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1
Nikazad, Touraj. "Algebraic Reconstruction Methods." Doctoral thesis, Linköping : Linköpings universitet, Department of Mathematics Scientific Computing, 2008. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-11670.
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Mohammadpour, Rahman. "New methods in forcing iteration and applications." Thesis, Université de Paris (2019-....), 2020. http://www.theses.fr/2020UNIP7050.
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Dans cette thèse on considère l’itération de forcing en utilisant des modèles virtuels comme conditions latérales. Le but ultime de ces techniques est de trouver un axiome de forcing supérieur. Dans le premier chapitre, nous présentons les matériaux nécessaires, y compris les définitions et les lemmes pour les chapitres suivants. Le deuxième chapitre contient quelques constructions avec des conditions latérales appelées scaffolding poset; c’est un échauffement pour les constructions compliquées des chapitres suivants. La notion de modèle virtuel et ses propriétés sont introduites et étudiées en détail dans le troisième chapitre, où nous étudions également la manière dont les modèles virtuels de différents types interagissent. Nous introduisons ensuite dans le chapitre quatre la notion de forcing qui consiste à les conditions latérales pures qui sont des ensembles finis de modèles dénombrables et de modèles Magidor. Dans le chapitre cinq, nous avons intégré des forcings dans la construction du chapitre quatre pour former une itération, nous analysons les propriétés de l’itération et de ses quotients par des modèle Magidors, par exemple la propriété de ω1-approximation. L’itération donne en effet un axiome de forcing pour une certaine classe de forcings propres qui est compatible avec 2ℵ0 > @2. Le dernier chapitre est consacré à l’étude des modèles d’estimation, nous introduisons certains principes combinatoires en termes de modèles de devinettes qui peuvent être considérés comme les conséquences d’un axiome de forcing supérieur. Nous montrons leur cohérence et énonçons leurs conséquences concernant l’idéal des points approchables, le principe de maximalité d’Abraham etcThis thesis concerns forcing iterations using virtual models as side conditions. The ultimate goal of such techniques is to achieve a higher forcing axiom. In the firstchapter, we present the necessary materials, including definitions and lemmata for the later chapters. The chapter two contains the scaffolding poset which is a warmup for the later constructions. The notion of a virtual model and its properties are introduced and investigated extensively in the third chapter, where we also study how the virtual models of different types interact. We then introduce, in the fourth chapter, the forcing notion consisting of pure side conditions which are finite sets of countable virtual models and Magidor models. In the chapter five, we plug forcings in our construction from the fourth chapter to form an iteration using virtual models, we analyze properties of our iteration and its quotients by Magidor models suchas the ω1-approximation. The iteration indeed gives a forcing axiom for a certain class of proper forcings which is compatible with 2ℵ0 > @2. The chapter six is devotedto the study of guessing models and their specialization, we introduce certain combinatorial principles in terms of guessing models which can be considered as consequencesof a higher forcing axiom. We shall show their consistency and state their consequences concerning the approachability ideal, Abraham’s maximality principle etc
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Garner, William Howard. "Iteration of the power operation." Virtual Press, 1995. http://liblink.bsu.edu/uhtbin/catkey/941367.
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This thesis is an investigation of the sequence of functions defined by fl (x) -xand fn+1 (x) -x , where the power is the principal value.In the case where the sequence is restricted to positive real this sequence of functions over thecomplex plane, we attack real numbers, the problem yields to the methods of analysis and we prove the behavior of the sequence.The more general problem of describing the behavior of both analytically and numerically. Though no full rigorous solution is given, the results presented suggest the behavior of the sequence over the complex plane is very interesting.Department of Mathematical Sciences
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Vogelgesang, Jonas [Verfasser]. "Semi-discrete iteration methods in x-ray tomography / Jonas Vogelgesang." Saarbrücken : Saarländische Universitäts- und Landesbibliothek, 2020. http://d-nb.info/1219068713/34.
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Freitag, Melina. "Inner-outer iterative methods for eigenvalue problems : convergence and preconditioning." Thesis, University of Bath, 2007. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.512248.
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Many methods for computing eigenvalues of a large sparse matrix involve shift-invert transformations which require the solution of a shifted linear system at each step. This thesis deals with shift-invert iterative techniques for solving eigenvalue problems where the arising linear systems are solved inexactly using a second iterative technique. This approach leads to an inner-outer type algorithm. We provide convergence results for the outer iterative eigenvalue computation as well as techniques for efficient inner solves. In particular eigenvalue computations using inexact inverse iteration, the Jacobi-Davidson method without subspace expansion and the shift-invert Arnoldi method as a subspace method are investigated in detail. A general convergence result for inexact inverse iteration for the non-Hermitian generalised eigenvalue problem is given, using only minimal assumptions. This convergence result is obtained in two different ways; on the one hand, we use an equivalence result between inexact inverse iteration applied to the generalised eigenproblem and modified Newton's method; on the other hand, a splitting method is used which generalises the idea of orthogonal decomposition. Both approaches also include an analysis for the convergence theory of a version of inexact Jacobi-Davidson method, where equivalences between Newton's method, inverse iteration and the Jacobi-Davidson method are exploited. To improve the efficiency of the inner iterative solves we introduce a new tuning strategy which can be applied to any standard preconditioner. We give a detailed analysis on this new preconditioning idea and show how the number of iterations for the inner iterative method and hence the total number of iterations can be reduced significantly by the application of this tuning strategy. The analysis of the tuned preconditioner is carried out for both Hermitian and non-Hermitian eigenproblems. We show how the preconditioner can be implemented efficiently and illustrate its performance using various numerical examples. An equivalence result between the preconditioned simplified Jacobi-Davidson method and inexact inverse iteration with the tuned preconditioner is given. Finally, we discuss the shift-invert Arnoldi method both in the standard and restarted fashion. First, existing relaxation strategies for the outer iterative solves are extended to implicitly restarted Arnoldi's method. Second, we apply the idea of tuning the preconditioner to the inner iterative solve. As for inexact inverse iteration the tuned preconditioner for inexact Arnoldi's method is shown to provide significant savings in the number of inner solves. The theory in this thesis is supported by many numerical examples.
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Safoutin, Michael John. "A methodology for empirical measurement of iteration in engineering design processes /." Thesis, Connect to this title online; UW restricted, 2003. http://hdl.handle.net/1773/7111.
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Scheben, Fynn. "Iterative methods for criticality computations in neutron transport theory." Thesis, University of Bath, 2011. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.545319.
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This thesis studies the so-called “criticality problem”, an important generalised eigenvalue problem arising in neutron transport theory. The smallest positive real eigenvalue of the problem contains valuable information about the status of the fission chain reaction in the nuclear reactor (i.e. the criticality of the reactor), and thus plays an important role in the design and safety of nuclear power stations. Because of the practical importance, efficient numerical methods to solve the criticality problem are needed, and these are the focus of this thesis. In the theory we consider the time-independent neutron transport equation in the monoenergetic homogeneous case with isotropic scattering and vacuum boundary conditions. This is an unsymmetric integro-differential equation in 5 independent variables, modelling transport, scattering, and fission, where the dependent variable is the neutron angular flux. We show that, before discretisation, the nonsymmetric eigenproblem for the angular flux is equivalent to a related eigenproblem for the scalar flux, involving a symmetric positive definite weakly singular integral operator(in space only). Furthermore, we prove the existence of a simple smallest positive real eigenvalue with a corresponding eigenfunction that is strictly positive in the interior of the reactor. We discuss approaches to discretise the problem and present discretisations that preserve the underlying symmetry in the finite dimensional form. The thesis then describes methods for computing the criticality in nuclear reactors, i.e. the smallest positive real eigenvalue, which are applicable for quite general geometries and physics. In engineering practice the criticality problem is often solved iteratively, using some variant of the inverse power method. Because of the high dimension, matrix representations for the operators are often not available and the inner solves needed for the eigenvalue iteration are implemented by matrix-free inneriterations. This leads to inexact iterative methods for criticality computations, for which there appears to be no rigorous convergence theory. The fact that, under appropriate assumptions, the integro-differential eigenvalue problem possesses an underlying symmetry (in a space of reduced dimension) allows us to perform a systematic convergence analysis for inexact inverse iteration and related methods. In particular, this theory provides rather precise criteria on how accurate the inner solves need to be in order for the whole iterative method to converge. The theory is illustrated with numerical examples on several test problems of physical relevance, using GMRES as the inner solver. We also illustrate the use of Monte Carlo methods for the solution of neutron transport source problems as well as for the criticality problem. Links between the steps in the Monte Carlo process and the underlying mathematics are emphasised and numerical examples are given. Finally, we introduce an iterative scheme (the so-called “method of perturbation”) that is based on computing the difference between the solution of the problem of interest and the known solution of a base problem. This situation is very common in the design stages for nuclear reactors when different materials are tested, or the material properties change due to the burn-up of fissile material. We explore the relation ofthe method of perturbation to some variants of inverse iteration, which allows us to give convergence results for the method of perturbation. The theory shows that the method is guaranteed to converge if the perturbations are not too large and the inner problems are solved with sufficiently small tolerances. This helps to explain the divergence of the method of perturbation in some situations which we give numerical examples of. We also identify situations, and present examples, in which the method of perturbation achieves the same convergence rate as standard shifted inverse iteration. Throughout the thesis further numerical results are provided to support the theory.
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Chidume, Chukwudi Soares de Souza Geraldo. "Iteration methods for approximation of solutions of nonlinear equations in Banach spaces." Auburn, Ala., 2008. http://repo.lib.auburn.edu/EtdRoot/2008/SUMMER/Mathematics_and_Statistics/Dissertation/Chidume_Chukwudi_33.pdf.
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Chand, Manoj. "Development of Efficient Numerical Methods for Solving Differential Equations using He's Variational Iteration Technique." Thesis, University of Louisiana at Lafayette, 2014. http://pqdtopen.proquest.com/#viewpdf?dispub=1553885.
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Differential equations play a prominent role in engineering and research fields in modeling engineering structures, describing important phenomena, and simulating mathematical behavior of engineering dynamical systems. Because of the increasing complexity of modern engineering systems, computationally efficient methods are demanded for solving these differential equations. In order to meet this challenge, this thesis presents two efficient algorithms for solving two types of differential equations: a one-dimensional heat equation with variable properties, and a one-dimensional parabolic equation, both of which are very popular and important in current engineering systems. In this study, the two equations were successfully solved using He’s variational iteration technique, and efficient algorithms have been developed. Detailed procedures for developing these algorithms are presented.
At first, a unique algorithm for solving the one-dimensional heat equations was developed by using the iteration variational approach. The accuracy of this algorithm was found by comparing the obtained solutions with the exact ones.
And similarly, using variational iteration approach, another efficient algorithm for solving the one-dimensional parabolic equation was developed. Three illustrative numerical problems were solved and the obtained results were compared with those yielded from the Adomian decomposition method (ADM) to verify the efficiency and accuracy of the developed algorithm.
With the encouraging results obtained from this study, it is expected that, in the future, developed algorithms can be extended to solve other differential equation systems, thus achieving a broader applicability in engineering and other research fields.
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Lohaka, Hippolyte O. "MAKING A GROUPED-DATA FREQUENCY TABLE: DEVELOPMENT AND EXAMINATION OF THE ITERATION ALGORITHM." Ohio : Ohio University, 2007. http://www.ohiolink.edu/etd/view.cgi?ohiou1194981215.
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Tiwari, Abhishek. "ANALYTICAL METHODS FOR TRANSPORT EQUATIONS IN SIMILARITY FORM." UKnowledge, 2007. http://uknowledge.uky.edu/gradschool_theses/457.
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We present a novel approach for deriving analytical solutions to transport equations expressedin similarity variables. We apply a fixed-point iteration procedure to these transformedequations by formally solving for the highest derivative term and then integrating to obtainan expression for the solution in terms of a previous estimate. We are able to analyticallyobtain the Lipschitz condition for this iteration procedure and, from this (via requirements forconvergence given by the contraction mapping principle), deduce a range of values for the outerlimit of the solution domain, for which the fixed-point iteration is guaranteed to converge.
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Junkermeier, Chad Everett. "Iteration Methods For Approximating The Lowest Order Energy Eigenstate of A Given Symmetry For One- and Two-Dimensional Systems." BYU ScholarsArchive, 2003. https://scholarsarchive.byu.edu/etd/85.
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Using the idea that a quantum mechanical system drops to its ground state as its temperature goes to absolute zero several operators are devised to enable the approximation of the lowest order energy eigenstate of a given symmetry; as well as an approximation to the energy eigenvalue of the same order.
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Massa, Julio Cesar. "Acceleration of convergence in solving the eigenvalue problem by matrix iteration using the power method." Thesis, Virginia Polytechnic Institute and State University, 1985. http://hdl.handle.net/10919/101452.
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A modification of the matrix iteration using the power method, in conjunction with Hotelling deflation, for the solution of the problem K.x = ω².M.x is here proposed. The problem can be written in the form D.x =λ.x, and the modification consists of raising the matrix D to an appropriate power p before carrying out the iteration process.
The selection of a satisfactory value of p is investigated, based on the spacing between the eigenvalues. The effect of p on the accuracy of the results is also discussed.M.S.
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Ali, Ali Hasan. "Modifying Some Iterative Methods for Solving Quadratic Eigenvalue Problems." Wright State University / OhioLINK, 2017. http://rave.ohiolink.edu/etdc/view?acc_num=wright1515029541712239.
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Archontakis, Ioannis Stylianos. "Agile development in the video game industry : Examining the effects of iteration and methods of limiting it." Thesis, Umeå universitet, Företagsekonomi, 2019. http://urn.kb.se/resolve?urn=urn:nbn:se:umu:diva-156211.
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This research is examining one of the most dominant managerial methods used in development in the video game industry, Agile development. More particularly, the thesis examines a certain attribute of Agile development, that of iteration. The thesis will set to examine how iteration affects several layers of development during the production of a video game and whether it can be replaced by other managerial technics.As a result, the purpose of this thesis is to raise a different viewpoint against the Agile’s iteration. Furthermore, this thesis aims to contribute to the academic research by concentrating on the video game industry, an industry that is often neglected by the academia.The theoretical framework and literature review concentrate on concepts of Agile development, overworking, development cycle in video games, definitions of project success and project failures and creative process in video game development.The thesis deploys qualitative methodology to address and research its data. The collected data belongs to two categories, data stemming from interviews conducted by the thesis’s author and data stemming from journalistic magazines.The results of both type of data are compared and act supplementary to each other, then they are analyzed to answer the research questions asked by this thesis. The results showcase that iteration has negative effects to video game developers in both a macroscale (company’s resources, annual revenue) and in a microscale (overworking, health issues) level. The results also highlight that Agile is an all-time favorite development methodology of developers in the video game industry.In conclusion, the thesis supports the notion that iteration should be suppressed and proposes a number of solutions for that matter. The suggestions are essentially encouragement towards developers: to seek higher interactivity with customers throughout the duration of all the development stages of a video game, to show more trust to established gameplay mechanics and to place more reliance on a franchise’s profit power and benefits. These measures can be used in a preventive manner in order to limit the appearance of iteration and as a result, to limit its’ negative effects.
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Rogozhin, Alexander. "Approximation Methods for Two Classes of Singular Integral Equations." Doctoral thesis, Universitätsbibliothek Chemnitz, 2003. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200300091.
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The dissertation consists of two parts. In the first part approximate methods for multidimensional weakly singular integral operators with operator-valued kernels are investigated. Convergence results and error estimates are given. There is considered an application of these methods to solving radiation transfer problems. Numerical results are presented, too. In the second part we consider a polynomial collocation method for the numerical solution of a singular integral equation over the interval. More precisely, the operator of our integral equation is supposed to be of the form \ $aI + b \mu^{-1} S \mu I $\ with \ $S$\ the Cauchy singular integral operator, with piecewise continuous coefficients \ $a$\ and \ $b,$\ and with a Jacobi weight \ $\mu.$\ To the equation we apply a collocation method, where the collocation points are the Chebyshev nodes of the first kind and where the trial space is the space of polynomials multiplied by another Jacobi weight. For the stability and convergence of this collocation method in weighted \ $L^2$\ spaces, we derive necessary and sufficient conditions. Moreover, the extension of these results to an algebra generated by the sequences of the collocation method applied to the mentioned singular integral operators is discussed and the behaviour of the singular values of the discretized operators is investigatedDie Dissertation beschäftigt sich insgesamt mit der numerischen Analysis singulärer Integralgleichungen, besteht aber aus zwei voneinander unabhängigen Teilen. Der este Teil behandelt Diskretisierungsverfahren für mehrdimensionale schwach singuläre Integralgleichungen mit operatorwertigen Kernen. Darüber hinaus wird hier die Anwendung dieser allgemeinen Resultate auf ein Strahlungstransportproblem diskutiert, und numerische Ergebnisse werden präsentiert. Im zweiten Teil betrachten wir ein Kollokationsverfahren zur numerischen Lösung Cauchyscher singulärer Integralgleichungen auf Intervallen. Der Operator der Integralgleichung hat die Form \ $aI + b \mu^{-1} S \mu I $\ mit dem Cauchyschen singulären Integraloperator \ $S,$\ mit stückweise stetigen Koeffizienten \ $a$\ und \ $b,$\ und mit einem klassischen Jacobigewicht \ $\mu.$\ Als Kollokationspunkte dienen die Nullstellen des n-ten Tschebyscheff-Polynoms erster Art und Ansatzfunktionen sind ein in einem geeigneten Hilbertraum orthonormales System gewichteter Tschebyscheff-Polynome zweiter Art. Wir erhalten notwendige und hinreichende Bedingungen für die Stabilität und Konvergenz dieses Kollokationsverfahrens. Außerdem wird das Stabilitätskriterium auf alle Folgen aus der durch die Folgen des Kollokationsverfahrens erzeugten Algebra erweitert. Diese Resultate liefern uns Aussagen über das asymptotische Verhalten der Singulärwerte der Folge der diskreten Operatoren
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Dinh-Truong, Thuy Linh. "An efficient algorithm combining cell multipole and multigrid methods for rapid evaluation of dipole iteration in polarizable force fields." Diss., Connect to a 24 p. preview or request complete full text in PDF format. Access restricted to UC campuses, 2007. http://wwwlib.umi.com/cr/ucsd/fullcit?p3238427.
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Thesis (Ph. D.)--University of California, San Diego, 2007.Title from first page of PDF file (viewed January 4, 2007). Available via ProQuest Digital Dissertations. Vita. Includes bibliographical references (p. 142-164).
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Junkermeier, Chad E. "Iteration methods for approximating the lowest order energy eigenstate of a given symmetry for one- and two-dimensional systems /." Diss., CLICK HERE for online access, 2003. http://contentdm.lib.byu.edu/ETD/image/etd226.pdf.
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Hua, Xiaoqin. "Studies on block coordinate gradient methods for nonlinear optimization problems with separable structure." 京都大学 (Kyoto University), 2015. http://hdl.handle.net/2433/199447.
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Rogozhin, Alexander. "Approximation methods for two classes of singular integral equations." Doctoral thesis, [S.l. : s.n.], 2002. http://deposit.ddb.de/cgi-bin/dokserv?idn=968783279.
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Penzl, T. "A cyclic low rank Smith method for large, sparse Lyapunov equations with applications in model reduction and optimal control." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199801035.
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We present a new method for the computation of low rank approximations
to the solution of large, sparse, stable Lyapunov equations. It is based
on a generalization of the classical Smith method and profits by the
usual low rank property of the right hand side matrix.
The requirements of the method are moderate with respect to both
computational cost and memory.
Hence, it provides a possibility to tackle large scale control
problems.
Besides the efficient solution of the matrix equation itself,
a thorough integration of the method into several control
algorithms can improve their performance
to a high degree.
This is demonstrated for algorithms
for model reduction and optimal control.
Furthermore, we propose a heuristic for determining a set of
suboptimal ADI shift parameters. This heuristic, which is based on a
pair of Arnoldi processes, does not require any a priori
knowledge on the spectrum of
the coefficient matrix of the Lyapunov equation.
Numerical experiments show the efficiency of the iterative scheme
combined with the heuristic for the ADI parameters.
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Palladino, Fabio Henrique. "Reconstrução 3D de imagens em tomografia por emissão de pósitrons com Câmaras de Cintilação." Universidade de São Paulo, 2004. http://www.teses.usp.br/teses/disponiveis/43/43134/tde-07032014-160312/.
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A Tomografia por Emissão de Pósitrons (PET) está se definindo como um dos métodos preferidos para diagnóstico e seguimento de inúmeras doenças em Oncologia, Neurologia e Cardiologia. Esta modalidade é realizada com sistemas dedicados e sistemas baseados em câmaras de cintilação, que podem ser também usados em tomografia por emissão de fótons únicos (SPECT). Neste trabalho, efetuamos uma avaliação dos fatores que favorecem a quantificação em imagens PET com câmaras de cintilação em coincidência, caracterizadas por urna menor sensibilidade em relação a sistemas dedicados. Avaliamos as condições de quantificação de imagens sob os modos 2D e 3D de aquisição, obtidas por métodos de reconstrução 2D e 3D diversos e correções associadas. Dados de aquisição foram simulados por método de Monte Carlo empregando parâmetros realistas. Objetos de interesse diversos foram modelados. Imagens foram reconstruídas pelos métodos FBP, ART, MLEM e OSEM e consideramos correções de sensibilidade, normalização de detector, espalhamento e atenuação de radiação. Estabelecemos uma metodologia de avaliação de detectabilidade e recuperação de contrastes em imagens que contemplam, a partir de dois parâmetros mensuráveis, os aspectos mais relevantes em quantificação. Análises visuais também foram consideradas. Verificamos que o modo 3D é mais adequado que 2D na recuperação de baixos contrastes no objeto com a aplicação das correções selecionadas. A detectabilidade de pequenas estruturas está limitada pelos efeitos de volume parcial e pela resolução espacial finita dos sistemas de detecção. Os métodos ART, MLEM e, em particular, OSEM com 8 subconjuntos, apresentam-se adequados para estudos quantitativos no modo 3D. Os parâmetros definidos para avaliação podem ser empregados como indicadores de condições propícias a estudos quantitativos.Volumetric reconstruction in gamma camera based PET imaging Positron Emission Tomography (PET) is considered as a very useful tool for diagnosing and following several diseases in Oncology, Neurology and Cardiology. Two types of systems are available for this imaging modality: the dedicated systems and those based on gamma camera technology. In this work, we assessed a number of factors affecting the quantitation of gamma camera based PET imaging, characterized by a lower sensitivity compared to those of dedicated systems. We also evaluated image quantitation conditions under 2D and 3D acquisition/reconstruction modes, for different reconstruction methods and associated corrections. Acquisition data were simulated by Monte Carla method, using realistic parameters. Several objects of interest were modelled. We reconstructed slices and volumes using FBP, ART, MLEM and OSEM and also included four corrections: detector sensitivity, detector normalization, scatter and attenuation of annihilation photons. We proposed a method to assess detectability and object contrast recovery by using two measurable parameters. Visual analysis was also considered. We found that 3D mode is more effective than 2D for low contrast recovery when the selected (J corrections are applied. Detectability of small structures is limited by partial volume effects and device finite spatial resolution. ART, MLEM and specially 8-subsets OSEM are the most adequate methods for quantitative studies in 3D mode. The parameter that we have defined may also be used as indicators of suitable conditions for quantitation in images.
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Detournay, Sylvie. "Méthodes multigrilles pour les jeux stochastiques à deux joueurs et somme nulle, en horizon infini." Phd thesis, Ecole Polytechnique X, 2012. http://pastel.archives-ouvertes.fr/pastel-00762010.
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Dans cette thèse, nous proposons des algorithmes et présentons des résultats numériques pour la résolution de jeux répétés stochastiques, à deux joueurs et somme nulle dont l'espace d'état est de grande taille. En particulier, nous considérons la classe de jeux en information complète et en horizon infini. Dans cette classe, nous distinguons d'une part le cas des jeux avec gain actualisé et d'autre part le cas des jeux avec gain moyen. Nos algorithmes, implémentés en C, sont principalement basés sur des algorithmes de type itérations sur les politiques et des méthodes multigrilles. Ces algorithmes sont appliqués soit à des équations de la programmation dynamique provenant de problèmes de jeux à deux joueurs à espace d'états fini, soit à des discrétisations d'équations de type Isaacs associées à des jeux stochastiques différentiels. Dans la première partie de cette thèse, nous proposons un algorithme qui combine l'algorithme des itérations sur les politiques pour les jeux avec gain actualisé à des méthodes de multigrilles algébriques utilisées pour la résolution des systèmes linéaires. Nous présentons des résultats numériques pour des équations d'Isaacs et des inéquations variationnelles. Nous présentons également un algorithme d'itérations sur les politiques avec raffinement de grilles dans le style de la méthode FMG. Des exemples sur des inéquations variationnelles montrent que cet algorithme améliore de façon non négligeable le temps de résolution de ces inéquations. Pour le cas des jeux avec gain moyen, nous proposons un algorithme d'itération sur les politiques pour les jeux à deux joueurs avec espaces d'états et d'actions finis, dans le cas général multichaine (c'est-à-dire sans hypothèse d'irréductibilité sur les chaînes de Markov associées aux stratégies des deux joueurs). Cet algorithme utilise une idée développée dans Cochet-Terrasson et Gaubert (2006). Cet algorithme est basé sur la notion de projecteur spectral non-linéaire d'opérateurs de la programmation dynamique de jeux à un joueur (lequel est monotone et convexe). Nous montrons que la suite des valeurs et valeurs relatives satisfont une propriété de monotonie lexicographique qui implique que l'algorithme termine en temps fini. Nous présentons des résultats numériques pour des jeux discrets provenant d'une variante des jeux de Richman et sur des problèmes de jeux de poursuite. Finalement, nous présentons de nouveaux algorithmes de multigrilles algébriques pour la résolution de systèmes linéaires singuliers particuliers. Ceux-ci apparaissent, par exemple, dans l'algorithme d'itérations sur les politiques pour les jeux stochastiques à deux joueurs et somme nulle avec gain moyen, décrit ci-dessus. Nous introduisons également une nouvelle méthode pour la recherche de mesures invariantes de chaînes de Markov irréductibles basée sur une approche de contrôle stochastique. Nous présentons un algorithme qui combine les itérations sur les politiques d'Howard et des itérations de multigrilles algébriques pour les systèmes linéaires singuliers.
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Wilkins, Bryce Daniel. "The E² Bathe subspace iteration method." Thesis, Massachusetts Institute of Technology, 2019. https://hdl.handle.net/1721.1/122238.
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Thesis: S.M., Massachusetts Institute of Technology, Department of Mechanical Engineering, 2019Cataloged from PDF version of thesis.
Includes bibliographical references (pages 91-93).
Since its development in 1971, the Bathe subspace iteration method has been widely-used to solve the generalized symmetric-definite eigenvalue problem. The method is particularly useful for solving large eigenvalue problems when only a few of the least dominant eigenpairs are sought. In reference [18], an enriched subspace iteration method was proposed that accelerated the convergence of the basic method by replacing some of the iteration vectors with more effective turning vectors. In this thesis, we build upon this recent acceleration effort and further enrich the subspace of each iteration by replacing additional iteration vectors with our new turning-of-turning vectors. We begin by reviewing the underpinnings of the subspace iteration methodology. Then, we present the steps of our new algorithm, which we refer to as the Enriched- Enriched (E2 ) Bathe subspace iteration method. This is followed by a tabulation of the number of floating point operations incurred during a general iteration of the E2 algorithm. Additionally, we perform a simplified convergence analysis showing that the E2 method converges asymptotically at a faster rate than the enriched method. Finally, we examine the results from several test problems that were used to illustrate the E2 method and to assess its potential computational savings compared to the enriched method. The sample results for the E2 method are consistent with the theoretical asymptotic convergence rate that was obtained in our convergence analysis. Further, the results from the CPU time tests suggest that the E2 method can often provide a useful reduction in computational effort compared to the enriched method, particularly when relatively few iteration vectors are used in comparison with the number of eigenpairs that are sought.
by Bryce Daniel Wilkins.
S.M.
S.M. Massachusetts Institute of Technology, Department of Mechanical Engineering
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Yan, Shu. "Efficient numerical methods for capacitance extraction based on boundary element method." Texas A&M University, 2005. http://hdl.handle.net/1969.1/3230.
Full textAbstract:
Fast and accurate solvers for capacitance extraction are needed by the VLSI industry
in order to achieve good design quality in feasible time. With the development
of technology, this demand is increasing dramatically. Three-dimensional capacitance
extraction algorithms are desired due to their high accuracy. However, the present
3D algorithms are slow and thus their application is limited. In this dissertation, we
present several novel techniques to significantly speed up capacitance extraction algorithms
based on boundary element methods (BEM) and to compute the capacitance
extraction in the presence of floating dummy conductors.
We propose the PHiCap algorithm, which is based on a hierarchical refinement
algorithm and the wavelet transform. Unlike traditional algorithms which result in
dense linear systems, PHiCap converts the coefficient matrix in capacitance extraction
problems to a sparse linear system. PHiCap solves the sparse linear system iteratively,
with much faster convergence, using an efficient preconditioning technique. We also
propose a variant of PHiCap in which the capacitances are solved for directly from a
very small linear system. This small system is derived from the original large linear
system by reordering the wavelet basis functions and computing an approximate LU
factorization. We named the algorithm RedCap. To our knowledge, RedCap is the
first capacitance extraction algorithm based on BEM that uses a direct method to solve a reduced linear system.
In the presence of floating dummy conductors, the equivalent capacitances among
regular conductors are required. For floating dummy conductors, the potential is unknown
and the total charge is zero. We embed these requirements into the extraction
linear system. Thus, the equivalent capacitance matrix is solved directly. The number
of system solves needed is equal to the number of regular conductors.
Based on a sensitivity analysis, we propose the selective coefficient enhancement
method for increasing the accuracy of selected coupling or self-capacitances with
only a small increase in the overall computation time. This method is desirable
for applications, such as crosstalk and signal integrity analysis, where the coupling
capacitances between some conductors needs high accuracy. We also propose the
variable order multipole method which enhances the overall accuracy without raising
the overall multipole expansion order. Finally, we apply the multigrid method to
capacitance extraction to solve the linear system faster.
We present experimental results to show that the techniques are significantly
more efficient in comparison to existing techniques.
26
Galbraith, Steven Douglas. "Iterations of elliptic curves." Thesis, Georgia Institute of Technology, 1991. http://hdl.handle.net/1853/28620.
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Choi, Yan-yu. "Residual Julia sets of Newton's maps and Smale's problems on the efficiency of Newton's method." Click to view the E-thesis via HKUTO, 2006. http://sunzi.lib.hku.hk/hkuto/record/B37680948.
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Choi, Yan-yu, and 蔡欣榆. "Residual Julia sets of Newton's maps and Smale's problems on the efficiency of Newton's method." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2006. http://hub.hku.hk/bib/B37680948.
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Karelius, Fanny. "Stationary iterative methods : Five methods and illustrative examples." Thesis, Karlstads universitet, Institutionen för matematik och datavetenskap (from 2013), 2017. http://urn.kb.se/resolve?urn=urn:nbn:se:kau:diva-69711.
Full textAbstract:
Systems of large sparse linear equations frequently arise in engineering and science. Therefore, there is a great need for methods that can solve these systems. In this thesis we will present three of the earliest and simplest iterative methods and also look at two more sophisticated methods. We will study their rate of convergence and illustrate them with examples.
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Claudio, Kleucio. "Elementos finitos com resolução simplificada de sistemas de equações lineares para dispositivos fotônicos." [s.n.], 2010. http://repositorio.unicamp.br/jspui/handle/REPOSIP/260408.
Full textAbstract:
Orientador: Hugo Enrique Hernández-FigueroaTese (doutorado) - Universidade Estadual de Campinas, Faculdade de Engenharia Elétrica e de Computação
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Resumo: O método de elementos finitos é largamente empregado na modelagem de problemas de eletromagnetismo. A modelagem implícita deste método recai em resolver sistemas de equações lineares esparsas, esta etapa é de alto custo computacional. Este trabalho propõe alternativas com o objetivo de melhorar o desempenho computacional das aplicações provenientes de formulações via elementos finitos, através do aproveitamento de soluções de sistemas de equações lineares por métodos direto e iterativo, para simular dispositivos ópticos com as características físicas alteradas constantemente. Na solução dos sistemas de equações, utilizou-se o método direto com Small Rank Adjustment e o método iterativo gradiente bi-conjugado estabilizado precondicionado com análises de reaproveitamento do precondicionador ILUT. Nos estudos desenvolvidos obteve-se um melhor desempenho computacional quando se utilizou o método iterativo. Estes resultados são de grande importância na área de otimização de dispositivos fotônicos tais como acopladores, filtros, demultiplexadores, etc, pois a otimização destes dispositivos consiste em avaliar várias configurações do espaço de busca, implicando em resolver vários sistemas de equações lineares similares provenientes do método de elementos finitos.
Abstract: The Finite Element Method is one of the most popular numerical tools in electromagnetics. Implicit schemes require the solution of sparse linear equation systems, this step demands a lot of computational time. This work proposes alternatives enhancements to obtain better computational performance of such implicit schemes. This was made through the improvement of direct and iterative methods, for problems which may be interpreted as perturbations of a given original one. This is very important specially in the optimization process of devices, due to the fact that one needs to solve many linear systems with little changes at each step, to explore the search space, so many perturbed linear systems are solved to obtain the optimum device. For direct methods the Small Rank Adjustment technique was used, while for iterative methods, the Preconditioned Gradient Stabilized Biconjugate Method reusing the preconditioner, were adopted. The applications were focused on the design of photonic devices, like couplers, filters, demultiplexers, etc.
Doutorado
Telecomunicações e Telemática
Doutor em Engenharia Elétrica
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Zhu, Qiwei. "High performance stationary iterative methods." Thesis, University of Manchester, 2009. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.498981.
Full textAbstract:
Iterative methods are well-established in the context of scientific computing. They solve a problem by finding successive approximations to the true solution starting from an initial guess. Iterative methods are preferred when dealing with large size problems, as direct methods would be prohibitively expensive. They are commonly used for solving polynomial systems, systems of linear equations, and partial differential equations. Iterative methods normally make heavy demands on computational resources, both in terms of computing power and data storage requirements, and are thus required to be partitioned and executed in parallel. However, their standard sequential order offers little opportunity for parallelism. Hence, it is necessary to re-order their execution in order to exploit the parallel computing power of the underlying computational resources.
32
MacHardy, William R. "Iterative methods for parameter estimation." Thesis, Monterey, California : Naval Postgraduate School, 1990. http://handle.dtic.mil/100.2/ADA246174.
Full textAbstract:
Thesis (M.S. in Electrical Engineering)--Naval Postgraduate School, December 1990.Thesis Advisor(s): Tummala, Murali. Second Reader: Therrien, Charles W. "December 1990." Description based on title screen as viewed on April 1, 2010. DTIC Identifier(s): Iterations, Parametric Analysis, Algorithms, Estimates, Theses, Computerized Simulation, Convergence. Author(s) subject terms: Finite Impulse Response, Infinite Impulse Response, Matrix Splitting, Matrix Portioning, Toeplitz, Symmetric, Condition Number. Includes bibliographical references (p. 90). Also available in print.
33
Lechner, Patrick O. "Iterative methods for heterogeneous media." Thesis, University of Bath, 2006. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.432374.
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McKay, Melanie. "Iterative methods for incompressible flow." Thesis, University of Ottawa (Canada), 2009. http://hdl.handle.net/10393/28063.
Full textAbstract:
The goal of this thesis is to illustrate the effectiveness of iterative methods on the discretized Navier Stokes equations. The standard lid-driven cavity in both 2-D and 3-D test cases are examined and compared with published results of the same type. The numerical results are obtained by reducing the partial differential equations (PDEs) to a system of algebraic equations with a stabilized P1-P1 Finite Element Method (FEM) in space. Gear's Backward Difference Formula (BDF2) and an adaptive time stepping scheme utilizing a first order Backward Euler (BE) startup and BDF2 are then utilized to discretizc the time derivative of the Javier-Stokes equations. The iterative method used is the Generalized Minimal Residual (GMRES) along with the selected preconditioners Incomplete LU Factorization (ILU), Jacobi preconditioner and the Block Jacobi preconditioner.
35
Neuman, Arthur James III. "Regularization Methods for Ill-posed Problems." Kent State University / OhioLINK, 2010. http://rave.ohiolink.edu/etdc/view?acc_num=kent1273611079.
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Chen, Fan. "DISTANCE FIELD TRANSFORM WITH AN ADAPTIVE ITERATION METHOD." Kent State University / OhioLINK, 2009. http://rave.ohiolink.edu/etdc/view?acc_num=kent1255727002.
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Biazotti, Herbert Antonio. "Soluções solitônicas por aproximantes de Padé via método iterativo de Taylor /." Guaratinguetá, 2018. http://hdl.handle.net/11449/157328.
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Orientador: Denis DalmaziCoorientador: Álvaro de Souza Dutra
Banca: Julio Marny Hoff da Silva
Banca: Rafael Augusto Couceiro Corrêa
Resumo: Certos sistemas físicos podem ser descritos por uma classe de equações não-lineares. Essas equações descrevem pacotes de onda chamado de sólitons que tem aplicações em diversas áreas, por exemplo, Óptica, Cosmologia, Matéria Condensada e Física de Partículas. Alguns métodos foram desenvolvidos ao longo dos anos para encontrar as soluções dessas equações. Buscaremos essas soluções usando o que chamamos de Método Iterativo de Taylor (MIT), que fornece uma solução aproximada em polinômio de Taylor de forma distinta do que se tem na literatura. Usaremos o MIT para calcular soluções por aproximantes de Padé que são razões entre dois polinômios e fornecem soluções melhores que o polinômio de Taylor que o gerou. Inicialmente resolveremos a equação de um modelo de um campo denominado λφ4 . Em seguida resolveremos um modelo com dois campos escalares acoplados e encontraremos uma solução analítica aproximada em casos onde não existe solução analítica, explorando a diversidade das soluções do modelo. Usando essa abordagem por aproximantes de Padé veremos que há algumas vantagens em relação a outros métodos
Abstract: Certain physical systems can be described by a class of non-linear differential equations. Those equations describe wave packets called solitons which have applications in several areas, for example, Optics, Cosmology, Condensed Matter, and Particle Physics. Some methods have been developed over the years to find solutions to these equations. We will look for those solutions using what we call the Taylor Iterative Method (TIM), which provides an approximate solution in terms of a Taylor's polynomial in a unusual way, regarding the present literature. We will use TIM to calculate solutions by Padé approximants, which are ratios between two polynomials and provide better solutions than the Taylor polynomial itself. We first solve the field equation of a model called λφ4 . Then we will solve a model with two coupled scalar fields and find an approximate analytic solution in cases where there is no known analytical solution, exploring the diversity of the solutions of the model. We will see that there are some advantages in using the Padè approximants as compared to other methods
Mestre
38
Helou, Neto Elias Salomão. "Algoritmos incrementais com aplicações em tomografia computadorizada." [s.n.], 2009. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307603.
Full textAbstract:
Orientador: Alvaro Rodolfo De PierroTese (doutorado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: O problema de viabilidade convexa é um campo fértil de pesquisa que deu origem a uma grande quantidade de algoritmos iterativos, tais como pocs, art, Cimmino e uma miríade de variantes. O motivo para tal interesse é o amplo leque de aplicabilidade que algoritmos gerais para a solução de problemas desse tipo podem alcançar. Dentre tais aplicações encontra-se a reconstrução de imagens em tomografia, caso que geralmente apresenta uma estrutura especial de esparsidade e tamanhos gigantescos. Também bastante estudados por seu interesse prático e teórico são problemas envolvendo a minimização irrestrita de funções convexas. Aqui, novamente, a variada gama de aplicações torna impossível mencionar uma lista minimamente abrangente. Dentre essas a tomografia é, outra vez, um exemplo de grande destaque. No presente trabalho desenvolvemos uma ponte que permite o uso de uma variedade de métodos para viabilidade em conjunto com algoritmos de otimização para obter a solução de problemas de otimização convexa com restrições. Uma teoria geral de convergência é apresentada e os resultados teóricos são especializados em métodos apropriados para problemas de grande porte. Tais métodos são testados em experimentos numéricos envolvendo reconstrução de imagens tomográficas. Esses testes utilizam-se da teoria de amostragem compressiva desenvolvida recentemente, através da qual conseguimos obter resultados sem par na reconstrução de imagens tomográficas a partir de uma amostragem angular altamente esparsa da transformada de Radon. Imagens obtidas a partir de dados simulados são recuperadas perfeitamente com menos de 1/20 das amostras classicamente necessárias. Testes com dados reais mostram que o tempo de uma leitura spect pode ser reduzido a até 1/3 do tempo normalmente utilizado, sem grande prejuízo para as reconstruções.
Abstract: The convex feasibility problem is a research field which has originated a large variety of iterative algorithms, such as pocs, art, Cimmino and a myriad of variants. The reason for such interest is the wide array of applicability that general algorithms for this kind of problem may reach. Among such applications there is tomographic image reconstruction, instance that generally presents a special sparsity structure and huge sizes. Also widely studied because its practical and theoretical interests are problems involving unconstrained minimization of convex functions. Here, again, the huge array of applications makes it impossible to mention even a minimal list. Among these, once more, tomography is a major example. In the present work we have developed a bridge that allows the use of a variety of methods for feasibility in conjunction with optimization algorithms in order to obtain the solution for convex optimization problems with restrictions. A general convergence theory is presented and the theoretical results are specialized into methods useful for large scale problems. These methods are tested in experiments involving tomographic image reconstruction. Such tests make use of the recently developed compressive sensing theory, through which we have been able to obtain unmatched results in tomographic image reconstruction from highly sparse angular sampling from the Radon transform. Images obtained from simulated data are perfectly reconstructed using less than 1/20 from the classically needed. Tests with real data show that the time of a spect scan can be reduced to 1/3 of the usual, without too much image deterioration.
Doutorado
Matematica Aplicada
Doutor em Matemática Aplicada
39
Kwan, Chun-kit, and 關進傑. "Fast iterative methods for image restoration." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2000. http://hub.hku.hk/bib/B31224520.
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何正華 and Ching-wah Ho. "Iterative methods for the Robbins problem." Thesis, The University of Hong Kong (Pokfulam, Hong Kong), 2000. http://hub.hku.hk/bib/B31222572.
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Kwan, Chun-kit. "Fast iterative methods for image restoration /." Hong Kong : University of Hong Kong, 2000. http://sunzi.lib.hku.hk:8888/cgi-bin/hkuto%5Ftoc%5Fpdf?B22956281.
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Ho, Ching-wah. "Iterative methods for the Robbins problem /." Hong Kong : University of Hong Kong, 2000. http://sunzi.lib.hku.hk/hkuto/record.jsp?B22054789.
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43
Wester, Roderick C. "Multidimensional spectral estimation using iterative methods." Thesis, Monterey, California : Naval Postgraduate School, 1990. http://handle.dtic.mil/100.2/ADA237025.
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Thesis (M.S. in Electrical Engineering)--Naval Postgraduate School, June 1990.Thesis Advisor(s): Therrien, Charles W. ; Tummala, Murali. "June 1990." Description based on title screen as viewed on October 15, 2009. DTIC Identifier(s): Iterations, Covariance, Regression Analysis, Estimates. Author(s) subject terms: Autoregressive Spectral Estimation, Covariance Method. Includes bibliographical references (p. 35). Also available in print.
44
Benbow, Steven James. "Iterative methods for augmented linear systems." Thesis, University of Bath, 1997. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.760703.
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Berti, Lilian Ferreira 1988. "Iteração continuada aplicada ao método de pontos interiores." [s.n.], 2012. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306753.
Full textAbstract:
Orientadores: Aurelio Ribeiro Leite de Oliveira, Carla Taviane Lucke da Silva GhidiniDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matemática, Estatística e Computação Científica
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Resumo: Os métodos de pontos interiores têm sido amplamente utilizados para determinar a solução de problemas de programação linear de grande porte. O método preditor corretor, dentre todas as variações de métodos de pontos interiores, é um dos que mais se destaca, devido à sua eficiência e convergência rápida. Este método, em cada iteração, necessita resolver dois sistemas lineares para determinar a direção preditora corretora. Resolver estes sistemas lineares corresponde ao passo que requer mais tempo de processamento, devendo assim ser realizada de forma eficiente. Para resolver estes sistemas lineares a abordagem mais utilizada é a fatoração de Cholesky. No entanto, realizar a fatoração de Cholesky em cada iteração tem um alto custo computacional. Dessa forma, na busca de redução de esforços, precisamente, na redução do número de iterações foi desenvolvida a iteração continuada. Iteração continuada é uma iteração subsequente, realizada após o cálculo da direção preditora corretora, onde é determinada uma nova direção sem que seja necessário realizar uma nova fatoração de Cholesky. Os resultados computacionais dos testes realizados, principalmente em problemas de médio e grande porte mostraram que esta abordagem obtém bom desempenho em comparação com o método preditor corretor
Abstract: Interior point methods have been widely used in the solution of large linear programming problems. The predictor corrector method, among ali interior point variants, is one of mostly used due to its efficiency and convergence properties. This method needs the solution of two linear systems to determine the predictor corrector direction, in each iteration. Solving such systems corresponds to the step which requires more processing time. Therefore, it should be done efficiently. The most common approach to solve the linear systems is the Cholesky factorization, demanding in each iteration a high computacional effort. Thus, in search of effort reduction, in particular, to reduce the iterations number continued iteration was developed. The continued iteration is a subsequent iteration performed after the predictor corrector direction is computed, where a new direction is calculated without need to of Cholesky refactorization. The numerical tests show that the continued iteration performs better in comparison with the preditor corretor method
Mestrado
Matematica Aplicada
Mestre em Matemática Aplicada
46
Padhy, Bijaya L. "NITSOL -- A Newton iterative solver for nonlinear systems a FORTRAN-to-MATLAB implementation." Link to electronic thesis, 2006. http://www.wpi.edu/Pubs/ETD/Available/etd-042806-161216/.
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Bai, Xianglan. "Non-Krylov Non-iterative Subspace Methods For Linear Discrete Ill-posed Problems." Kent State University / OhioLINK, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=kent1627042947894919.
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SINGH, ONKAR DEEP. "ITERATIVE SOLVERS FOR DISCONTINUOUS GALERKIN FINITE ELEMENT METHODS." University of Cincinnati / OhioLINK, 2004. http://rave.ohiolink.edu/etdc/view?acc_num=ucin1093023928.
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Begiato, Rodolfo Gotardi 1980. "Um metodo Newton-Inexato com estrategia hibrida para globalização." [s.n.], 2007. http://repositorio.unicamp.br/jspui/handle/REPOSIP/305941.
Full textAbstract:
Orientador: Marcia Aparecida Gomes RuggieroDissertação (mestrado) - Universidade Estadual de Campinas, Instituto de Matematica, Estatistica e Computação Cientifica
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Resumo: o principal objetivo deste trabalho é a proposta de uma estratégia híbrida de globalização para o método de Newton-inexato. Assim como o método de Newton, o método de Newton-inexato tem sua convergência garantida somente em vizinhanças adequadas da solução do sistema e uma estratégia de globalização deve, portanto, ser incorporada. Estratégias de globalização se baseiam na minimização de funções de mérito e duas abordagens podem ser consideradas: busca linear e regiões de confiança. Neste trabalho optamos pelo uso conjunt0 das duas abordagens, resultando numa estratégia híbrida, envolvendo inicialmente uma seqüência de buscas lineares, e se necessário, prossegue-se com uma variação da estratégia Dogleg, proposta por Powell em 1970. Para a resolução aproximada de sistemas lineares foi utilizado o método GMRES, que faz parte de métodos de projeções sobre subespaços de Krylov. Este método possibilita a implementação com a estratégia matrix-free. Para reduzir o uso de requerimentos de memória, optamos ainda pelo uso do método GMRES com recomeços. A eficiência dos algoritmos desenvolvidos foi avaliada através da resolução -de um conjunto de sistemas não lineares acadêmicos e um conjunto de sistemas sistemas não-lineares resultantes' da discretização de problemas de valor de contorno. Estes testes compravaram a eficiência da estratégia híbrida empregada no processo de globalização
Abstract: The main objective of this work is to propose a hybrid globalization strategie for inexact-Newton method. Globalization strategies are based on line search or trust region procedures. In this work, we choose a hybrid strategy which involves a cycle of line search and a variation of Powell dogleg trust region. For solving the linear systems we chose the GMRES method with restarts and to avoid the calculation of Jacobian matrices we used a matrix-free strategie. The numerical performance of the algorithms was evaluated by means a set of academic problems and a set of nonlinear systems of boundary value problem discretization. These results showed the good performance of hybrid globalization strategy
Mestrado
Otimização Matematica
Mestre em Matemática Aplicada
50
Grau, Gotés Ma Àngela (Maria Àngela). "On iterative methods to solve nonlinear equations." Doctoral thesis, Universitat Politècnica de Catalunya, 2015. http://hdl.handle.net/10803/396684.
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Many of the problems in experimental sciences and other disciplines can be expressed in the form of nonlinear equations. The solution of these equations is rarely obtained in closed form. With the development of computers, these problems can be addressed by numerical algorithms that approximate the solution. Specifically, fixed point iterative methods are used, which generate a convergent sequence presumably to the solution of the equation or system of equations. Since J.F. Traub (Iterative methods for the solution of equations, Prentice-Hall, N.J. 1964) initiated the qualitative as well the quantitative analysis of iterative methods in the 1960s, iterative methods for nonlinear systems has been a constantly interesting field of study for numerical analysts.
Our contribution to this field is the analysis and construction of new iterative methods, by improving the order of convergence and computational efficiency either of these or other known methods. To study the new iterative methods that we have proposed, we reviewed analyzed and improved classic concepts of computational order of convergence, the error equation, and the computational cost of an iterative method for both an equation and a system of nonlinear equations. Specifically, we have worked on the following points:
- We computed the local order of convergence for known two-step and new multi-step iterative methods by means of expansions in formal developments in power series of the function F, the Jacobian operator, the inverse Jacobian operator and the divided difference operator and its inverse operator.
- We generated some measures that approximate the order of convergence. Four new variants for computing the computational order of convergence (COC) are given: one requires the value of the root, whilst the other three do not.
- We constructed families of iterative schemes that are variants of Newton’s method and Chebyshev’s method and improve the order and the efficiency.
- We studied several families of the modified Secant method (Secant, Kurchatov and Steffensen), evaluated variants of these methods and chose the most efficient.
- We generalized the concepts of efficiency index and computational efficiency for nonlinear equations to systems of nonlinear equations. This has been termed the computational efficiency index (CEI).
- We considered that in iterative process using variable precision, the accuracy will increase as the computation proceeds. The final result will be obtained as precisely as possible, depending on the computer and the software.
- We expressed the cost of evaluating elementary functions in terms of products. This cost depends on the computer, the software and the arithmetic that we used. The above numerical calculations were performed in the algebraic system called MAPLE.
- We presented a new way of comparing elapsed time for different iterative schemes. This consists of estimating the time required to achieve a correct decimal of the solution by the method selected. That is, we measured the relationship between the time to fulfill the stop criterion and the total number of correct decimals obtained by method. The five papers selected for this compendium were published in scientific journals in the area of applied mathematics. The impact factor of these journals is, in all cases, in the first third according to the classification of the Journal of Citation Reports. There are four preceding papers that no are part of this report by its publication date.Gran parte de los problemas en ciencias experimentales y otras disciplinas se pueden expresar en forma de ecuaciones no lineales. La solución de estas ecuaciones rara vez se obtiene en forma cerrada; con el desarrollo de los ordenadores, estos problemas pueden ser abordados por algoritmos numéricos que aproximan la solución. Concretamente, se utilizan métodos iterativos de punto fijo, que generan una secuencia convergente presumiblemente a la solución de la ecuación o sistema de ecuaciones. Desde J.F. Traub, (Iterative methods for the solution of equations, Prentice-Hall, N.J. 1964) inició el estudio cualitativo y el análisis cuantitativo de éstos métodos iterativos en la década de los sesenta, los métodos iterativos para sistemas no lineales ha sido un área de constante estudio para los analistas numéricos. La contribución que presenta este compendio en este campo es el análisis y la construcción de nuevos métodos iterativos mejorando ya sea el orden de convergencia o ya sea la eficiencia computacional de éstos o de otros ya conocidos. Para el estudio de nuevos métodos iterativos, se ha revisado, analizado y en algun caso redefinido los conceptos clásicos de orden de convergencia computacional, de ecuación del error y de coste computacional de un método iterativo, tanto para una ecuación como para un sistema de ecuaciones no linealesEn concreto, se ha trabajado en los siguientes puntos: - El cálculo del orden local de convergencia para métodos conocidos de dos pasos y para nuevos métodos iterativos multipaso se realiza haciendo uso de desarrollos formales en serie de potencias del error. Se ha desarrollado la función F, el operador Jacobiano, el operador Jacobiano inverso, el operador diferencia dividida y su operador inverso. -Se generan algunas medidas que aproximan el orden local de convergencia del método iterativo. Se presentan cuatro nuevas variantes para el cálculo del orden de convergencia computacional (COC, computational order of convergence); un parámetro que necesita el valor de la solución o raíz, y tres parámetros que no requieren de éste valor. - Construcción de familias, los esquemas iterativos de las cuáles son variantes del método de Newton y del método de Chebyshev, mejorando el orden y la eficiencia de éstos. - Estudio de diversas familias, derivadas del método de la Secante (Secante, Kurchatov y Steffensen), variantes de estos métodos y elección de los más eficientes. - Generalización de los conceptos de índice de eficiencia y de eficiencia computacional para ecuaciones a sistemas de ecuaciones no lineales. Se ha denominado índice de eficiencia computacional (CEI, Computational Efficiency Index). - Análisis y construcción de procesos iterativos de precisión variable. La precisión aumenta a medida que la computación avanza, y el resultado final se obtiene con la máxima precisión posible, dependiendo del ordenador y el software disponibles. - Expresión del coste de la evaluación de las funciones elementales en términos de productos. Este coste depende del ordinador, el software y la aritmética que se utiliza. Los cálculos numéricos mencionados se ha realizado con el sistema algebraico MAPLE. - Una nueva forma de comparar el tiempo de ejecución dedicado al cálculo por los diferentes esquemas iterativos. Consiste en calcular el tiempo necesario para conseguir un decimal correcto de la solución por el método escogido. Concretamente, se mide la relación entre el tiempo transcurrido para cumplir el criterio de parada y el número total de decimales correctos obtenidos por el algoritmo. Los cinco trabajos seleccionados para constituir este compendio fueron publicados en revistas científicas del área de matemática aplicada. El factor de impacto de éstas se encuentra en el primer tercio de acuerdo con la clasificación del Journal of Citation Reports. Además, he publicado cuatro artículos previos, que no forman parte de esta memoria por fecha de publicación, válidos para un sexenio el año 2011.