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

Métodos iterativos para la resolución de problemas aplicados transformados a sistemas no lineales

[ES] La resolución de ecuaciones y sistemas no lineales es un tema de gran interés teórico-práctico, pues muchos modelos matemáticos de la ciencia o de la industria se expresan mediante sistemas no lineales o ecuaciones diferenciales o integrales que, mediante técnicas de discretización, dan lugar a dichos sistemas. Dado que generalmente es difícil, si no imposible, resolver analíticamente las ecuaciones no lineales, la herramienta más extendida son los métodos iterativos, que tratan de obtener aproximaciones cada vez más precisas de las soluciones partiendo de determinadas estimaciones iniciales. Existe una variada literatura sobre los métodos iterativos para resolver ecuaciones y sistemas, que abarca conceptos como, eficiencia, optimalidad, estabilidad, entre otros importantes temas. En este estudio obtenemos nuevos métodos iterativos que mejoran algunos conocidos en términos de orden o eficiencia, es decir que obtienen mejores aproximaciones con menor coste computacional. La convergencia de los métodos iterativos suele estudiarse desde el punto de vista local. Esto significa que se obtienen resultados de convergencia imponiendo condiciones a la ecuación en un entorno de la solución. Obviamente, estos resultados no son aplicables si no la conocemos. Otro punto de vista, que abordamos en este trabajo, es el estudio semilocal que, imponiendo condiciones en un entorno de la estimación inicial, proporciona un entorno de dicho punto que contiene la solución y garantiza la convergencia del método iterativo a la misma. Finalmente, desde un punto de vista global, estudiamos el comportamiento de los métodos iterativos en función de la estimación inicial, mediante el estudio de la dinámica de las funciones racionales asociadas a estos métodos. La presente memoria recoge los resultados de varios artículos de nuestra autoría, en los que se tratan distintos aspectos de la materia, como son, las peculiaridades de la convergencia en el caso de raíces múltiples, la posibilidad de aumentar el orden de un método óptimo de orden cuatro a orden ocho, manteniendo la optimalidad en el caso de raíces múltiples, el estudio de la convergencia semilocal en un método de alto orden, así como el comportamiento dinámico de algunos métodos iterativos.[CA] La resolució d'equacions i sistemes no lineals és un tema de gran interés teoricopràctic, perquè molts models matemàtics de la ciència o de la indústria s'expressen mitjançant sistemes no lineals o equacions diferencials o integrals que, mitjançant tècniques de discretizació, donen lloc a aquests sistemes. Atés que generalment és difícil, si no impossible, resoldre analíticament les equacions no lineals, l'eina més estesa són els mètodes iteratius, que tracten d'obtindre aproximacions cada vegada més precises de les solucions partint de determinades estimacions inicials. Existeix una variada literatura sobre els mètodes iteratius per a resoldre equacions i sistemes, que abasta conceptes com ordre d'aproximació, eficiència, optimalitat, estabilitat, entre altres importants temes. En aquest estudi obtenim nous mètodes iteratius que milloren alguns coneguts en termes d'ordre o eficiència, és a dir que obtenen millors aproximacions amb menor cost computacional. La convergència dels mètodes iteratius sol estudiar-se des del punt de vista local. Això significa que s'obtenen resultats de convergència imposant condicions a l'equació en un entorn de la solució. Òbviament, aquests resultats no són aplicables si no la coneixem. Un altre punt de vista, que abordem en aquest treball, és l'estudi semilocal que, imposant condicions en un entorn de l'estimació inicial, proporciona un entorn d'aquest punt que conté la solució i garanteix la convergència del mètode iteratiu a aquesta. Finalment, des d'un punt de vista global, estudiem el comportament dels mètodes iteratius en funció de l'estimació inicial, mitjançant l'estudi de la dinàmica de les funcions racionals associades a aquests mètodes. La present memòria recull els resultats de diversos articles de la nostra autoria, en els quals es tracten diferents aspectes de la matèria, com són, les peculiaritats de la convergència en el cas d'arrels múltiples, la possibilitat d'augmentar l'ordre d'un mètode òptim d'ordre quatre a ordre huit, mantenint l'optimalitat en el cas d'arrels múltiples, l'estudi de la convergència semilocal en un mètode d'alt ordre, així com el comportament dinàmic d'alguns mètodes iteratius.[EN] The resolution of nonlinear equations and systems is a subject of great theoretical and practical interest, since many mathematical models in science or industry are expressed through nonlinear systems or differential or integral equations that, by means of discretization techniques, give rise to such systems. Since it is generally difficult, if not impossible, to solve nonlinear equations analytically, the most widely used tool is iterative methods, which try to obtain increasingly precise approximations of the solutions based on certain initial estimates. There is a varied literature on iterative methods for solving equations and systems, which covers concepts of order of approximation, efficiency, optimality, stability, among other important topics. In this study we obtain new iterative methods that improve some known ones in terms of order or efficiency, that is, they obtain better approximations with lower computational cost. The convergence of iterative methods is usually studied locally. This means that convergence results are obtained by imposing conditions on the equation in a neighbourhood of the solution. Obviously, these results are not applicable if we do not know it. Another point of view, which we address in this work, is the semilocal study that, by imposing conditions in a neighbourhood of the initial estimation, provides an environment of this point that contains the solution and guarantees the convergence of the iterative method to it. Finally, from a global point of view, we study the behaviour of iterative methods as a function of the initial estimation, by studying the dynamics of the rational functions associated with these methods. This report collects the results of several articles of our authorship, in which different aspects of the matter are dealt with, such as the peculiarities of convergence in the case of multiple roots, the possibility of increasing the order of an optimal method from order four to order eight, maintaining optimality in the case of multiple roots, the study of semilocal convergence in a high-order method, as well as the dynamic behaviour of some iterative methods.Cevallos Alarcón, FA. (2023). Métodos iterativos para la resolución de problemas aplicados transformados a sistemas no lineales [Tesis doctoral]. Universitat Politècnica de València. https://doi.org/10.4995/Thesis/10251/19349

Atomistic and ab initio prediction and optimization of thermoelectric and photovoltaic properties

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Physics, 2009.Cataloged from PDF version of thesis.Includes bibliographical references (p. 123-130).The accurate prediction of physical properties in the vast spaces of nanoscale structures and chemical compounds is made increasingly possible through the use of atomistic and ab initio computation. In this thesis we investigate lattice thermal conductivities KL and electronic band gaps E,, which are relevant to thermoelectric and photovoltaic applications, respectively, and develop or modify computational tools for predicting and optimizing these properties. For lattice thermal conductivity, we study SiGe nanostructures, which are technologically important for thermoelectric applications. From computing aL for various SiGe nanostructures, we establish that the Kubo-Green approach using classical molecular dynamics (MD) gives additional quantitative predictions not available from phenomenological models, such as the existences of a minimum value of KL as the nanostructure size is varied and of configurational dependence of KL. We carry out the minimizatin of KL in the space of atomic configurations in SiGe alloy nanowires and demonstrate the feasibility of using the cluster expansion technique to parameterize KL. We find that the use of coarse graining and a meta cluster expansion approach is effective, in conjunction with a genetic algorithm, to find configurations which drastically lower KL. The low values of KL obtained, close to the bulk amorphous limit, are due to the absence of long-range order, and such absence allows a local cluster expansion approach to optimize KL. We examine ab initio bandgap prediction for semiconductor compounds, and address the large errors of Kohn-Sham band gaps in density functional theory (DFT).(cont.) We apply corrections using the self-energy approach in the GW approximation, which includes non-local screened exchange and correlation, and find that the G₀W₀ approximation significantly reduces prediction errors compared to Kohn-Sham band gaps, though at much higher computational cost. We propose a new method involving total energies in DFT to predict the fundamental gap, by use of the properties of the screening or exchange-correlation hole in an electron gas. With this method, we are able to efficiently predict band gaps that are in agreement with experimental values.by Maria Kai Yee Chan.Ph.D

SciCADE 95: International conference on scientific computation and differential equations

This report consists of abstracts from the conference. Topics include algorithms, computer codes, and numerical solutions for differential equations. Linear and nonlinear as well as boundary-value and initial-value problems are covered. Various applications of these problems are also included

Evaluation of process systems operating envelopes

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Chemical Engineering, 2013.This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.Cataloged from student-submitted PDF version of thesis.Includes bibliographical references (p. 229-238).This thesis addresses the problem of worst-case steady-state design of process systems under uncertainty, also known as robust design. Designing for the worst case is of great importance when considering systems for deployment in extreme and hostile environments, where operational failures cannot be risked due to extraordinarily high economic and/or environmental expense. For this unique scenario, the cost of "over-designing" the process far outweighs the cost associated with operational failure. Hence, it must be guaranteed that the process is sufficiently robust in order to avoid operational failures. Many engineering, economic, and operations research applications are concerned with worst-case scenarios. Classically, these problems give rise to a type of leader-follower game, or Stackelberg game, commonly known as the "minimax" problem, or more precisely as a max-min or min-max optimization problem. However, since the application here is to steady-state design, the problem formulation results in a more general nonconvex equality-constrained min-max program, for which no previously available algorithm can solve effectively. Under certain assumptions, the equality constraints, which correspond to the steady-state model, can be eliminated from the problem by solving them for the state variables as implicit functions of the control variables and uncertainty parameters. This approach eliminates explicit functional dependence on the state variables, and in turn reduces the dimensionality of the original problem. However, this embeds implicit functions in the program, which have no explicit algebraic form and can only be approximated using numerical methods. By doing this, the max-min program can be reformulated as a more computationally tractable semi-infinite program, with the caveat that there are embedded implicit functions. Semi-infinite programming with embedded implicit functions is a new approach to modeling worst-case design problems. Furthermore, modeling process systems--especially those associated with chemical engineering--often results in highly nonconvex functions. The primary contribution of this thesis is a mathematical tool for solving implicit semi-infinite programs and assessing robust feasibility of process systems using a rigorous model-based approach. This tool has the ability to determine, with mathematical certainty, whether or not a physical process system based on the proposed design will fail in the worst case by taking into account uncertainty in the model parameters and uncertainty in the environment.by Matthew David Stuber.Ph.D

Toward Realistic DFT Description of Complex Systems: Ethylene Epoxidation on Ag-Cu Alloys and RPA Correlation in van der Waals Molecules

In this thesis we have studied two different aspects of Density Functional Theory (DFT): (i) the application of DFT with the generalized gradient approximation (GGA) functional for exchange-correlation energy in modeling an heterogeneous catalysis problem, and (ii) the development of a new self-consistent field (scf) strategy to solve the Kohn-Sham (KS) equations that allows to improve the accuracy of DFT method with exact exchange (EXX) and RPA correlation energy functionals in the description of weak chemical interactions. Ethylene epoxidation, one of the largest-scale catalytic processes in the chemical industry, were studied in Chapter 2 of this thesis. The formation of the desired product ethylene oxide (EO) in this reaction is promoted by a Ag-Cu alloy catalyst. In this study, the oxidation of ethylene is considered to occur on the Ag-Cu structures formed by thin copper-oxide layers on an Ag slab. These structures have been determined by theoretical and experimental works to be the favorable structures on the surfaces of Ag-Cu alloys in the high pressure and temperature conditions relevant to experiment. According to the calculations for reaction pathways, we found that the structures of Ag-Cu alloys are selective towards the formation of the EO final product, rather than the undesired product acetaldehyde (Ac) which is readily converted to carbon dioxide. The selectivity of Ag-Cu alloys is found to be higher than pure Ag, in agreement with experimental results. To do this, we carried out a study of the stability of the surface structures in thermodynamic equilibrium conditions ( at T = 600 K and pO2 = 1 atm), and we have shown that the higher selectivities relate to the formation of copper-oxide layers on the Ag slab. Moreover, our theoretical results show that the high selectivity of a copperoxide layer is maintained even when the thickness of the oxide is increased to two layers. In particular, we have found that a very high selectivity could be obtained by structure containing 1.25 ML of Cu and 0.25 ML of sub-surface oxygen. Another important result is the finding of a selectivity indicator that allows to determine the selectivity of the pure metals and alloy catalysts even with the thin oxide structures in ethylene epoxidation reaction. In further works, this indicator could be applied to predict the selectivity of other Ag-based alloys such as Ag-Pd, Ag-Pt, etc. These alloys were found experimentally to be selective catalysts towards the formation of EO. In spite of the great success of DFT when employing the well-known approximations such as LDA or GGA exchange-correlation functionals, the standard DFT approaches exhibit several serious shortcomings, and one of them is the poor or even wrong evaluation of long-range dispersion interactions (i.e., van der Waals interactions). Calculations with the EXX/RPA-correlation energy within the adiabatic connection uctuation-dissipation theorem (ACFDT) formalism have shown as a promising approach that can give the correct description not only of weak bonds but also of systems with covalent bonds. In Chapter 3, we developed the complete scf procedure that enables the optimization of KS systems whose total energy is computed with the EXX/RPA-correlation energy functionals. The implementation has been applied to the study of some simple molecules. In future work, EXX/RPA calculations could be applied to heterogeneous catalysis systems, where the role of van der Waals interactions is still largely unknown. Moreover, improvement of the accuracy of EXX/RPA calculations is also needed. According to ACFDT, one can go beyond the RPA formalism by taking into account higher-level approximations of the exchange-correlation kernel in the Dyson equation such as the time-dependent EXX kernel

MS FT-2-2 7 Orthogonal polynomials and quadrature: Theory, computation, and applications

Quadrature rules find many applications in science and engineering. Their analysis is a classical area of applied mathematics and continues to attract considerable attention. This seminar brings together speakers with expertise in a large variety of quadrature rules. It is the aim of the seminar to provide an overview of recent developments in the analysis of quadrature rules. The computation of error estimates and novel applications also are described

Generalized averaged Gaussian quadrature and applications

A simple numerical method for constructing the optimal generalized averaged Gaussian quadrature formulas will be presented. These formulas exist in many cases in which real positive GaussKronrod formulas do not exist, and can be used as an adequate alternative in order to estimate the error of a Gaussian rule. We also investigate the conditions under which the optimal averaged Gaussian quadrature formulas and their truncated variants are internal

Theoretical prediction of solubility, reactivity and degradation pathways of selected azo disperse dyes

Abstract :Ph.D. (Chemistry