Difference equations and iterative processes

Divergence equations and iterative processe

Estudio sobre convergencia y dinámica de los métodos de Newton, Stirling y alto orden

Las matemáticas, desde el origen de esta ciencia, han estado al servicio de la sociedad tratando de dar respuesta a los problemas que surgían. Hoy en día sigue siendo así, el desarrollo de las matemáticas está ligado a la demanda de otras ciencias que necesitan dar solución a situaciones concretas y reales. La mayoría de los problemas de ciencia e ingeniería no pueden resolverse usando ecuaciones lineales, es por tanto que hay que recurrir a las ecuaciones no lineales para modelizar dichos problemas (Amat, 2008; véase también Argyros y Magreñán, 2017, 2018), entre otros. El conflicto que presentan las ecuaciones no lineales es que solo en unos pocos casos es posible encontrar una solución única, por tanto, en la mayor parte de los casos, para resolverlas hay que recurrir a los métodos iterativos. Los métodos iterativos generan, a partir de un punto inicial, una sucesión que puede converger o no a la solución

Parametric classification and variable selection by the minimum integrated squared error criterion

This thesis presents a robust solution to the classification and variable selection problem when the dimension of the data, or number of predictor variables, may greatly exceed the number of observations. When faced with the problem of classifying objects given many measured attributes of the objects, the goal is to build a model that makes the most accurate predictions using only the most meaningful subset of the available measurements. The introduction of [cursive l] 1 regularized model titling has inspired many approaches that simultaneously do model fitting and variable selection. If parametric models are employed, the standard approach is some form of regularized maximum likelihood estimation. While this is an asymptotically efficient procedure under very general conditions, it is not robust. Outliers can negatively impact both estimation and variable selection. Moreover, outliers can be very difficult to identify as the number of predictor variables becomes large. Minimizing the integrated squared error, or L 2 error, while less efficient, has been shown to generate parametric estimators that are robust to a fair amount of contamination in several contexts. In this thesis, we present a novel robust parametric regression model for the binary classification problem based on L 2 distance, the logistic L 2 estimator (L 2 E). To perform simultaneous model fitting and variable selection among correlated predictors in the high dimensional setting, an elastic net penalty is introduced. A fast computational algorithm for minimizing the elastic net penalized logistic L 2 E loss is derived and results on the algorithm's global convergence properties are given. Through simulations we demonstrate the utility of the penalized logistic L 2 E at robustly recovering sparse models from high dimensional data in the presence of outliers and inliers. Results on real genomic data are also presented

Nondifferentiable Optimization: Motivations and Applications

IIASA has been involved in research on nondifferentiable optimization since 1976. The Institute's research in this field has been very productive, leading to many important theoretical, algorithmic and applied results. Nondifferentiable optimization has now become a recognized and rapidly developing branch of mathematical programming. To continue this tradition and to review developments in this field IIASA held this Workshop in Sopron (Hungary) in September 1984. This volume contains selected papers presented at the Workshop. It is divided into four sections dealing with the following topics: (I) Concepts in Nonsmooth Analysis; (II) Multicriteria Optimization and Control Theory; (III) Algorithms and Optimization Methods; (IV) Stochastic Programming and Applications

Majorizing sequences for Newton's method from initial value problems

The most restrictive condition used by Kantorovich for proving the semilocal convergence of Newton's method in Banach spaces is relaxed in this paper, providing we can guarantee the semilocal convergence in situations that Kantorovich cannot. To achieve this, we use Kantorovich's technique based on majorizing sequences, but our majorizing sequences are obtained differently, by solving initial value problems. © 2011 Elsevier B.V. All rights reserved

The Music Sound

A guide for music: compositions, events, forms, genres, groups, history, industry, instruments, language, live music, musicians, songs, musicology, techniques, terminology , theory, music video. Music is a human activity which involves structured and audible sounds, which is used for artistic or aesthetic, entertainment, or ceremonial purposes. The traditional or classical European aspects of music often listed are those elements given primacy in European-influenced classical music: melody, harmony, rhythm, tone color/timbre, and form. A more comprehensive list is given by stating the aspects of sound: pitch, timbre, loudness, and duration. Common terms used to discuss particular pieces include melody, which is a succession of notes heard as some sort of unit; chord, which is a simultaneity of notes heard as some sort of unit; chord progression, which is a succession of chords (simultaneity succession); harmony, which is the relationship between two or more pitches; counterpoint, which is the simultaneity and organization of different melodies; and rhythm, which is the organization of the durational aspects of music

Problemas de valor inicial en la construcción de sucesiones mayorizantes para el método de Newton en espacios de Banach

Es bien conocido que la resolución de ecuaciones no lineales de la forma F(x) = 0, donde F es un operador no lineal definido entre espacios de Banach, es un problema habitual en las ciencias e ingenierías. Habitualmente se buscan aproximaciones numéricas de las raíces de la ecuación anterior, puesto que encontrar raíces exactas suele ser difícil. Para aproximar una raíz de la ecuación anterior se utilizan habitualmente métodos iterativos, de entre los que destaca el método de Newton por su sencillez, fácil aplicación y eficiencia. El primer resultado de convergencia semilocal para el método de Newton en espacios de Banach fue dado por Kantorovich. En esta memoria se analiza la convergencia semilocal del método de Newton en espacios de Banach y cuyo principal objetivo es darle una mayor generalidad al problema de aproximar las raíces de una ecuación no lineal mediante el método de Newton, de manera que se pueda extender la aplicabilidad de este método a situaciones en las que la teoría clásica de Kantorovich no la puede garantizar. Para ello, se utiliza el conocido principio de la mayorante, que se basa en la construcción de sucesiones reales mayorizantes, y que, bajo nuevas condiciones de convergencia semilocal de tipo Kantorovich, permite generalizar las condiciones clásicas de Kantorovich. Aquí juega un papel importante la construcción ad hoc de sucesiones mayorizantes a partir de la resolución de problemas de valor inicial. Se ilustra todo lo anterior con diferentes tipos de ecuaciones no lineales, destacando las ecuaciones integrales de tipo Hammerstein mixto y la ecuación de Bratu, que tienen su origen en diversos problemas del mundo real, tal y como se pone de manifiesto a largo de la Memoria.It is well known that solving nonlinear equations of the form F(x)=0, where F is a nonlinear operator defined between Banach spaces, is a common problem in science and engineering. Usually numerical approximations of the roots of the previous equation are looking for, since finding exact roots is often difficult. To approximate a root of the previous equation is commonly used iterative methods, among which Newton's method is important for its simplicity, easy implementation and efficiency. The first result of semilocal convergence for Newton's method in Banach spaces was given by Kantorovich. In this dissertation it is analyze the semilocal convergence of Newton's method in Banach spaces, whose principal aim is to give greater generality to the problem of approximating the roots of a nonlinear equation by Newton's method, so that it can extend the applicability of this method to situations where the classical theory of Kantorovich cannot. To do this, we use the majorant principle, based on the construction of majorizing sequences, and, under new semilocal convergence conditions of Kantorovich-type, that allows us to generalize the classical conditions of Kantorovich. This plays an important role in the ad hoc construction of majorizing sequences from solving initial value problems. We illustrate the above with different types of nonlinear equations, highlighting the Hammerstein integral equations of mixed type and Bratu's equation, which have their origin in various real-world problems, such as it is evidenced in this dissertation