Dissertations / Theses on the topic 'Sparse bound'
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Shabara, Yahia. "Establishing Large-Scale MIMO Communication: Coding for Channel Estimation." The Ohio State University, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=osu1618578732285999.
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Price, Eric (Eric C. ). "Algorithms and lower bounds for sparse recovery." Thesis, Massachusetts Institute of Technology, 2010. http://hdl.handle.net/1721.1/62668.
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Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2010.Cataloged from PDF version of thesis.
Includes bibliographical references (p. 69-71).
We consider the following k-sparse recovery problem: design a distribution of m x n matrix A, such that for any signal x, given Ax with high probability we can efficiently recover x satisfying IIx - x l, by Eric Price.
M.Eng.
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Do, Ba Khanh. "Algorithms and lower bounds in the streaming and sparse recovery models." Thesis, Massachusetts Institute of Technology, 2012. http://hdl.handle.net/1721.1/75629.
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Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2012.Cataloged from PDF version of thesis.
Includes bibliographical references (p. 52-56).
In the data stream computation model, input data is given to us sequentially (the data stream), and our goal is to compute or approximate some function or statistic on that data using a sublinear (in both the length of the stream and the size of the universe of items that can appear in the stream) amount of space; in particular, we can store neither the entire stream nor a counter for each possible item we might see. In the sparse recovery model (also known as compressed sensing), input data is a large but sparse vector x [epsilon] Rn, and our goal is to design an m x n matrix [Phi]D, where m << n, such that for any sufficiently sparse x we can efficiently recover a good approximation of x from [Phi]x. Although at first glance these two models may seem quite different, they are in fact intimately related. In the streaming model, most statistics of interest are order-invariant, meaning they care only about the frequency of each item in the stream and not their position. For these problems, the data in the stream can be viewed as an n-dimensional vector x, where xi is the number of occurrences of item i. Using this representation, one of the high-level tools that have proven most popular has been the linear sketch, where for some m x n matrix {Phi]we maintain {Phi]x (the sketch) for the partial vector x as we progress along the stream. The linearity of the mapping D allows us to efficiently do incremental updates on our sketch, and as in its use in sparse recovery, the linear sketch turns out to be surprisingly powerful. In this thesis, we try to answer some questions of interest in each model, illustrating both the power and the limitations of the linear sketch. In Chapter 2, we provide an efficient sketch for estimating the (planar) Earth-Mover Distance (EMD) between two multisets of points. The EMD between point sets A, B R2 of the same size is defined as the minimum cost of a perfect matching between them, with each edge contributing a cost equal to its (Euclidean) length. As immediate consequences, we give an improved algorithm for estimating EMD between point sets given over a stream, and an improved algorithm for the approximate nearest neighbor problem under EMD. In Chapter 3, we prove tight lower bounds for sparse recovery in the number of rows in the matrix [Phi] (i.e., the number of measurements) in order to achieve any of the three most studied recovery guarantees. Specifically, consider a matrix [Phi] and an algorithm R such that for any signal x, R can recover an approximation & from [Phi] satisfying ... where (1) p= q= 1 and C= O(1), (2) p= q= 2 and C = O(1), or (3) p =2, q = 1 and C = O(k-1/ 2 ). We show that any such [Phi] I must have at least [Omega](k log(n/k)) rows. This is known to be optimal in cases (1) and (2), and near optimal for (3). In Chapter 4, we propose a variant of sparse recovery that incorporates some additional knowledge about the signal that allows the above lower bound to be broken. In particular, we consider the scenario where, after measurements are taken, we are given a set S of size s < n (s is known beforehand) that is supposed to contain most of the "large" coefficients of x. The goal is then to recover i satisfying ... We refer to this formulation as the sparse recovery with partial support knowledge problem (SRPSK). We focus on the guarantees where p = q = 1 or 2 and C = 1 + e, for which we provide lower bounds as well as a method of converting algorithms for "standard" sparse recovery into ones for SRPSK. We also make use of one of the reductions to give an optimal algorithm for SRPSK for the guarantee where p = q = 2.
by Khanh Do Ba.
Ph.D.
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Brennan, Matthew (Matthew Stewart). "Reducibility and computational lower bounds for problems with planted sparse structure." Thesis, Massachusetts Institute of Technology, 2018. http://hdl.handle.net/1721.1/118062.
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Thesis: S.M. in Computer Science and Engineering, Massachusetts Institute of Technology, Department of Electrical Engineering and Computer Science, 2018.Cataloged from PDF version of thesis.
Includes bibliographical references (pages 145-155).
Recently, research in unsupervised learning has gravitated towards exploring statistical-computational gaps induced by sparsity. A line of work initiated by Berthet and Rigollet has aimed to explain these gaps through reductions to conjecturally hard problems from complexity theory. However, the delicate nature of average-case reductions has limited the development of techniques and often led to weaker hardness results that only apply to algorithms that are robust to different noise distributions or that do not need to know the parameters of the problem. We introduce several new techniques to give a web of average-case reductions showing strong computational lower bounds based on the planted clique conjecture for planted independent set, planted dense subgraph, biclustering, sparse rank-1 submatrix, sparse PCA and the subgraph stochastic block model. Our results demonstrate that, despite the delicate nature of average-case reductions, using natural problems as intermediates can often be beneficial, as is the case in worst-case complexity. Our main technical contribution is to introduce a set of techniques for average-case reductions that: (1) maintain the level of signal in an instance of a problem; (2) alter its planted structure; and (3) map two initial high-dimensional distributions simultaneously to two target distributions approximately under total variation. We also give algorithms matching our lower bounds and identify the information-theoretic limits of the models we consider.
by Matthew Brennan.
S.M. in Computer Science and Engineering
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Gasiorowski, Pawel. "Individual and group dynamic behaviour patterns in bound spaces." Thesis, London Metropolitan University, 2017. http://repository.londonmet.ac.uk/1447/.
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The behaviour analysis of individual and group dynamics in closed spaces is a subject of extensive research in both academia and industry. However, despite recent technological advancements the problem of implementing the existing methods for visual behaviour data analysis in production systems remains difficult and the applications are available only in special cases in which the resourcing is not a problem. Most of the approaches concentrate on direct extraction and classification of the visual features from the video footage for recognising the dynamic behaviour directly from the source. The adoption of such an approach allows recognising directly the elementary actions of moving objects, which is a difficult task on its own. The major factor that impacts the performance of the methods for video analytics is the necessity to combine processing of enormous volume of video data with complex analysis of this data using and computationally resourcedemanding analytical algorithms. This is not feasible for many applications, which must work in real time. In this research, an alternative simulation-based approach for behaviour analysis has been adopted. It can potentially reduce the requirements for extracting information from real video footage for the purpose of the analysis of the dynamic behaviour. This can be achieved by combining only limited data extracted from the original video footage with a symbolic data about the events registered on the scene, which is generated by 3D simulation synchronized with the original footage. Additionally, through incorporating some physical laws and the logics of dynamic behaviour directly in the 3D model of the visual scene, this framework allows to capture the behavioural patterns using simple syntactic pattern recognition methods. The extensive experiments with the prototype implementation prove in a convincing manner that the 3D simulation generates sufficiently rich data to allow analysing the dynamic behaviour in real-time with sufficient adequacy without the need to use precise physical data, using only a limited data about the objects on the scene, their location and dynamic characteristics. This research can have a wide applicability in different areas where the video analytics is necessary, ranging from public safety and video surveillance to marketing research to computer games and animation. Its limitations are linked to the dependence on some preliminary processing of the video footage which is still less detailed and computationally demanding than the methods which use directly the video frames of the original footage.
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Seeger, Matthias. "Bayesian Gaussian process models : PAC-Bayesian generalisation error bounds and sparse approximations." Thesis, University of Edinburgh, 2003. http://hdl.handle.net/1842/321.
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Non-parametric models and techniques enjoy a growing popularity in the field of machine learning, and among these Bayesian inference for Gaussian process (GP) models has recently received significant attention. We feel that GP priors should be part of the standard toolbox for constructing models relevant to machine learning in the same way as parametric linear models are, and the results in this thesis help to remove some obstacles on the way towards this goal. In the first main chapter, we provide a distribution-free finite sample bound on the difference between generalisation and empirical (training) error for GP classification methods. While the general theorem (the PAC-Bayesian bound) is not new, we give a much simplified and somewhat generalised derivation and point out the underlying core technique (convex duality) explicitly. Furthermore, the application to GP models is novel (to our knowledge). A central feature of this bound is that its quality depends crucially on task knowledge being encoded faithfully in the model and prior distributions, so there is a mutual benefit between a sharp theoretical guarantee and empirically well-established statistical practices. Extensive simulations on real-world classification tasks indicate an impressive tightness of the bound, in spite of the fact that many previous bounds for related kernel machines fail to give non-trivial guarantees in this practically relevant regime. In the second main chapter, sparse approximations are developed to address the problem of the unfavourable scaling of most GP techniques with large training sets. Due to its high importance in practice, this problem has received a lot of attention recently. We demonstrate the tractability and usefulness of simple greedy forward selection with information-theoretic criteria previously used in active learning (or sequential design) and develop generic schemes for automatic model selection with many (hyper)parameters. We suggest two new generic schemes and evaluate some of their variants on large real-world classification and regression tasks. These schemes and their underlying principles (which are clearly stated and analysed) can be applied to obtain sparse approximations for a wide regime of GP models far beyond the special cases we studied here.
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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'imagesThis 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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Cherief-Abdellatif, Badr-Eddine. "Contributions to the theoretical study of variational inference and robustness." Electronic Thesis or Diss., Institut polytechnique de Paris, 2020. http://www.theses.fr/2020IPPAG001.
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Cette thèse de doctorat traite de l'inférence variationnelle et de la robustesse en statistique et en machine learning. Plus précisément, elle se concentre sur les propriétés statistiques des approximations variationnelles et sur la conception d'algorithmes efficaces pour les calculer de manière séquentielle, et étudie les estimateurs basés sur le Maximum Mean Discrepancy comme règles d'apprentissage qui sont robustes à la mauvaise spécification du modèle.Ces dernières années, l'inférence variationnelle a été largement étudiée du point de vue computationnel, cependant, la littérature n'a accordé que peu d'attention à ses propriétés théoriques jusqu'à très récemment. Dans cette thèse, nous étudions la consistence des approximations variationnelles dans divers modèles statistiques et les conditions qui assurent leur consistence. En particulier, nous abordons le cas des modèles de mélange et des réseaux de neurones profonds. Nous justifions également d'un point de vue théorique l'utilisation de la stratégie de maximisation de l'ELBO, un critère numérique qui est largement utilisé dans la communauté VB pour la sélection de modèle et dont l'efficacité a déjà été confirmée en pratique. En outre, l'inférence Bayésienne offre un cadre d'apprentissage en ligne attrayant pour analyser des données séquentielles, et offre des garanties de généralisation qui restent valables même en cas de mauvaise spécification des modèles et en présence d'adversaires. Malheureusement, l'inférence Bayésienne exacte est rarement tractable en pratique et des méthodes d'approximation sont généralement employées, mais ces méthodes préservent-elles les propriétés de généralisation de l'inférence Bayésienne ? Dans cette thèse, nous montrons que c'est effectivement le cas pour certains algorithmes d'inférence variationnelle (VI). Nous proposons de nouveaux algorithmes tempérés en ligne et nous en déduisons des bornes de généralisation. Notre résultat théorique repose sur la convexité de l'objectif variationnel, mais nous soutenons que notre résultat devrait être plus général et présentons des preuves empiriques à l'appui. Notre travail donne des justifications théoriques en faveur des algorithmes en ligne qui s'appuient sur des méthodes Bayésiennes approchées.Une autre question d'intérêt majeur en statistique qui est abordée dans cette thèse est la conception d'une procédure d'estimation universelle. Cette question est d'un intérêt majeur, notamment parce qu'elle conduit à des estimateurs robustes, un thème d'actualité en statistique et en machine learning. Nous abordons le problème de l'estimation universelle en utilisant un estimateur de minimisation de distance basé sur la Maximum Mean Discrepancy. Nous montrons que l'estimateur est robuste à la fois à la dépendance et à la présence de valeurs aberrantes dans le jeu de données. Nous mettons également en évidence les liens qui peuvent exister avec les estimateurs de minimisation de distance utilisant la distance L2. Enfin, nous présentons une étude théorique de l'algorithme de descente de gradient stochastique utilisé pour calculer l'estimateur, et nous étayons nos conclusions par des simulations numériques. Nous proposons également une version Bayésienne de notre estimateur, que nous étudions à la fois d'un point de vue théorique et d'un point de vue computationnelThis PhD thesis deals with variational inference and robustness. More precisely, it focuses on the statistical properties of variational approximations and the design of efficient algorithms for computing them in an online fashion, and investigates Maximum Mean Discrepancy based estimators as learning rules that are robust to model misspecification.In recent years, variational inference has been extensively studied from the computational viewpoint, but only little attention has been put in the literature towards theoretical properties of variational approximations until very recently. In this thesis, we investigate the consistency of variational approximations in various statistical models and the conditions that ensure the consistency of variational approximations. In particular, we tackle the special case of mixture models and deep neural networks. We also justify in theory the use of the ELBO maximization strategy, a model selection criterion that is widely used in the Variational Bayes community and is known to work well in practice.Moreover, Bayesian inference provides an attractive online-learning framework to analyze sequential data, and offers generalization guarantees which hold even under model mismatch and with adversaries. Unfortunately, exact Bayesian inference is rarely feasible in practice and approximation methods are usually employed, but do such methods preserve the generalization properties of Bayesian inference? In this thesis, we show that this is indeed the case for some variational inference algorithms. We propose new online, tempered variational algorithms and derive their generalization bounds. Our theoretical result relies on the convexity of the variational objective, but we argue that our result should hold more generally and present empirical evidence in support of this. Our work presents theoretical justifications in favor of online algorithms that rely on approximate Bayesian methods. Another point that is addressed in this thesis is the design of a universal estimation procedure. This question is of major interest, in particular because it leads to robust estimators, a very hot topic in statistics and machine learning. We tackle the problem of universal estimation using a minimum distance estimator based on the Maximum Mean Discrepancy. We show that the estimator is robust to both dependence and to the presence of outliers in the dataset. We also highlight the connections that may exist with minimum distance estimators using L2-distance. Finally, we provide a theoretical study of the stochastic gradient descent algorithm used to compute the estimator, and we support our findings with numerical simulations. We also propose a Bayesian version of our estimator, that we study from both a theoretical and a computational points of view
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Lashkaripour, Rahmatollah. "Lower bounds and norms of operators on Lorentz sequence spaces." Thesis, Lancaster University, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.364315.
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Rammea, Lisema. "Computations and bounds for surfaces in weighted projective four-spaces." Thesis, University of Bath, 2009. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.507235.
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Many researchers have constructed examples of non general type surfaces in weighted projective spaces in various dimensions. Most of these constructions so far have been concentrated on complete intersections, and in the past three decades there has been a lot of success in this direction. Nowadays we have seen use of computer algebra systems to handle examples that are too cumbersome to do by hand. All smooth projective surfaces can be embedded in P5, but only few of them in P4. The most amazing fact is that the numerical invariants of any smooth surface in P4 must satisfy the double point formula. A natural question is whether there are any non general type surfaces in four dimensional weighted projective space, P4(w), which are not complete intersections. We believe that the answer is \yes, but they are not abundant". This thesis shows the �rst part, and justi�es the second part. That is, this thesis has two distinctive parts. First we prove that families of non general type surfaces in weighted projective four{space, P4(w) are rare by showing that their corresponding covers in straight P4, which are usually general type surfaces, are rare. In the second part we construct explicit examples of these rare objects in P4 using a technique involving sheaf cohomology and the Beilinson monad. We concentrate on the case of weights w = (1; 1; 1; 1; 2) for our particular examples. We present three explicit examples, one of which is symmetric. The main computer algebra system used is Macaulay2, Version 1.1 developed by D. Grayson and M. Stillman.
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Nancarrow, Cindy. "Bound to the borders: Representing refugees in the Australian space." Thesis, Queensland University of Technology, 2014. https://eprints.qut.edu.au/72792/4/Cindy_Nancarrow_Thesis.pdf.
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This project consists of a novel and an exegesis that explore the use of fiction to counter negative hegemonic representations of refugees in Australia. The possibilities of using Australian spaces, including border spaces, to reveal tensions surrounding refugee belonging and to highlight the reconfiguration of border sites in the Australian imaginary, is a particular focus of this work.
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Renesse, Max-K. von. "Comparison properties of diffusion semigroups on spaces with lower curvature bounds." Bonn : Mathematisches Institut der Universität Bonn, 2003. http://catalog.hathitrust.org/api/volumes/oclc/52348149.html.
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Forster, Jürgen. "Some results concerning arrangements of half spaces and relative loss bounds." [S.l.] : [s.n.], 2002. http://deposit.ddb.de/cgi-bin/dokserv?idn=964520389.
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Samuel, Javed K. K. "Lower bounds for embedding the Earth Mover Distance metric into normed spaces." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/33346.
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Thesis (M. Eng.)--Massachusetts Institute of Technology, Dept. of Electrical Engineering and Computer Science, 2005.Includes bibliographical references (p. 71-73).
This thesis presents a lower bounds for embedding the Earth Mover Distance (EMID) metric into normed spaces. The EMID is a metric over two distributions where one is a mass of earth spread out in space and the other is a collection of holes in that same space. The EMD between these two distributions is defined as the least amount of work needed to fill the holes with earth. The EMD metric is used in a number of applications, for example in similarity searching and for image retrieval. We present a simple construction of point sets in the ENID metric space over two dimensions that cannot be embedded from the ED metric exactly into normed spaces, namely l1 and the square of l2. An embedding is a mapping f : X --> V with X a set of points in a metric space and ' Va set of points in some normed vector space. When the Manhattan distance is used as the underlying metric for the EMD, it can be shown that this example is isometric to K2,4 which has distortion equal to 1.25 when it is embedded into I and( 1.1180 when embedded into the square of 12. Other constructions of points sets in the EMID metric space over three and higher dimensisions are also discussed..
by Javed K.K. Samuel.
M.Eng.
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Sahovic, Arman. "Spectral bounds for infinite dimensional polydiagonal symmetric matrix operators on discrete spaces." Thesis, Imperial College London, 2013. http://hdl.handle.net/10044/1/18065.
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In this thesis, we prove a variety of discrete Agmon Kolmogorov inequalities and apply them to prove Lieb Thirring inequalities for discrete Schrodinger operators on ℓ[superscript 2](ℤ). We generalise these results in two ways: Firstly, to higher order difference operators, leading to spectral bounds for Tri-, Penta- and Polydiagonal Jacobi-type matrix operators. Secondly, to ℓ[superscript 2]-spaces on higher dimensional domains, specifically on ℓ[superscript 2](ℤ[superscript 2]), ℓ[superscript 2](ℤ[superscript 3]) and finally ℓ[superscript 2](ℤ[superscript d]). In the Introduction we discuss previous work on Landau Kolmogorov inequalities on a variety of Banach Spaces, Lieb Thirring inequalities in ℓ[superscript 2](ℝ[superscript d]), and the use of Jacobi Matrices in relation to the discrete Schrodinger Operator. We additionally give our main results with some introduction to the notation at hand. Chapters 2, 3 and 4 follow a similar structure. We first introduce the relevant difference operators and examine their properties. We then move on to prove the Agmon Kolmogorov and Generalised Sobolev inequalities over ℤ of order 1, 2 and σ respectively. Furthermore, we prove the Lieb Thirring inequality for the respective discrete Schrodinger-type operators, which we subsequently lift to arbitrary moments. Finally we apply this inequality to obtain spectral bounds for tri-, penta- and polydiagonal matrices. In Chapter 5, we prove a variety of Agmon Kolmogorov inequalities on ℓ[superscript 2](ℤ[superscript 2]) and ℓ[superscript 2](ℤ[superscript 3]). We use these intuitive ideas to obtain 2[superscript d-1] Agmon Kolmogorov inequalities on ℓ[superscript 2](ℤ[superscript d]). We continue from here in the same manner as before and prove the discrete Generalised Sobolev and Lieb Thirring inequalities for a variety of exponent combinations on ℓ[superscript 2](ℤ[superscript d]).
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Schneck, Arne. "Bounds for optimization of the reflection coefficient by constrained optimization in hardy spaces." Karlsruhe Univ.-Verl. Karlsruhe, 2009. http://d-nb.info/995244383/04.
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Minsker, Stanislav. "Non-asymptotic bounds for prediction problems and density estimation." Diss., Georgia Institute of Technology, 2012. http://hdl.handle.net/1853/44808.
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This dissertation investigates the learning scenarios where a high-dimensional parameter has to be estimated from a given sample of fixed size, often smaller than the dimension of the problem. The first part answers some open questions for the binary classification problem in the framework of active learning.
Given a random couple (X,Y) with unknown distribution P, the goal of binary classification is to predict a label Y based on the observation X. Prediction rule is constructed from a sequence of observations sampled from P. The concept of active learning can be informally characterized as follows: on every iteration, the algorithm is allowed to request a label Y for any instance X which it considers to be the most informative. The contribution of this work consists of two parts: first, we provide the minimax lower bounds for the performance of active learning methods. Second, we propose an active learning algorithm which attains nearly optimal rates over a broad class of underlying distributions and is adaptive with respect to the unknown parameters of the problem.
The second part of this thesis is related to sparse recovery in the framework of dictionary learning. Let (X,Y) be a random couple with unknown distribution P. Given a collection of functions H, the goal of dictionary learning is to construct a prediction rule for Y given by a linear combination of the elements of H. The problem is sparse if there exists a good prediction rule that depends on a small number of functions from H. We propose an estimator of the unknown optimal prediction rule based on penalized empirical risk minimization algorithm. We show that the proposed estimator is able to take advantage of the possible sparse structure of the problem by providing probabilistic bounds for its performance.
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Erkursun, Nazife. "Convergence Of Lotz-raebiger Nets On Banach Spaces." Phd thesis, METU, 2010. http://etd.lib.metu.edu.tr/upload/3/12612108/index.pdf.
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The concept of LR-nets was introduced and investigated firstly by H.P. Lotz in [27] and by F. Raebiger in [30]. Therefore we call such nets Lotz-Raebiger nets, shortly LR-nets. In this thesis
we treat two problems on asymptotic behavior of these operator nets.
First problem is to generalize well known theorems for Ces`aro averages of a single operator to LR-nets, namely to generalize the Eberlein and Sine theorems. The second problem is related
to the strong convergence of Markov LR-nets on L1-spaces. We prove that the existence of a lower-bound functions is necessary and sufficient for asymptotic stability of LR-nets of
Markov operators.
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Salvato, Maria. "On the solvability of linear PDEs in weighted Sobolev spaces." Doctoral thesis, Universita degli studi di Salerno, 2012. http://hdl.handle.net/10556/335.
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Tameling, Carla [Verfasser], Axel [Akademischer Betreuer] Munk, Axel [Gutachter] Munk, and Anja [Gutachter] Sturm. "Empirical Optimal Transport on Discrete Spaces: Limit Theorems, Distributional Bounds and Applications / Carla Tameling ; Gutachter: Axel Munk, Anja Sturm ; Betreuer: Axel Munk." Göttingen : Niedersächsische Staats- und Universitätsbibliothek Göttingen, 2019. http://d-nb.info/1201548608/34.
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Tamanini, Luca. "Analysis and Geometry of RCD spaces via the Schrödinger problem." Thesis, Paris 10, 2017. http://www.theses.fr/2017PA100082/document.
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Le but principal de ce manuscrit est celui de présenter une nouvelle méthode d'interpolation entre des probabilités inspirée du problème de Schrödinger, problème de minimisation entropique ayant des liens très forts avec le transport optimal. À l'aide de solutions au problème de Schrödinger, nous obtenons un schéma d'approximation robuste jusqu'au deuxième ordre et différent de Brenier-McCann qui permet d'établir la formule de dérivation du deuxième ordre le long des géodésiques Wasserstein dans le cadre de espaces RCD* de dimension finie. Cette formule était inconnue même dans le cadre des espaces d'Alexandrov et nous en donnerons quelques applications. La démonstration utilise un ensemble remarquable de nouvelles propriétés pour les solutions au problème de Schrödinger dynamique :- une borne uniforme des densités le long des interpolations entropiques ;- la lipschitzianité uniforme des potentiels de Schrödinger ;- un contrôle L2 uniforme des accélérations. Ces outils sont indispensables pour explorer les informations géométriques encodées par les interpolations entropiques. Les techniques utilisées peuvent aussi être employées pour montrer que la solution visqueuse de l'équation d'Hamilton-Jacobi peut être récupérée à travers une méthode de « vanishing viscosity », comme dans le cas lisse.Dans tout le manuscrit, plusieurs remarques sur l'interprétation physique du problème de Schrödinger seront mises en lumière. Cela pourra aider le lecteur à mieux comprendre les motivations probabilistes et physiques du problème, ainsi qu'à les connecter avec la nature analytique et géométrique de la dissertationMain aim of this manuscript is to present a new interpolation technique for probability measures, which is strongly inspired by the Schrödinger problem, an entropy minimization problem deeply related to optimal transport. By means of the solutions to the Schrödinger problem, we build an efficient approximation scheme, robust up to the second order and different from Brenier-McCann's classical one. Such scheme allows us to prove the second order differentiation formula along geodesics in finite-dimensional RCD* spaces. This formula is new even in the context of Alexandrov spaces and we provide some applications.The proof relies on new, even in the smooth setting, estimates concerning entropic interpolations which we believe are interesting on their own. In particular we obtain:- equiboundedness of the densities along the entropic interpolations,- equi-Lipschitz continuity of the Schrödinger potentials,- a uniform weighted L2 control of the Hessian of such potentials. These tools are very useful in the investigation of the geometric information encoded in entropic interpolations. The techniques used in this work can be also used to show that the viscous solution of the Hamilton-Jacobi equation can be obtained via a vanishing viscosity method, in accordance with the smooth case. Throughout the whole manuscript, several remarks on the physical interpretation of the Schrödinger problem are pointed out. Hopefully, this will allow the reader to better understand the physical and probabilistic motivations of the problem as well as to connect them with the analytical and geometric nature of the dissertation
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Rebs, Christian [Verfasser], Albrecht [Akademischer Betreuer] Böttcher, Steffen [Gutachter] Roch, and Stefan [Gutachter] Kunis. "Asymptotic bounds and values for the norm of the Laplace operator and other partial differential operators on spaces of polynomials / Christian Rebs ; Gutachter: Steffen Roch, Stefan Kunis ; Betreuer: Albrecht Böttcher." Chemnitz : Technische Universität Chemnitz, 2020. http://d-nb.info/1223416488/34.
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Sridhar, Vijay Sridhar. "On the effect of asymmetry and dimension on computational geometric problems." The Ohio State University, 2018. http://rave.ohiolink.edu/etdc/view?acc_num=osu1531362300593304.
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Pecatte, Timothée. "Bornes inférieures et algorithmes de reconstruction pour des sommes de puissances affines." Thesis, Lyon, 2018. http://www.theses.fr/2018LYSEN029/document.
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Le cadre général de cette thèse est l'étude des polynômes comme objets de modèles de calcul. Cette approche permet de définir de manière précise la complexité d'évaluation d'un polynôme, puis de classifier des familles de polynômes en fonction de leur difficulté dans ce modèle. Dans cette thèse, nous nous intéressons en particulier au modèle AffPow des sommes de puissance de forme linéaire, i.e. les polynômes qui s'écrivent $f = \sum_{i = 1}^s \alpha_i \ell_i^{e_i}$, avec $\deg \ell_i = 1$. Ce modèle semble assez naturel car il étend à la fois le modèle de Waring $f = \sum \alpha_i \ell_i^d$ et le modèle du décalage creux $f = \sum \alpha_i \ell^{e_i}$, mais peu de résultats sont connus pour cette généralisation.Nous avons pu prouver des résultats structurels pour la version univarié de ce modèle, qui nous ont ensuite permis d'obtenir des bornes inférieures et des algorithmes de reconstruction, qui répondent au problème suivant : étant donné $f = \sum \alpha_i (x-a_i)^{e_i}$ par la liste de ses coefficients, retrouver les $\alpha_i, a_i, e_i$ qui apparaissent dans la décomposition optimale de $f$.Nous avons aussi étudié plus en détails la version multivarié du modèle, qui avait été laissé ouverte par nos précédents algorithmes de reconstruction, et avons obtenu plusieurs résultats lorsque le nombre de termes dans une expression optimale est relativement petit devant le nombre de variables ou devant le degré du polynômeThe general framework of this thesis is the study of polynomials as objects of models of computation. This approach allows to define precisely the evaluation complexity of a polynomial, and then to classify families of polynomials depending on their complexity. In this thesis, we focus on the study of the model of sums of affine powers, that is polynomials that can be written as $f = \sum_{i = 1}^s \alpha_i \ell_i^{e_i}$, with $\deg \ell_i = 1$.This model is quite natural, as it extends both the Waring model $f = \sum \alpha_i \ell_i^d$ , and the sparsest shift model $f = \sum \alpha_i \ell^{e_i}$, but it is still not well known.In this work, we obtained structural results for the univariate variant of this model, which allow us to obtain lower bounds and reconstruction algorithms, that solve the following problem : given $f = \sum \alpha_i (x-a_i)^{e_i}$ as a list of its coefficient, find the values of the $\alpha_i$’s, $e_i$’s and $a_i$’s in the optimal decomposition of $f$.We also studied the multivariate case and obtained several reconstruction algorithms that work whenever the number of terms in the optimal expression is small in terms of the number of variable or the degree of the polynomial
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Kamari, Halaleh. "Qualité prédictive des méta-modèles construits sur des espaces de Hilbert à noyau auto-reproduisant et analyse de sensibilité des modèles complexes." Thesis, université Paris-Saclay, 2020. http://www.theses.fr/2020UPASE010.
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Ce travail porte sur le problème de l'estimation d'un méta-modèle d'un modèle complexe, noté m. Le modèle m dépend de d variables d'entrées X1,...,Xd qui sont indépendantes et ont une loi connue. Le méta-modèle, noté f∗, approche la décomposition de Hoeffding de m et permet d'estimer ses indices de Sobol. Il appartient à un espace de Hilbert à noyau auto-reproduisant (RKHS), noté H, qui est construit comme une somme directe d'espaces de Hilbert (Durrande et al. (2013)). L'estimateur du méta-modèle, noté f^, est calculé en minimisant un critère des moindres carrés pénalisé par la somme de la norme de Hilbert et de la norme empirique L2 (Huet and Taupin (2017)). Cette procédure, appelée RKHS ridge groupe sparse, permet à la fois de sélectionner et d'estimer les termes de la décomposition de Hoeffding, et donc de sélectionner les indices de Sobol non-nuls et de les estimer. Il permet d'estimer les indices de Sobol même d'ordre élevé, un point connu pour être difficile à mettre en pratique.Ce travail se compose d'une partie théorique et d'une partie pratique. Dans la partie théorique, j'ai établi les majorations du risque empirique L2 et du risque quadratique de l'estimateur f^ d'un modèle de régression où l'erreur est non-gaussienne et non-bornée. Il s'agit des bornes supérieures par rapport à la norme empirique L2 et à la norme L2 pour la distance entre le modèle m et son estimation f^ dans le RKHS H. Dans la partie pratique, j'ai développé un package R appelé RKHSMetaMod, pour la mise en œuvre des méthodes d'estimation du méta-modèle f∗ de m. Ce package s'applique indifféremment dans le cas où le modèle m est calculable et le cas du modèle de régression. Afin d'optimiser le temps de calcul et la mémoire de stockage, toutes les fonctions de ce package ont été écrites en utilisant les bibliothèques GSL et Eigen de C++ à l'exception d'une fonction qui est écrite en R. Elles sont ensuite interfacées avec l'environnement R afin de proposer un package facilement exploitable aux utilisateurs. La performance des fonctions du package en termes de qualité prédictive de l'estimateur et de l'estimation des indices de Sobol, est validée par une étude de simulationIn this work, the problem of estimating a meta-model of a complex model, denoted m, is considered. The model m depends on d input variables X1 , ..., Xd that are independent and have a known law. The meta-model, denoted f ∗ , approximates the Hoeffding decomposition of m, and allows to estimate its Sobol indices. It belongs to a reproducing kernel Hilbert space (RKHS), denoted H, which is constructed as a direct sum of Hilbert spaces (Durrande et al. (2013)). The estimator of the meta-model, denoted f^, is calculated by minimizing a least-squares criterion penalized by the sum of the Hilbert norm and the empirical L2-norm (Huet and Taupin (2017)). This procedure, called RKHS ridge group sparse, allows both to select and estimate the terms in the Hoeffding decomposition, and therefore, to select the Sobol indices that are non-zero and estimate them. It makes possible to estimate the Sobol indices even of high order, a point known to be difficult in practice.This work consists of a theoretical part and a practical part. In the theoretical part, I established upper bounds of the empirical L2 risk and the L2 risk of the estimator f^. That is, upper bounds with respect to the L2-norm and the empirical L2-norm for the f^ distance between the model m and its estimation f into the RKHS H. In the practical part, I developed an R package, called RKHSMetaMod, that implements the RKHS ridge group sparse procedure and a spacial case of it called the RKHS group lasso procedure. This package can be applied to a known model that is calculable in all points or an unknown regression model. In order to optimize the execution time and the storage memory, except for a function that is written in R, all of the functions of the RKHSMetaMod package are written using C++ libraries GSL and Eigen. These functions are then interfaced with the R environment in order to propose an user friendly package. The performance of the package functions in terms of the predictive quality of the estimator and the estimation of the Sobol indices, is validated by a simulation study
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Gerchinovitz, Sébastien. "Prédiction de suites individuelles et cadre statistique classique : étude de quelques liens autour de la régression parcimonieuse et des techniques d'agrégation." Phd thesis, Université Paris Sud - Paris XI, 2011. http://tel.archives-ouvertes.fr/tel-00653550.
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Cette thèse s'inscrit dans le domaine de l'apprentissage statistique. Le cadre principal est celui de la prévision de suites déterministes arbitraires (ou suites individuelles), qui recouvre des problèmes d'apprentissage séquentiel où l'on ne peut ou ne veut pas faire d'hypothèses de stochasticité sur la suite des données à prévoir. Cela conduit à des méthodes très robustes. Dans ces travaux, on étudie quelques liens étroits entre la théorie de la prévision de suites individuelles et le cadre statistique classique, notamment le modèle de régression avec design aléatoire ou fixe, où les données sont modélisées de façon stochastique. Les apports entre ces deux cadres sont mutuels : certaines méthodes statistiques peuvent être adaptées au cadre séquentiel pour bénéficier de garanties déterministes ; réciproquement, des techniques de suites individuelles permettent de calibrer automatiquement des méthodes statistiques pour obtenir des bornes adaptatives en la variance du bruit. On étudie de tels liens sur plusieurs problèmes voisins : la régression linéaire séquentielle parcimonieuse en grande dimension (avec application au cadre stochastique), la régression linéaire séquentielle sur des boules L1, et l'agrégation de modèles non linéaires dans un cadre de sélection de modèles (régression avec design fixe). Enfin, des techniques stochastiques sont utilisées et développées pour déterminer les vitesses minimax de divers critères de performance séquentielle (regrets interne et swap notamment) en environnement déterministe ou stochastique.
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Hait, Sourav. "Sparse bounds for various spherical maximal functions." Thesis, 2020. https://etd.iisc.ac.in/handle/2005/4551.
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Harmonic analysis mainly deals with the qualitative and quantitative properties of functions
and transforms of those functions. It has applications in various areas of Mathematics like PDE,
Differential geometry, Ergodic theory etc and also in several areas of Physics like Classical and
Quantum mechanics etc and this makes it a very attractive area of study. The theory of spherical
means plays a very crucial role in the field of Classical harmonic analysis. In 1976, E.M.Stein
first studied the boundedness properties of maximal function associated to spherical means taken
over the Euclidean sphere. Theory of spherical means taken over geodesic spheres in different Lie
groups and Symmetric spaces has received considerable attention in the last few decades. In this
thesis, we consider various versions of spherical maximal function, mainly on Euclidean space and
its non-commutative neighbour Heisenberg groupNBHM
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"Bayesian Framework for Sparse Vector Recovery and Parameter Bounds with Application to Compressive Sensing." Master's thesis, 2019. http://hdl.handle.net/2286/R.I.55639.
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abstract: Signal compressed using classical compression methods can be acquired using brute force (i.e. searching for non-zero entries in component-wise). However, sparse solutions require combinatorial searches of high computations. In this thesis, instead, two Bayesian approaches are considered to recover a sparse vector from underdetermined noisy measurements. The first is constructed using a Bernoulli-Gaussian (BG) prior distribution and is assumed to be the true generative model. The second is constructed using a Gamma-Normal (GN) prior distribution and is, therefore, a different (i.e. misspecified) model. To estimate the posterior distribution for the correctly specified scenario, an algorithm based on generalized approximated message passing (GAMP) is constructed, while an algorithm based on sparse Bayesian learning (SBL) is used for the misspecified scenario. Recovering sparse signal using Bayesian framework is one class of algorithms to solve the sparse problem. All classes of algorithms aim to get around the high computations associated with the combinatorial searches. Compressive sensing (CS) is a widely-used terminology attributed to optimize the sparse problem and its applications. Applications such as magnetic resonance imaging (MRI), image acquisition in radar imaging, and facial recognition. In CS literature, the target vector can be recovered either by optimizing an objective function using point estimation, or recovering a distribution of the sparse vector using Bayesian estimation. Although Bayesian framework provides an extra degree of freedom to assume a distribution that is directly applicable to the problem of interest, it is hard to find a theoretical guarantee of convergence. This limitation has shifted some of researches to use a non-Bayesian framework. This thesis tries to close this gab by proposing a Bayesian framework with a suggested theoretical bound for the assumed, not necessarily correct, distribution. In the simulation study, a general lower Bayesian Cram\'er-Rao bound (BCRB) bound is extracted along with misspecified Bayesian Cram\'er-Rao bound (MBCRB) for GN model. Both bounds are validated using mean square error (MSE) performances of the aforementioned algorithms. Also, a quantification of the performance in terms of gains versus losses is introduced as one main finding of this report.Dissertation/Thesis
Masters Thesis Computer Engineering 2019
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Abdelfattah, Ahmad. "Accelerating Scientific Applications using High Performance Dense and Sparse Linear Algebra Kernels on GPUs." Diss., 2015. http://hdl.handle.net/10754/346955.
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High performance computing (HPC) platforms are evolving to more heterogeneous configurations to support the workloads of various applications. The current hardware landscape is composed of traditional multicore CPUs equipped with hardware accelerators that can handle high levels of parallelism. Graphical Processing Units (GPUs) are popular high performance hardware accelerators in modern supercomputers. GPU programming has a different model than that for CPUs, which means that many numerical kernels have to be redesigned and optimized specifically for this architecture. GPUs usually outperform multicore CPUs in some compute intensive and massively parallel applications that have regular processing patterns. However, most scientific applications rely on crucial memory-bound kernels and may witness bottlenecks due to the overhead of the memory bus latency. They can still take advantage of the GPU compute power capabilities, provided that an efficient architecture-aware design is achieved.
This dissertation presents a uniform design strategy for optimizing critical memory-bound kernels on GPUs. Based on hierarchical register blocking, double buffering and latency hiding techniques, this strategy leverages the performance of a wide range of standard numerical kernels found in dense and sparse linear algebra libraries. The work presented here focuses on matrix-vector multiplication kernels (MVM) as repre-
sentative and most important memory-bound operations in this context. Each kernel inherits the benefits of the proposed strategies. By exposing a proper set of tuning parameters, the strategy is flexible enough to suit different types of matrices, ranging from large dense matrices, to sparse matrices with dense block structures, while high performance is maintained. Furthermore, the tuning parameters are used to maintain the relative performance across different GPU architectures. Multi-GPU acceleration is proposed to scale the performance on several devices. The performance experiments show improvements ranging from 10% and up to more than fourfold speedup against competitive GPU MVM approaches. Performance impacts on high-level numerical libraries and a computational astronomy application are highlighted, since such memory-bound kernels are often located in innermost levels of the software chain. The excellent performance obtained in this work has led to the adoption of code in NVIDIAs widely distributed cuBLAS library.
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Tameling, Carla. "Empirical Optimal Transport on Discrete Spaces: Limit Theorems, Distributional Bounds and Applications." Doctoral thesis, 2018. http://hdl.handle.net/11858/00-1735-0000-002E-E552-1.
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"On density theorems, connectedness results and error bounds in vector optimization." 2001. http://library.cuhk.edu.hk/record=b5890681.
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Yung Hon-wai.Thesis (M.Phil.)--Chinese University of Hong Kong, 2001.
Includes bibliographical references (leaves 133-139).
Abstracts in English and Chinese.
Chapter 0 --- Introduction --- p.1
Chapter 1 --- Density Theorems in Vector Optimization --- p.7
Chapter 1.1 --- Preliminary --- p.7
Chapter 1.2 --- The Arrow-Barankin-Blackwell Theorem in Normed Spaces --- p.14
Chapter 1.3 --- The Arrow-Barankin-Blackwell Theorem in Topolog- ical Vector Spaces --- p.27
Chapter 1.4 --- Density Results in Dual Space Setting --- p.32
Chapter 2 --- Density Theorem for Super Efficiency --- p.45
Chapter 2.1 --- Definition and Criteria for Super Efficiency --- p.45
Chapter 2.2 --- Henig Proper Efficiency --- p.53
Chapter 2.3 --- Density Theorem for Super Efficiency --- p.58
Chapter 3 --- Connectedness Results in Vector Optimization --- p.63
Chapter 3.1 --- Set-valued Maps --- p.64
Chapter 3.2 --- The Contractibility of the Efficient Point Sets --- p.67
Chapter 3.3 --- Connectedness Results in Vector Optimization Prob- lems --- p.83
Chapter 4 --- Error Bounds In Normed Spaces --- p.90
Chapter 4.1 --- Error Bounds of Lower Semicontinuous Functionsin Normed Spaces --- p.91
Chapter 4.2 --- Error Bounds of Lower Semicontinuous Convex Func- tions in Reflexive Banach Spaces --- p.100
Chapter 4.3 --- Error Bounds with Fractional Exponents --- p.105
Chapter 4.4 --- An Application to Quadratic Functions --- p.114
Bibliography --- p.133
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Bonciocat, Anca-Iuliana [Verfasser]. "Curvature bounds and heat kernels: discrete versus continuous spaces / vorgelegt von Anca-Iuliana Bonciocat." 2008. http://d-nb.info/989966135/34.
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Larose, Avery. "Mysterious geographies : the scaped spaces of Eleanor Bond, Janet Cardiff and Brenda Pelkey." Thesis, 2005. http://spectrum.library.concordia.ca/8593/1/MR10311.pdf.
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This thesis explores how the works of Eleanor Bond, Brenda Pelkey and Janet Cardiff emphasize the inseparability of represented space from human perception, and the crucial role played by such physical and cultural concerns as the body, memories, stories and dreams in the shaping of spaces. Their works refer, therefore, not simply to physical spaces, but to conditions or states of space, often determined by layers of metaphorical significance which these artists bring to light. It is these ephemeral transformations and psychic interventions which irrevocably alter our environments that constitute the scape, a concept which captures this building up of a network of human responses to one's environment which in turn change the way it is perceived and lived. In their insistence upon the impact of the elusive and metaphorical, these works constitute a decided intervention in the tradition of Canadian landscape art while remaining firmly embedded in discussions of space, situation and environment. By subverting hegemonic modes of spatial representation, Bond, Pelkey and Cardiff construct their own visual assertions of what spaces were, can, or could be. The creation of scapes becomes part of a process for re-shaping both perceived and lived space, collapsing binaries such as self and other, public and private, nature and culture, and reality and fiction, in order to encompass the range of concrete and metaphorical attempts to shape the spaces we inhabit.
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Forster, Jürgen [Verfasser]. "Some results concerning arrangements of half spaces and relative loss bounds / vorgelegt von Jürgen Forster." 2002. http://d-nb.info/964520389/34.
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CAVALLUCCI, Nicola. "Packing conditions in metric spaces with curvature bounded above and applications." Doctoral thesis, 2021. http://hdl.handle.net/11573/1486078.
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General metric spaces satisfying weak and synthetic notions of upper and lower curvature bounds will be studied. The relations between upper and lower bounds will be pointed out, especially the interactions between a packing condition and different forms of convexity of the metric. The main tools will be a new and flexible definition of entropy on metric spaces and a version of the Tits Alternative for groups of isometries of the metric spaces under consideration. The applications can be divided into classical and new results: the former consist in generalizations to a wider context of the theory of negatively curved Riemannian manifolds, while the latter include several compactness and continuity results.
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Pediconi, Francesco. "Geometric aspects of locally homogeneous Riemannian spaces." Doctoral thesis, 2020. http://hdl.handle.net/2158/1197175.
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The subject of this thesis is the study of some geometric problems arising in the context of locally and globally homogeneous Riemannian spaces. In particular, we are mainly interested in investigate the interplay between curvature conditions and the compactness of some classes of locally homogeneous spaces, with respect to appropriate topologies.
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Rebs, Christian. "Asymptotic bounds and values for the norm of the Laplace operator and other partial differential operators on spaces of polynomials." 2019. https://monarch.qucosa.de/id/qucosa%3A72803.
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In der vorliegenden Dissertation werden endlichdimensionale Räume multivariater Polynome in N Variablen mit der Laguerre-, Hermite- bzw. Legendrenorm versehen.
Dabei sei der Höchstgrad der Polynome oder die Summe der Grade der Variablen durch eine natürliche Zahl n nach oben beschränkt. Wir betrachten auf diesen Räumen den Laplaceoperator und zwei weitere partielle Differentialoperatoren
und interessieren uns für das Verhalten der von den Polynomnormen induzierten Operatornormen dieser Operatoren, wenn n gegen unendlich strebt.
Im Fall der Laguerre- und Legendrenorm werden asymptotische obere und untere Schranken der Operatornormen hergeleitet. Im Fall der Hermitenorm kann sogar eine asymptotische Formel gezeigt werden, wenn man voraussetzt, dass der Höchstgrad der Poynome duch n beschränkt ist.
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Dhara, Raj Narayan. "Existence and regularity theory in weighted Sobolev spaces and applications." Doctoral thesis, 2016. https://depotuw.ceon.pl/handle/item/2051.
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In the thesis we discuss several questions related to the study of degenerate, possibly nonlinear PDEs of elliptic type. At first we discuss the equivalent conditions between the validity of weighted Poincar\'e inequalities, structure of the functionals on weighted Sobolev spaces, isoperimetric inequalities and the existence and uniqueness of solutions to the degenerate nonlinear elliptic PDEs with nonhomogeneous boundary condition, having the form:\begin{eqnarray}\label{eqn:abs}\left\{\begin{array}{lll}{\rm div} \left( \rho (x)|\nabla u|^{p-2}\nabla u\right) =x^*,\\~~~~~~~~~~~~u-w \in W^{1,p}_{\rho,0} (\Omega),\end{array}\right.\end{eqnarray}involving any given $x^*\in (W^{1,p}_{\rho,0} (\Omega))^*$ and $w\in W^{1,p}_{\rho} (\Omega)$, where $u\in W^{1,p}_{\rho} (\Omega)$ and $W^{1,p}_{\rho} (\Omega)$ denotes certain weighted Sobolev space, $W^{1,p}_{\rho,0} (\Omega)$ is the completion of $\mathcal{C}_{0}^{\infty}(\Omega)$. As a next step, we undertake a natural question how to interpret the nonhomogenous boundary conditions in weighted Sobolev spaces, when the natural analytical tools, like trace embedding theorems, are missing. Our further goal is to contribute to solvability and uniqueness for degenerate elliptic PDEs with nonhomogenous boundary condition being the extension of~\eqref{eqn:abs}. In addition to the monotonicity method used in the first step of our discussion for the problem~\eqref{eqn:abs}, we also exploit Lax-Miligram theorem to treat the linear problem like:\begin{equation*}\begin{cases}-{\rm div} (A(x)\nabla u(x)) + B(x)\cdot\nabla u(x) + C(x)u(x) = x^{*}\ \ \text{for a.e.}\ x\in \Omega, \\~~~~~~~~~~~~~~~~~~ u(x) = g(x) \ \ \text{for a.e. }\ x\in \partial\Omega ,\end{cases}\end{equation*}as well as Ekeland's Variational Principle and Boccardo-Murat techniques to consider problem like:\begin{align*} \begin{cases} - {\rm div} \left( \rho (x)|\nabla u|^{p-2}\nabla u\right) - \lambda\, b(x)| u|^{p-2} u = x^*,\\~~~~~~~~~~~~~~~~~~~u-z \in X , \end{cases}\end{align*}where $p>1,\ \lambda>0$, and the operator $\mathcal{L}_{\lambda} u:= - {\rm div} \left( \rho (x)|\nabla u|^{p-2}\nabla u\right) - \lambda\, b(x)| u|^{p-2} u $ is non-monotone.For the study of the nonhomogeneous BVPs, we apply recent results due to Ka\l{}amajska and myself, where we constructed trace extension operator from weighted Orlicz-Slobodetskii spaces defined on the boundary of the domain to weighted Orlicz-Sobolev spaces in the domain. Information on the spectrum of the corresponding differential operator is also derived. Moreover, some nonexistence and nonuniqueness results are also analyzed.
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Medvedev, Vladimir. "Conformal spectra, moduli spaces and the Friedlander-Nadirahvili invariants." Thesis, 2020. http://hdl.handle.net/1866/24805.
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Dans cette thèse, nous étudions le spectre conforme d'une surface fermée et le spectre de Steklov conforme d'une surface compacte à bord et leur application à la géométrie conforme et à la topologie. Soit (Σ, c) une surface fermée munie d'une classe conforme c. Alors la k-ième valeur propre conforme est définie comme Λ_k(Σ,c)=sup{λ_k(g) Aire(Σ,g)| g ∈ c), où λ_k(g) est la k-ième valeur propre de l'operateur de Laplace-Beltrami de la métrique g sur Σ. Notons que nous commeçons par λ_0(g) = 0. En prennant le supremum sur toutes les classes conformes C sur Σ on obtient l'invariant topologique suivant de Σ: Λ_k(Σ)=sup{Λ_k(Σ,c)| c ∈ C}. D'après l'article [65], les quantités Λ_k(Σ, c) et Λ_k(Σ) sont bien définies. Si une métrique g sur Σ satisfait λ_k(g) Aire(Σ, g) = Λ_k(Σ), alors on dit que g est maximale pour la fonctionnelle λ_k(g) Aire(Σ, g). Dans l'article [73], il a été montré que les métriques maximales pour λ_1(g) Aire(Σ, g) peuvent au pire avoir des singularités coniques. Dans cette thèse nous montrons que les métriques maximales pour les fonctionnelles λ_1(g) Aire(T^2, g) et λ_1(g) Aire(KL, g), où T^2 et KL dénotent le 2-tore et la bouteille de Klein, ne peuvent pas avoir de singularités coniques. Ce résultat découle d'un théorème de classification de classes conformes par des métriques induites d'une immersion minimale ramifiée dans une sphère ronde aussi montré dans cette thèse. Un autre invariant que nous étudions dans cette thèse est le k-ième invariant de Friedlander-Nadirashvili défini comme: I_k(Σ) = inf{Λ_k(Σ, c)| c ∈ C}. L'invariant I_1(Σ) a été introduit dans l'article [34]. Dans cette thèse nous montrons que pour toute surface orientable et pour toute surface non-orientable de genre impaire I_k(Σ)=I_k(S^2) et pour toute surface non-orientable de genre paire I_k(RP^2) ≥ I_k(Σ)>I_k(S^2). Ici S^2 et RP^2 dénotent la 2-sphère et le plan projectif. Nous conjecturons que I_k(Σ) sont des invariants des cobordismes des surfaces fermées. Le spectre de Steklov conforme est défini de manière similaire. Soit (Σ, c) une surface compacte à bord non vide ∂Σ, alors les k-ièmes valeurs propres de Steklov conformes sont définies comme: σ*_k(Σ, c)=sup{σ_k(g) Longueur(∂Σ, g)| g ∈ c}, où σ_k(g) est la k-ième valeur propre de Steklov de la métrique g sur Σ. Ici nous supposons que σ_0(g) = 0. De façon similaire au problème fermé, on peut définir les quantités suivantes: σ*_k(Σ)=sup{σ*_k(Σ, c)| c ∈ C} et I^σ_k(Σ)=inf{σ*_k(Σ, c)| c ∈ C}. Les résultats de l'article [16] impliquent que toutes ces quantités sont bien définies. Dans cette thèse on obtient une formule pour la limite de σ*_k(Σ, c_n) lorsque la suite des classes conformes c_n dégénère. Cette formule implique que pour toute surface à bord I^σ_k(Σ)= I^σ_k(D^2), où D^2 dénote le 2-disque. On remarque aussi que les quantités I^σ_k(Σ) sont des invariants des cobordismes de surfaces à bord. De plus, on obtient une borne supérieure pour la fonctionnelle σ^k(g) Longueur(∂Σ, g), où Σ est non-orientable, en terme de son genre et le nombre de composants de bord.In this thesis, we study the conformal spectrum of a closed surface and the conformal Steklov spectrum of a compact surface with boundary and their application to conformal geometry and topology. Let (Σ,c) be a closed surface endowed with a conformal class c then the k-th conformal eigenvalue is defined as Λ_k(Σ,c)=sup{λ_k(g) Aire(Σ,g)| g ∈ c), where λ_k(g) is the k-th Laplace-Beltrami eigenvalue of the metric g on Σ. Note that we start with λ_0(g) = 0 Taking the supremum over all conformal classes C on Σ one gets the following topological invariant of Σ: Λ_k(Σ)=sup{Λ_k(Σ,c)| c ∈ C}. It follows from the paper [65] that the quantities Λ_k(Σ, c) and Λ_k(Σ) are well-defined. Suppose that for a metric g on Σ the following identity holds λ_k(g) Aire(Σ, g) = Λ_k(Σ). Then one says that the metric g is maximal for the functional λ_k(g) Aire(Σ, g). In the paper [73] it was shown that the maximal metrics for the functional λ_1(g) Aire(Σ, g) at worst can have conical singularities. In this thesis we show that the maximal metrics for the functionals λ_1(g) Aire(T^2, g) and λ_1(g) Aire(KL, g), where T^2 and KL stand for the 2-torus and the Klein bottle respectively, cannot have conical singularities. This result is a corollary of a conformal class classification theorem by metrics induced from a branched minimal immersion into a round sphere that we also prove in the thesis. Another invariant that we study in this thesis is the k-th Friedlander-Nadirashvili invariant defined as: I_k(Σ) = inf{Λ_k(Σ, c)| c ∈ C}. The invariant I_1(Σ) was introduced in the paper [34]. In this thesis we prove that for any orientable surface and any non-orientable surface of odd genus I_k(Σ)=I_k(S^2) and for any non-orientable surface of even genus I_k(RP^2) ≥ I_k(Σ)>I_k(S^2). Here S^2 and RP^2 denote the 2-sphere and the projective plane respectively. We also conjecture that I_k(Σ) are invariants of cobordisms of closed manifolds. The conformal Steklov spectrum is defined in a similar way. Let (Σ, c) be a compact surface with non-empty boundary ∂Σ then the k-th conformal Steklov eigenvalues is defined by the formula: σ*_k(Σ, c)=sup{σ_k(g) Longueur(∂Σ, g)| g ∈ c}, where σ_k(g) is the k-th Steklov eigenvalue of the metric g on Σ. Here we suppose that σ_0(g) = 0. Similarly to the closed problem one can define the following quantities: σ*_k(Σ)=sup{σ*_k(Σ, c)| c ∈ C} and I^σ_k(Σ)=inf{σ*_k(Σ, c)| c ∈ C}. The results of the paper [16] imply that all these quantities are well-defined. In this thesis we obtain a formula for the limit of the k-th conformal Steklov eigenvalue when the sequence of conformal classes degenerates. Using this formula we show that for any surface with boundary I^σ_k(Σ)= I^σ_k(D^2), where D^2 stands for the 2-disc. We also notice that I^σ_k(Σ) are invariants of cobordisms of surfaces with boundary. Moreover, we obtain an upper bound for the functional σ^k(g) Longueur(∂Σ, g), where Σ is non-orientable, in terms of its genus and the number of boundary components.
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Rieger, Christian. "Sampling Inequalities and Applications." Doctoral thesis, 2008. http://hdl.handle.net/11858/00-1735-0000-0006-B3B9-0.
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