Recurrence relations for a third-order family of methods in Banach Spaces

Recently, PArida and Gupta (J. Comp. Appl.Math. 2-6 (2007), 873-877) used Rall's recurrence relations approach (from 1961) to approximate roots of nonlinear equations, by developing several methods, the latest of which is free of second derivative and it is of third order. In this paper, we use an idea of Kou and Li (appl. Math. Comp. 187 (2007), 1027-1032) and modify the approach of Parida and Gupta, obtaining yet another third-order method to approximate a solution of a non-linear equation in a Banach space. We give several applications to our method

Mathematical Aspects of Hopfield Models

Diese Dissertation behandelt zwei Modelle aus der statistischen Mechanik ungeordneter Systeme. Beide sind Varianten des Hopfield-Modells und gehören zur Klasse der Molekularfeldmodelle. Im ersten Teil behandeln wir den Fall mit p-Spin-Wechselwirkungen (p größer als 4 und gerade) und superextensiv vielen Mustern (deren Anzahl M wie die p-te Potenz der Systemgröße N wächst), wobei wir zwei verschiedene Energiefunktionen betrachten. Wir beweisen die Existenz einer kritischen Temperatur, bei welcher der sogenannte Replikaüberlapp von Null auf einen strikt positiven Wert springt. Wir geben obere und untere Schranken für ihren Wert an und zeigen, daß für die eine Wahl der Hamiltonfunktion beide gegen die kritische Temperatur (bis auf einen konstanten Faktor) des Random Energy Model konvergieren, falls p gegen Unendlich strebt. Diese kritische Temperatur fällt mit der kleinsten Temperatur zusammen, für welche die ausgeglühte freie Energie und der Erwartungswert der abgeschreckten freien Energie identisch sind. Der Zusammenhang zwischen diesen beiden Resultaten wird durch eine partielle Integrationsformel geliefert, welche mit Hilfe einer Störungsentwicklung der Boltzmannfaktoren bewiesen wird. Außerdem berechnen wir die Fluktuationen der freien Energie und erhalten, daß sie von der Ordnung Quadratwurzel von N sind. Weiterhin beweisen wir die Existenz einer kritischen Proportionalitätskonstanten für die Anzahl Muster, oberhalb welcher das Minimum der Hamiltonfunktion mit großer Wahrscheinlichkeit nicht in der Nähe eines der Muster angenommen wird. Dies bedeutet, daß, obwohl das Gibbsmaß sich bei kleinen Temperaturen auf einer kleinen Teilmenge des Zustandsraumes konzentriert, dies nicht in der Nähe der Muster geschieht. In einem zweiten Teil beweisen wir obere Schranken für die Norm von gewissen zufälligen Matrizen mit abhängigen Einträgen. Diese Abschätzungen werden im ersten Teil zur Berechnung der Fluktuationen der freien Energie benutzt. Sie werden mit der Chebyshev-Markov-Ungleichung, angewandt auf die Spur von hohen Potenzen der Matrizen, bewiesen. Das zentrale Resultat dazu ist eine Darstellung der Spur von diesen hohen Potenzen als Wege auf gewissen bipartiten Graphen. Dies transformiert das Berechnen des Erwartungswertes der Spur auf das kombinatorische Problem, die maximale Anzahl kreisförmiger Teilgraphen eines gegebenen Eulergraphen zu bestimmen. Die Resultate zeigen, dass die Abhängigkeit zwischen den Einträgen eine wichtige Rolle spielt und nicht vernachlässigt werden kann. Im letzten Teil schließlich betrachten wir ein Hopfield-Modell mit zwei Gauß'schen Mustern. Wir zeigen, da$szlig; überabzählbar viele extremale Gibbszustände existieren, welche durch den Einheitskreis indiziert werden. Diese Symmetrie wird zufällig gebrochen im Sinne, daß der Metazustand von einem Kontinuum von Paaren von extremalen Gibbsmaßen getragen wird, welche durch eine globale Spinsymmetrie verknüpft sind. Wir beweisen diese Resultate mit Hilfe der zufälligen Ratenfunktion des induzierten Maßes auf den Überlapparametern. Insbesondere zeigen wir, daß die Symmetriebrechung durch die Fluktuationen der Ratenfunktion auf den (entarteten) Minima ihrer Erwartung erzwungen wird. Diese Fluktuationen werden durch einen zufälligen Prozeß auf dem Einheitskreis beschrieben, dessen globale Minima die Menge (schlussendlich ein Paar) der extremalen Zustände auswählen.This thesis is concerned with two models from equilibrium statistical mechanics of disordered systems. Both of them are variants of the Hopfield model, and belong to the class of mean-field models. In the first part, we treat the case of p-spin interactions (with p larger than 4 and even) and super-extensively many patterns (their number M scaling as the (p-1)th power of the system size N). We consider two choices of the Hamiltonians. We find that there exists a critical temperature, at which the replica overlap changes from 0 to a strictly positive value. We give upper and lower bounds for its value, and show that for one choice of the Hamiltonian, both of them converge as p tends to infinity to the critical temperature (up to a constant factor) of the random energy model. This critical temperature coincides with the minimum temperature for which annealed free energy and mean of the quenched free energy are equal. The relation between the two results is furnished by an integration by parts formula that is proved by perturbative expansion of the Boltzmann factors. We also calculate the fluctuations of the free energy which are shown to be of the order of the square root of the system size N. Furthermore, we find that there exists a critical scaling constant for the number of patterns above which with large probability the minimum of the Hamiltonian is not realized in the vicinity of any of the patterns. This means that while there is a condensation for low temperatures, the Gibbs measure does not concentrate around the patterns. In a second part of the thesis, we prove upper bounds on the norm of certain random matrices with dependent entries. These estimates are used in Part I to prove the fluctuations of the free energy. They are proved by the Chebyshev-Markov inequality, applied to the trace of large powers of the matrices. The key ingredient is a representation of the trace of these large powers in terms of walks on an appropriate bipartite graph. This reduces the calculation of the expectation of the trace to the combinatorial problem of counting the maximum number of sub-circuits of a given circuit. The results show that the dependence between the entries cannot be neglected. Finally, in the last part, we consider a two pattern Hopfield model with Gaussian patterns. We show that there are uncountably many pure states indexed by the circle. This symmetry is randomly broken in the sense that the metastate is supported on a continuum of pairs of pure states that are related by a global (spin-flip) symmetry. We prove these results by studying the random rate function of the induced measure on the overlap parameters. In particular, the breaking of the symmetry is shown to be due to the fluctuations of this rate function at the (degenerate) minima of its expectation. These fluctuations are described by a random process on the circle whose global minima determine the chosen set (eventually a pair) of states

Asymptotic equivalence for nonparametric generalized linear models

We establish that a non-Gaussian nonparametric regression model is asymptotically equivalent to a regression model with Gaussian noise. The approximation is in the sense of Le Cam's deficiency distance Δ; the models are then asymptotically equivalent for all purposes of statistical decision with bounded loss. Our result concerns a sequence of independent but not identically distributed observations with each distribution in the same real-indexed exponential family. The canonical parameter is a value ƒ(ti) of a regression function ƒ at a grid point ti (nonparametric GLM). When ƒ is in a Hölder ball with exponent β > 1⁄2, we establish global asymptotic equivalence to observations of a signal Γ(f(t)) in Gaussian white noise, where Γ is related to a variance stabilizing transformation in the exponential family. The result is a regression analog of the recently established Gaussian approximation for the i.i.d. model. The proof is based on a functional version of the Hungarian construction for the partial sum process

Asymptotic equivalence of density estimation and white noise.

Signal recovery in Gaussian white noise with variance tending to zero has served for some time as a representative model for nonparametric curve estimation, having all the essential traits in a purified form. The equivalence has mostly been stated informally, but an approximation in the sense of Le Cam's deficiency distance Δ would make it precise. Then two models are asymptotically equivalent for all purposes of statistical decision with bounded loss. In nonparametrics, a first result of this kind has recently been established for Gaussian regression (Brown and Low, 1992). We consider the analogous problem for the experiment given by n i. i. d. observations having density ƒ on the unit interval. Our basic result concerns the parameter space of densities which are in a Sobolev class of order 4 and uniformly bounded away from zero. We show that an i. i. d. sample of size n with density ƒ is globally asymptotically equivalent to a white noise experiment with trend ƒ1/2 and variance 1⁄4n-1. This represents a nonparametric analog of Le Cam's heteroskedastic Gaussian approximation in the finite dimensional case. The proof utilizes empirical process techniques, especially the Hungarian construction. White noise models on ƒ and log ƒ are also considered, allowing for various "automatic" asymptotic risk bounds in the i. i. d. model from white noise. As first applications we discuss linear wavelet estimators of a density and exact constants for Hellinger loss

Scalable Performance Analysis of Massively Parallel Stochastic Systems

The accurate performance analysis of large-scale computer and communication systems is directly inhibited by an exponential growth in the state-space of the underlying Markovian performance model. This is particularly true when considering massively-parallel architectures such as cloud or grid computing infrastructures. Nevertheless, an ability to extract quantitative performance measures such as passage-time distributions from performance models of these systems is critical for providers of these services. Indeed, without such an ability, they remain unable to offer realistic end-to-end service level agreements (SLAs) which they can have any confidence of honouring. Additionally, this must be possible in a short enough period of time to allow many different parameter combinations in a complex system to be tested. If we can achieve this rapid performance analysis goal, it will enable service providers and engineers to determine the cost-optimal behaviour which satisfies the SLAs. In this thesis, we develop a scalable performance analysis framework for the grouped PEPA stochastic process algebra. Our approach is based on the approximation of key model quantities such as means and variances by tractable systems of ordinary differential equations (ODEs). Crucially, the size of these systems of ODEs is independent of the number of interacting entities within the model, making these analysis techniques extremely scalable. The reliability of our approach is directly supported by convergence results and, in some cases, explicit error bounds. We focus on extracting passage-time measures from performance models since these are very commonly the language in which a service level agreement is phrased. We design scalable analysis techniques which can handle passages defined both in terms of entire component populations as well as individual or tagged members of a large population. A precise and straightforward specification of a passage-time service level agreement is as important to the performance engineering process as its evaluation. This is especially true of large and complex models of industrial-scale systems. To address this, we introduce the unified stochastic probe framework. Unified stochastic probes are used to generate a model augmentation which exposes explicitly the SLA measure of interest to the analysis toolkit. In this thesis, we deploy these probes to define many detailed and derived performance measures that can be automatically and directly analysed using rapid ODE techniques. In this way, we tackle applicable problems at many levels of the performance engineering process: from specification and model representation to efficient and scalable analysis

Computational and Statistical Approaches for Large-Scale Genome-Wide Association Studies

Over the past decade, genome-wide association studies (GWAS) have proven successful at shedding light on the underlying genetic variations that affect the risk of human complex diseases, which can be translated to novel preventative and therapeutic strategies. My research aims at identifying novel disease-associated genetic variants through large-scale GWAS and developing computational and statistical pipelines and methods to improve power and accuracy of GWAS. Bicuspid aortic valve (BAV) is a congenital heart defect characterized by fusion of two of the normal three leaflets of the aortic valve. As the most common cardiovascular malformation in humans, BAV is moderately heritable and is an important risk factor for valvulopathy and aortopathy, but its genetic origins remain elusive. In Chapter 2, we present the first large-scale GWAS study to identify novel genetic variants associated with BAV. We report association with a non-coding variant 151kb from the gene encoding the cardiac-specific transcription factor, GATA4, and near-significance for p.Ser377Gly in GATA4. We used multiple bioinformatics approaches to demonstrate that the GATA4 gene is a plausible biological candidate. In the subsequent functional follow-up, GATA4 was interrupted by CRISPR-Cas9 in induced pluripotent stem cells from healthy donors. The disruption of GATA4 significantly impaired the transition from endothelial cells into mesenchymal cells, a critical step in heart valve development. Genotype imputation is widely used in GWAS to perform in silico genotyping, leading to higher power to identify novel genetic signals. When multiple reference panels are not consented to combine together, it is unclear how to combine the imputation results to optimize the power of genetic association tests. In Chapter 3, we compared the accuracy of 9,265 Norwegian genomes imputed from three reference panels – 1000 Genomes Phase 3 (1000G), Haplotype Reference Consortium (HRC), and a reference panel containing 2,201 Norwegian participants from the HUNT study with low-pass genome sequencing. We observed that the overall imputation accuracy from the population-specific panel was substantially higher than 1000G and was comparable with HRC, despite HRC being 15-fold larger. We also evaluated different strategies to utilize multiple sets of imputed genotypes to increase the power of association studies. We propose that testing association for all variants imputed from any panel results in higher power to detect association than the alternative strategy of testing only the version of each genetic variant with the highest imputation quality metric. In phenome-wide GWAS by large biobanks, most binary traits have substantially fewer cases than controls. Both of the widely used approaches, linear mixed model and the recently proposed logistic mixed model, perform poorly -- producing large type I error rates -- in the analysis of phenotypes with unbalanced case-control ratios. In Chapter 4, we propose a scalable and accurate generalized mixed model association test that uses the saddlepoint approximation (SPA) to calibrate the distribution of score test statistics. This method, SAIGE, provides accurate p-values even when case-control ratios are extremely unbalanced. It utilizes state-of-art optimization strategies to reduce computational time and memory cost of generalized mixed model. The computation cost linearly depends on sample size, and hence can be applicable to GWAS for thousands of phenotypes by large biobanks. Through the analysis of UK Biobank data of 408,961 white British European-ancestry samples for 1,403 dichotomous phenotypes, we show that SAIGE can efficiently analyze large sample data, controlling for unbalanced case-control ratios and sample relatedness.PHDBioinformaticsUniversity of Michigan, Horace H. Rackham School of Graduate Studieshttps://deepblue.lib.umich.edu/bitstream/2027.42/144097/1/zhowei_1.pd

Heterogeneous neural networks: theory and applications

Aquest treball presenta una classe de funcions que serveixen de models neuronals generalitzats per ser usats en xarxes neuronals artificials. Es defineixen com una mesura de similitud que actúa com una definició flexible de neurona vista com un reconeixedor de patrons. La similitud proporciona una marc conceptual i serveix de cobertura unificadora de molts models neuronals de la literatura i d'exploració de noves instàncies de models de neurona. La visió basada en similitud porta amb naturalitat a integrar informació heterogènia, com ara quantitats contínues i discretes (nominals i ordinals), i difuses ó imprecises. Els valors perduts es tracten de manera explícita. Una neurona d'aquesta classe s'anomena neurona heterogènia i qualsevol arquitectura neuronal que en faci ús serà una Xarxa Neuronal Heterogènia.En aquest treball ens concentrem en xarxes neuronals endavant, com focus inicial d'estudi. Els algorismes d'aprenentatge són basats en algorisms evolutius, especialment extesos per treballar amb informació heterogènia. En aquesta tesi es descriu com una certa classe de neurones heterogènies porten a xarxes neuronals que mostren un rendiment molt satisfactori, comparable o superior al de xarxes neuronals tradicionals (com el perceptró multicapa ó la xarxa de base radial), molt especialment en presència d'informació heterogènia, usual en les bases de dades actuals.This work presents a class of functions serving as generalized neuron models to be used in artificial neural networks. They are cast into the common framework of computing a similarity function, a flexible definition of a neuron as a pattern recognizer. The similarity endows the model with a clear conceptual view and serves as a unification cover for many of the existing neural models, including those classically used for the MultiLayer Perceptron (MLP) and most of those used in Radial Basis Function Networks (RBF). These families of models are conceptually unified and their relation is clarified. The possibilities of deriving new instances are explored and several neuron models --representative of their families-- are proposed. The similarity view naturally leads to further extensions of the models to handle heterogeneous information, that is to say, information coming from sources radically different in character, including continuous and discrete (ordinal) numerical quantities, nominal (categorical) quantities, and fuzzy quantities. Missing data are also explicitly considered. A neuron of this class is called an heterogeneous neuron and any neural structure making use of them is an Heterogeneous Neural Network (HNN), regardless of the specific architecture or learning algorithm. Among them, in this work we concentrate on feed-forward networks, as the initial focus of study. The learning procedures may include a great variety of techniques, basically divided in derivative-based methods (such as the conjugate gradient)and evolutionary ones (such as variants of genetic algorithms).In this Thesis we also explore a number of directions towards the construction of better neuron models --within an integrant envelope-- more adapted to the problems they are meant to solve.It is described how a certain generic class of heterogeneous models leads to a satisfactory performance, comparable, and often better, to that of classical neural models, especially in the presence of heterogeneous information, imprecise or incomplete data, in a wide range of domains, most of them corresponding to real-world problems.Postprint (published version

Mathematical modelling: Applied problems of mathematical physics

Krājums veltīts skaitlisko metožu izstrādei, pamatošanai un pielietošanai konkrētu matemātiskās fizikas problēmu risināšan