Publication: CONVERGENCIA DE SOLUCIONES PARA SISTEMAS DE SEGUNDO ORDEN DE FAMILIAS DE OPERADORES COCOERCIVOS
Date
2019
Authors
ASTUDILLO COLINA, RUBÉN URBANO
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Abstract
Esta tesis tiene como objetivo analizar el comportamiento asintótico de sistemas de segundo ordenu (t) + yu (t) + (t)A(t)u(t) = 0; t > t0 (0.1)cuando el operador A sea maximal monótono, (t) 2 C1, 2 R y la relación de estas con el conjunto deceros A-1 1f0g. A partir de esta formulación se obtiene una discretización implícita que posea resultadosanálogos(uk+1 - 2uk + uk-1)/s2 + (uk+1 - uk)/s + kAkuk+1 = 0; k > k0:Se deriva un método iterativo basado en RIPA a partir de este. Este es un resultado de importanciaen distintas áreas como optimización, teoría de equilibrio y economía.En el primer capítulo se dará una motivación al estudio de este problema. Ira acompañada por unareseña de los desarrollos que condujeron a (0.1). Se señala algunas propiedades de la formulación quela hacen atractiva a la hora de estudiar inclusiones 0 2 Ax de operadores maximales monótonos. Enel segundo capítulo se dan algunas definiciones preliminares a nuestro trabajo.El tercer capítulo contiene la formulación continua de nuestro resultado. Este está presentado medianteun lema valido para formulación más generales y un teorema específico para nuestra representación demanera de resaltar la utilidad de las hipótesis. El resultado esta acompañado por una sección con lemasclaves de la demostración. Se incluyen experimentos numéricos comparado las tasas de convergenciasde este modelo en diferentes casos.El cuarto capítulo contiene una variante discreta implícita del teorema anterior. Mostrando que lasimplementaciones numéricas preservaran el comportamiento de sus contrapartes continúas. Se adjuntauna sección de con el calculo de términos intermedios de la demostración. Se propone un método iterativobasado en RIPA [5] de implementación sencilla. Se plantean una serie de experimentos numéricoscomparado la formulación RIPA para diversos operadores y situaciones, mostrando sus fortalezas.Finalmente, en el quinto capítulo se habla un poco de las direcciones futuras de este trabajo y bibliografía
The objective of this thesis is to analyze the asymptotic behavior of the second order systemu (t) + yu (t) + (t)A(t)u(t) = 0; t > t0 (0.2)when A is a maximally monotone operator, (t) 2 C1, 2 R and its relation with set of zeros A-1 1f0g.From this, we obtain an implicit discretization that has related results(uk+1 - 2uk + uk-1)/s2 + (uk+1 - uk)/s + kAkuk+1 = 0; k > k0:We obtain an iterative method based on RIPA from this last formulation. This result has applicationof different areas such as optimization, equilibrium theory and economics.On the first chapter, we’ll give motivation to the study of this problem. It will followed by a reviewof the developments that led to (0.2). We will highlight some properties that make this formulationattractive when studying inclusions of the form 0 2 Ax on maximally monotone operators. On thesecond chapter we’ll give some definitions related to our work.The third chapter contains the continuous formulation of our result. It will be presented via a lemmathat is valid on a more general setting and a specific theorem for our formulation, highlighting theutility of the different hypothesis. We attach the necessary lemmas for the proof. We also attachnumerical experiments that compare the convergence rate of this formulation on different cases.The fourth chapter contain an implicit discrete variant of the previous theorem. Showing that thediscrete implementation preserves the behavior of their continuous counterparts. We attach a sectionwith the calculation terms participating on the demonstration. We propose a iterative method basedon RIPA [5] of simple implementation. We include a series of numerical experiments comparing theRIPA formulation for different operators and situation, showing their strengths.Finally on the fifth chapter, we speak of the future work directions and the bibliography.
The objective of this thesis is to analyze the asymptotic behavior of the second order systemu (t) + yu (t) + (t)A(t)u(t) = 0; t > t0 (0.2)when A is a maximally monotone operator, (t) 2 C1, 2 R and its relation with set of zeros A-1 1f0g.From this, we obtain an implicit discretization that has related results(uk+1 - 2uk + uk-1)/s2 + (uk+1 - uk)/s + kAkuk+1 = 0; k > k0:We obtain an iterative method based on RIPA from this last formulation. This result has applicationof different areas such as optimization, equilibrium theory and economics.On the first chapter, we’ll give motivation to the study of this problem. It will followed by a reviewof the developments that led to (0.2). We will highlight some properties that make this formulationattractive when studying inclusions of the form 0 2 Ax on maximally monotone operators. On thesecond chapter we’ll give some definitions related to our work.The third chapter contains the continuous formulation of our result. It will be presented via a lemmathat is valid on a more general setting and a specific theorem for our formulation, highlighting theutility of the different hypothesis. We attach the necessary lemmas for the proof. We also attachnumerical experiments that compare the convergence rate of this formulation on different cases.The fourth chapter contain an implicit discrete variant of the previous theorem. Showing that thediscrete implementation preserves the behavior of their continuous counterparts. We attach a sectionwith the calculation terms participating on the demonstration. We propose a iterative method basedon RIPA [5] of simple implementation. We include a series of numerical experiments comparing theRIPA formulation for different operators and situation, showing their strengths.Finally on the fifth chapter, we speak of the future work directions and the bibliography.
Description
Catalogado desde la version PDF de la tesis.
Keywords
COCOERCIVO , MAXIMAL MONOTONO , SISTEMAS DE SEGUNDO ORDEN