. https://digital.library.unt.edu/ark:/67531/metadc1011753/.
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Spaces such as the closed interval [0, 1] do not have the property of being homogeneous, strongly locally homogeneous (SLH) or countable dense homogeneous (CDH), but the Hilbert cube has all three properties. We investigate subsets X of real numbers to determine when their countable product is homogeneous, SLH, or CDH. We give necessary and sufficient conditions for the product to be homogeneous. We also prove that the product is SLH if and only if X is zero-dimensional or an interval. And finally we show that for a Borel subset X of real numbers the product is CDH iff X is a G-delta zero-dimensional set or an interval.
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Hofmann, B., and O. Scherzer. "Local Ill-Posedness and Source Conditions of Operator Equations in Hilbert Spaces." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199800957.
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The characterization of the local ill-posedness and the local degree of nonlinearity are of particular importance for the stable solution of nonlinear ill-posed problems. We present assertions concerning the interdependence between the ill-posedness of the nonlinear problem and its linearization. Moreover, we show that the concept of the degree of nonlinearity com bined with source conditions can be used to characterize the local ill-posedness and to derive a posteriori estimates for nonlinear ill-posed problems. A posteriori estimates are widely used in finite element and multigrid methods for the solution of nonlinear partial differential equations, but these techniques are in general not applicable to inverse an ill-posed problems. Additionally we show for the well-known Landweber method and the iteratively regularized Gauss-Newton method that they satisfy a posteriori estimates under source conditions; this can be used to prove convergence rates results.
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Kotsakis, Christophoros. "Multiresolution aspects of linear approximation methods in Hilbert spaces using gridded data." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk1/tape4/PQDD_0016/NQ54794.pdf.
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Hansmann, Marcel [Verfasser]. "On the discrete spectrum of linear operators in Hilbert spaces / Marcel Hansmann." Clausthal-Zellerfeld : Universitätsbibliothek Clausthal, 2010. http://d-nb.info/1001898664/34.
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Maruhn, Jan Hendrik. "An augmented Lagrangian algorithm for optimization with equality constraints in Hilbert spaces." Thesis, Virginia Tech, 2001. http://hdl.handle.net/10919/32098.
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Since augmented Lagrangian methods were introduced by Powell and Hestenes, this class of methods has been investigated very intensively. While the finite dimensional case has been treated in a satisfactory manner, the infinite dimensional case is studied much less.
The general approach to solve an infinite dimensional optimization problem subject to equality constraints is as follows: First one proves convergence for a basic algorithm in the Hilbert space setting. Then one discretizes the given spaces and operators in order to make numerical computations possible. Finally, one constructs a discretized version of the infinite dimensional method and tries to transfer the convergence results to the finite dimensional version of the basic algorithm.
In this thesis we discuss a globally convergent augmented Lagrangian algorithm and discretize it in terms of functional analytic restriction operators. Given this setting, we prove global convergence of the discretized version of this algorithm to a stationary point of the infinite dimensional optimization problem. The proposed algorithm includes an explicit rule of how to update the discretization level and the penalty parameter from one iteration to the next one - questions that had been unanswered so far. In particular the latter update rule guarantees that the penalty parameters stay bounded away from zero which prevents the Hessian of the discretized augmented Lagrangian functional from becoming more and more ill conditioned.<br>Master of Science
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Srithharan, T. "Theory and applications of Hilbert's and Thompson's metrics to positive operators in ordered spaces." Thesis, University of Sussex, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.262302.
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Chui, Nelson Loong Chik. "Subspace methods and informative experiments for system identification." Thesis, University of Cambridge, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.298794.
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Santiago, Landerson Bezerra. "O nÃcleo do calor em uma variedade riemanniana." Universidade Federal do CearÃ, 2011. http://www.teses.ufc.br/tde_busca/arquivo.php?codArquivo=5674.
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Em uma variedade riemanniana conexa e compacta introduziremos o conceito de espectro do operador laplaciano.
Utilizando a existÃncia e a unicidade do nÃcleo do calor em uma variedade riemanniana,provaremos o teorema de decomposiÃÃo de Hodge. Este teorema afirma que o espaÃo de Hilbert L2(M, g) se decompÃe em uma soma direta de subespaÃos de dimensÃo finita, onde cada subespaÃo à o auto-espaÃo associado a um autovalor do laplaciano. AlÃm disso, os autovalores formam uma sequÃncia nÃo-negativa
que acumula somente no infinito.
Em seguida iniciaremos a construÃÃo do nÃcleo do calor e, por fim, mostraremos que se duas variedades riemannianas sÃo isospectrais entÃo elas possuem o mesmo volume.<br>In a connected and compact Riemannian Manifold we will introduce the concept of spectre of Laplace operator. Using the existence and unicity of the heat kernel
in Riemannian manifold we proof the Hodge composition theorem. This theorem states that the Hilbert space L2(M, g) decompose in direct sum of subspaces with finite dimesion, where each subspace is the eigen-space relative of a eigenvalue of the laplacian. Furthermore, the eigenvalues form a nonnegative sequence the
accumulate only in the infinity.
After that we begin the construction of the heat kernel and, finally, we show that two isospetral Riemannian manifolds have the same volume.
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Hofmann, B. "On Ill-Posedness and Local Ill-Posedness of Operator Equations in Hilbert Spaces." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199801185.
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In this paper, we study ill-posedness concepts of nonlinear and linear inverse problems
in a Hilbert space setting. We define local ill-posedness of a nonlinear operator
equation $F(x) = y_0$ in a solution point $x_0$ and the interplay between the nonlinear
problem and its linearization using the Frechet derivative $F\acent(x_0)$ . To find an
appropriate ill-posedness concept for the linarized equation we define intrinsic
ill-posedness for linear operator equations $Ax = y$ and compare this approach with
the ill-posedness definitions due to Hadamard and Nashed.
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Montgomery, Jason W. "Condition-dependent Hilbert Spaces for Steepest Descent and Application to the Tricomi Equation." Thesis, University of North Texas, 2014. https://digital.library.unt.edu/ark:/67531/metadc699977/.
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A steepest descent method is constructed for the general setting of a linear differential equation paired with uniqueness-inducing conditions which might yield a generally overdetermined system. The method differs from traditional steepest descent methods by considering the conditions when defining the corresponding Sobolev space. The descent method converges to the unique solution to the differential equation so that change in condition values is minimal. The system has a solution if and only if the first iteration of steepest descent satisfies the system. The finite analogue of the descent method is applied to example problems involving finite difference equations. The well-posed problems include a singular ordinary differential equation and Laplace’s equation, each paired with respective Dirichlet-type conditions. The overdetermined problems include a first-order nonsingular ordinary differential equation with Dirichlet-type conditions and the wave equation with both Dirichlet and Neumann conditions. The method is applied in an investigation of the Tricomi equation, a long-studied equation which acts as a prototype of mixed partial differential equations and has application in transonic flow. The Tricomi equation has been studied for at least ninety years, yet necessary and sufficient conditions for existence and uniqueness of solutions on an arbitrary mixed domain remain unknown. The domains of interest are rectangular mixed domains. A new type of conditions is introduced. Ladder conditions take the uncommon approach of specifying information on the interior of a mixed domain. Specifically, function values are specified on the parabolic portion of a mixed domain. The remaining conditions are specified on the boundary. A conjecture is posed and states that ladder conditions are necessary and sufficient for existence and uniqueness of a solution to the Tricomi equation. Numerical experiments, produced by application of the descent method, provide strong evidence in support of the conjecture. Ladder conditions allow for a continuous deformation from Dirichlet conditions to initial-boundary value conditions. Such a deformation is applied to a class of Tricomi-type equations which transition from degenerate elliptic to degenerate hyperbolic. A conjecture is posed and states that each problem is uniquely solvable and the solutions vary continuously as the differential equation and corresponding conditions vary continuously. If the conjecture holds true, the result will provide a method of unifying elliptic Dirichlet problems and hyperbolic initial-boundary value problem. Numerical evidence in support of the conjecture is presented.
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Paiva, António R. C. "Reproducing kernel Hilbert spaces for point processes, with applications to neural activity analysis." [Gainesville, Fla.] : University of Florida, 2008. http://purl.fcla.edu/fcla/etd/UFE0022471.
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Bhujwalla, Yusuf. "Nonlinear System Identification with Kernels : Applications of Derivatives in Reproducing Kernel Hilbert Spaces." Thesis, Université de Lorraine, 2017. http://www.theses.fr/2017LORR0315/document.
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Cette thèse se concentrera exclusivement sur l’application de méthodes non paramétriques basées sur le noyau à des problèmes d’identification non-linéaires. Comme pour les autres méthodes non-linéaires, deux questions clés dans l’identification basée sur le noyau sont les questions de comment définir un modèle non-linéaire (sélection du noyau) et comment ajuster la complexité du modèle (régularisation). La contribution principale de cette thèse est la présentation et l’étude de deux critères d’optimisation (un existant dans la littérature et une nouvelle proposition) pour l’approximation structurale et l’accord de complexité dans l’identification de systèmes non-linéaires basés sur le noyau. Les deux méthodes sont basées sur l’idée d’intégrer des contraintes de complexité basées sur des caractéristiques dans le critère d’optimisation, en pénalisant les dérivées de fonctions. Essentiellement, de telles méthodes offrent à l’utilisateur une certaine souplesse dans la définition d’une fonction noyau et dans le choix du terme de régularisation, ce qui ouvre de nouvelles possibilités quant à la facon dont les modèles non-linéaires peuvent être estimés dans la pratique. Les deux méthodes ont des liens étroits avec d’autres méthodes de la littérature, qui seront examinées en détail dans les chapitres 2 et 3 et formeront la base des développements ultérieurs de la thèse. Alors que l’analogie sera faite avec des cadres parallèles, la discussion sera ancrée dans le cadre de Reproducing Kernel Hilbert Spaces (RKHS). L’utilisation des méthodes RKHS permettra d’analyser les méthodes présentées d’un point de vue à la fois théorique et pratique. De plus, les méthodes développées seront appliquées à plusieurs «études de cas» d’identification, comprenant à la fois des exemples de simulation et de données réelles, notamment : • Détection structurelle dans les systèmes statiques non-linéaires. • Contrôle de la fluidité dans les modèles LPV. • Ajustement de la complexité à l’aide de pénalités structurelles dans les systèmes NARX. • Modelisation de trafic internet par l’utilisation des méthodes à noyau<br>This thesis will focus exclusively on the application of kernel-based nonparametric methods to nonlinear identification problems. As for other nonlinear methods, two key questions in kernel-based identification are the questions of how to define a nonlinear model (kernel selection) and how to tune the complexity of the model (regularisation). The following chapter will discuss how these questions are usually dealt with in the literature. The principal contribution of this thesis is the presentation and investigation of two optimisation criteria (one existing in the literature and one novel proposition) for structural approximation and complexity tuning in kernel-based nonlinear system identification. Both methods are based on the idea of incorporating feature-based complexity constraints into the optimisation criterion, by penalising derivatives of functions. Essentially, such methods offer the user flexibility in the definition of a kernel function and the choice of regularisation term, which opens new possibilities with respect to how nonlinear models can be estimated in practice. Both methods bear strong links with other methods from the literature, which will be examined in detail in Chapters 2 and 3 and will form the basis of the subsequent developments of the thesis. Whilst analogy will be made with parallel frameworks, the discussion will be rooted in the framework of Reproducing Kernel Hilbert Spaces (RKHS). Using RKHS methods will allow analysis of the methods presented from both a theoretical and a practical point-of-view. Furthermore, the methods developed will be applied to several identification ‘case studies’, comprising of both simulation and real-data examples, notably: • Structural detection in static nonlinear systems. • Controlling smoothness in LPV models. • Complexity tuning using structural penalties in NARX systems. • Internet traffic modelling using kernel methods