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Dissertations / Theses on the topic 'Iteration method'
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1
Wilkins, Bryce Daniel. "The E² Bathe subspace iteration method." Thesis, Massachusetts Institute of Technology, 2019. https://hdl.handle.net/1721.1/122238.
Full textAbstract:
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
2
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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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.
4
Kim, Ki-Tae Ph D. Massachusetts Institute of Technology. "The enriched subspace iteration method and wave propagation dynamics with overlapping finite elements." Thesis, Massachusetts Institute of Technology, 2018. http://hdl.handle.net/1721.1/119346.
Full textAbstract:
Thesis: Ph. D., Massachusetts Institute of Technology, Department of Mechanical Engineering, 2018.Cataloged from PDF version of thesis.
Includes bibliographical references (pages 133-137).
In structural dynamic problems, the mode superposition method is the most widely used solution approach. The largest computational effort (about 90% of the total solution time) in the mode superposition method is spent on calculating the required eigenpairs and it is of critical importance to develop effective eigensolvers. We present in this thesis a novel solution scheme for the generalized eigenvalue problem. The scheme is an extension of the Bathe subspace iteration method and a significant reduction in computational time is achieved. For the solution of wave propagation problems, the finite element method with direct time integration has been extensively employed. However, using the traditional finite element solution approach, accurate solutions can only be obtained of rather simple one-dimensional wave propagation problems. In this thesis, we investigate the solution characteristics of a solution scheme using 'overlapping finite elements', disks and novel elements, enriched with harmonic functions and the Bathe implicit time integration method to solve transient wave propagation problems. The proposed solution scheme shows two important properties: monotonic convergence of calculated solutions with decreasing time step size and a solution accuracy almost independent of the direction of wave travel through uniform, or distorted, meshes. These properties make the scheme promising to solve general wave propagation problems in complex geometries involving multiple waves.
by Ki-Tae Kim.
Ph. D.
5
Altintan, Derya. "An Extension To The Variational Iteration Method For Systems And Higher-order Differential Equations." Phd thesis, METU, 2011. http://etd.lib.metu.edu.tr/upload/12613864/index.pdf.
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It is obvious that differential equations can be used to model real-life problems. Although it is possible to obtain analytical solutions of some of them, it is in general
difficult to find closed form solutions of differential equations. Finding thus approximate solutions has been the subject of many researchers from different areas.
In this thesis, we propose a new approach to Variational Iteration Method (VIM) to obtain the solutions of systems of first-order differential equations. The main
contribution of the thesis to VIM is that proposed approach uses restricted variations only for the nonlinear terms and builds up a matrix-valued Lagrange multiplier that leads to the extension of the method (EVIM).
Close relation between the matrix-valued Lagrange multipliers and fundamental solutions of the differential equations highlights the relation between the extended version of the variational iteration method and the classical variation of parameters formula.
It has been proved that the exact solution of the initial value problems for (nonhomogenous) linear differential equations can be obtained by such a generalisation using
only a single variational step.
Since higher-order equations can be reduced to first-order systems, the proposed approach is capable of solving such equations tooindeed, without such a reduction, variational iteration method is also extended to higher-order scalar equations. Further, the close connection with the associated first-order systems is presented. Such extension of the method to higher-order equations is then applied to solve boundary value problems: linear and nonlinear ones. Although the corresponding Lagrange multiplier resembles the Green&rsquo
s function, without the need of the latter, the extended approach to the variational iteration method is systematically applied to solve boundary value problems, surely in the nonlinear case as well. In order to show the applicability of the method, we have applied the EVIM to various real-life problems: the classical Sturm-Liouville eigenvalue problems, Brusselator reaction-diffusion, and chemical master equations. Results show that the method is simple, but powerful and effective.
6
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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Tan, Li. "A Computational Iteration Method to Analyze Mechanics of Timing Belt Systems with Non-Circular Pulleys." Thesis, Virginia Tech, 2018. http://hdl.handle.net/10919/84991.
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Timing belt systems, usually consisting of a toothed belt and multiple pulleys, are used in many mechanical devices, especially in the internal combustion engine to synchronize the rotation of the crankshafts and the camshafts. When the system operates, the belt teeth will be transmitted by the pulley teeth meshed with them. Timing belt drives can make sure that the engine' s valves open and close properly due to their precise transmission ratio. In this thesis, a quasi-static computational model is developed to calculate the belt load distributions and the torques around pulleys for different timing belt systems. The simplest system is a two-pulley system with one oval pulley and one circular pulley. This computational model is then extended to a two-pulley system with one special-shaped pulley and finally generalized to determine the load conditions for a multi-pulley system with multiple special-shaped pulleys. Belt tooth deflections, tooth loads, belt tension distributions, friction forces, and the effect of friction hysteresis have been taken into consideration. Results of these quantities are solved by a nested numerical iteration method. Periodic torques generated by the varied radius of noncircular pulley are calculated by this computational model to cancel the undesired external cyclic torque, which will increase the life of timing belts.Master of Science
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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.
9
Andersson, Tomas. "An iterative solution method for p-harmonic functions on finite graphs with an implementation." Thesis, Linköping University, Department of Mathematics, 2009. http://urn.kb.se/resolve?urn=urn:nbn:se:liu:diva-18162.
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In this paper I give a description and derivation of Dirichlet's problem, a boundary value problem, for p-harmonic functions on graphs and study an iterative method for solving it.The method's convergence is proved and some preliminary results about its speed of convergence are presented.There is an implementation accompanying this thesis and a short description of the implementation is included. The implementation will be made available on the internet at http://www.mai.liu.se/~anbjo/pharmgraph/ for as long as possible.
10
Byers, R., C. He, and V. Mehrmann. "The Matrix Sign Function Method and the Computation of Invariant Subspaces." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199800619.
Full textAbstract:
A perturbation analysis shows that if a numerically stable
procedure is used to compute the matrix sign function, then it is competitive
with conventional methods for computing invariant subspaces.
Stability analysis of the Newton iteration improves an earlier result of Byers
and confirms that ill-conditioned iterates may cause numerical
instability. Numerical examples demonstrate the theoretical results.
11
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.
Full textAbstract:
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.
12
Zinzani, Filippo. "Calculation of the eigenfunctions of the two-group neutron diffusion equation and application to modal decomposition of BWR instabilities." Bachelor's thesis, Alma Mater Studiorum - Università di Bologna, 2007. http://amslaurea.unibo.it/594/.
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In this thesis, numerical methods aiming at determining the eigenfunctions, their adjoint and the corresponding eigenvalues of the two-group neutron diffusion equations representing any heterogeneous system are investigated. First, the classical power iteration method is modified so that the calculation of modes higher than the fundamental mode is possible. Thereafter, the Explicitly-Restarted Arnoldi method, belonging to the class of Krylov subspace methods, is touched upon. Although the modified power iteration method is a computationally-expensive algorithm, its main advantage is its robustness, i.e. the method always converges to the desired eigenfunctions without any need from the user to set up any parameter in the algorithm. On the other hand, the Arnoldi method, which requires some parameters to be defined by the user, is a very efficient method for calculating eigenfunctions of large sparse system of equations with a minimum computational effort. These methods are thereafter used for off-line analysis of the stability of Boiling Water Reactors. Since several oscillation modes are usually excited (global and regional oscillations) when unstable conditions are encountered, the characterization of the stability of the reactor using for instance the Decay Ratio as a stability indicator might be difficult if the contribution from each of the modes are not separated from each other. Such a modal decomposition is applied to a stability test performed at the Swedish Ringhals-1 unit in September 2002, after the use of the Arnoldi method for
pre-calculating the different eigenmodes of the neutron flux throughout the reactor. The modal decomposition clearly demonstrates the excitation of both the global and regional oscillations. Furthermore, such oscillations are found to be intermittent with a time-varying phase shift between the first and second azimuthal modes.
13
Ouyang, Guang. "Study of the variability in brain potentials and responses : development of a new method for electroencephalography (EEG) analysis - residue iteration decomposition (RIDE)." HKBU Institutional Repository, 2013. https://repository.hkbu.edu.hk/etd_ra/1529.
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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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Benner, Peter, Enrique Quintana-Ortí, and Gregorio Quintana-Ortí. "Solving Linear Matrix Equations via Rational Iterative Schemes." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601460.
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We investigate the numerical solution of stable Sylvester equations via iterative schemes proposed for computing the sign function of a matrix. In particular, we discuss how the rational iterations for the matrix sign function can efficiently be adapted to the special structure implied by the Sylvester equation. For Sylvester equations with factored constant term as those arising in model reduction or image restoration, we derive an algorithm that computes the solution in factored form directly. We also suggest convergence criteria for the resulting iterations and compare the accuracy and performance of the resulting methods with existing Sylvester solvers. The algorithms proposed here are easy to parallelize. We report on the parallelization of those algorithms and demonstrate their high efficiency and scalability using experimental results obtained on a cluster of Intel Pentium Xeon processors.
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Павленко, Іван Володимирович, Иван Владимирович Павленко, Ivan Volodymyrovych Pavlenko, Віта Володимирівна Павленко, Вита Владимировна Павленко, and Vita Volodymyrivna Pavlenko. "Застосування методу послідовних наближень при розв'язанні позиційних задач нарисної геометрії." Thesis, Сумський державний університет, 2013. http://essuir.sumdu.edu.ua/handle/123456789/30604.
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Перевагами запропонованого методу є можливість контролю точності розв'язання позиційних задач нарисної геометрії на кожному ітераційному кроці, відносно невелика кількість геометричних побудов, простий алгоритм чисельної реалізації, інваріантність алгоритму до типу поверхні тіла.Метою роботи є створення, обгрунтування і чисельна реалізація якісно нового способу розв'язання позиційних задач нарисної геометрії, який має ряд переваг порівняно з існуючими методами проекційного креслення. При цитуванні документа, використовуйте посилання http://essuir.sumdu.edu.ua/handle/123456789/30604
Целью работы является создание, обоснование и численная реализация качественно нового способа решения позиционных задач начертательной геометрии, который имеет ряд преимуществ по сравнению с существующими методами проекционного черчения. При цитировании документа, используйте ссылку http://essuir.sumdu.edu.ua/handle/123456789/30604
The aim of this article is creation and numerical implementation of a qualitatively new way of solving positional problems of descriptive geometry, which has some advantages over existing methods of engineering graphics. When you are citing the document, use the following link http://essuir.sumdu.edu.ua/handle/123456789/30604
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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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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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Cai, Shang-Gui. "Computational fluid-structure interaction with the moving immersed boundary method." Thesis, Compiègne, 2016. http://www.theses.fr/2016COMP2276/document.
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Dans cette thèse, une nouvelle méthode de frontières immergées a été développée pour la simulation d'interaction fluide-structure, appelée la méthode de frontières immergées mobiles (en langage anglo-saxon: MIBM). L'objectif principal de cette nouvelle méthode est de déplacer arbitrairement les solides à géométrie complexe dans un fluide visqueux incompressible, sans remailler le domaine fluide. Cette nouvelle méthode a l'avantage d'imposer la condition de non-glissement à l'interface d'une manière exacte via une force sans introduire des constantes artificielles modélisant la structure rigide. Cet avantage conduit également à la satisfaction de la condition CFL avec un pas de temps plus grand. Pour un calcul précis de la force induite par les frontières mobiles, un système linéaire a été introduit et résolu par la méthode de gradient conjugué. La méthode proposée peut être intégrée facilement dans des solveurs résolvant les équations de Navier-Stokes. Dans ce travail la MIBM a été mise en œuvre en couplage avec un solveur fluide utilisant une méthode de projection adaptée pour obtenir des solutions d'ordre deux en temps et en espace. Le champ de pression a été obtenu par l'équation de Poisson qui a été résolue à l'aide de la méthode du gradient conjugué préconditionné par la méthode multi-grille. La combinaison de ces deux méthodes a permis un gain de temps considérable par rapport aux méthodes classiques de la résolution des systèmes linéaires. De plus le code de calcul développé a été parallélisé sur l'unité graphique GPU équipée de la bibliothèque CUDA pour aboutir à des hautes performances de calcul. Enfin, comme application de nos travaux sur la MIBM, nous avons étudié le couplage "fort" d'interaction fluide-structure (IFS). Pour ce type de couplage, un schéma implicite partitionné a été adopté dans lequel les conditions à l'interface sont satisfaites via un schéma de type "point fixe". Pour réduire le temps de calcul inhérent à cette application, un nouveau schéma de couplage a été proposé pour éviter la résolution de l'équation de Poisson durant les itérations du "point fixe". Cette nouvelle façon de résoudre les problèmes IFS a montré des performances prometteuses pour des systèmes en IFS complexeIn this thesis a novel non-body conforming mesh formulation is developed, called the moving immersed boundary method (MIBM), for the numerical simulation of fluid-structure interaction (FSI). The primary goal is to enable solids of complex shape to move arbitrarily in an incompressible viscous fluid, without fitting the solid boundary motion with dynamic meshes. This novel method enforces the no-slip boundary condition exactly at the fluid-solid interface with a boundary force, without introducing any artificial constants to the rigid body formulation. As a result, large time step can be used in current method. To determine the boundary force more efficiently in case of moving boundaries, an additional moving force equation is derived and the resulting system is solved by the conjugate gradient method. The proposed method is highly portable and can be integrated into any fluid solver as a plug-in. In the present thesis, the MIBM is implemented in the fluid solver based on the projection method. In order to obtain results of high accuracy, the rotational incremental pressure correction projection method is adopted, which is free of numerical boundary layer and is second order accurate. To accelerate the calculation of the pressure Poisson equation, the multi-grid method is employed as a preconditioner together with the conjugate gradient method as a solver. The code is further parallelized on the graphics processing unit (GPU) with the CUDA library to enjoy high performance computing. At last, the proposed MIBM is applied to the study of two-way FSI problem. For stability and modularity reasons, a partitioned implicit scheme is selected for this strongly coupled problem. The interface matching of fluid and solid variables is realized through a fixed point iteration. To reduce the computational cost, a novel efficient coupling scheme is proposed by removing the time-consuming pressure Poisson equation from this fixed point interaction. The proposed method has shown a promising performance in modeling complex FSI system
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Popuzin, Vitaly. "Methods of fast Fourier transform in diffraction problems of elastic and acoustic waves with applications to crack mechanics." Doctoral thesis, Università di Catania, 2014. http://hdl.handle.net/10761/1548.
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The activity is connected with a development of modern fast computational methods applied to problems in wave dynamics, acoustics, and boundary-value problems of mechanics with mixed boundary conditions. This includes:
1) Development of fast methods for integral equations with convolution kernels arising in these fields of application. Such integral equations, after discretization, may be reduced to linear algebraic systems with matrix of Toepliz or circulant form. For both the types there can be applied fast iteration methods founded on Conjugate Gradient method with a preconditioning. This leads to a quasi-linear numerical algorithm.
2) Applications are constructed in crack mechanics. Two problems are studied with their mechanical conclusions: (i) Diffraction of a planar acoustic wave by a planar crack in the classical linear elastic isotropic space; (ii) static problem for linear cracks in the non-classical porous material of a Cowin-Nunziato type.
3) In the case when the diffraction is happen by a general-form object whose shape is neither linear nor circular, the problem can be reduced to an integral equation with a general-form kernel. There is developed a new approach, which permits an iteration scheme with a convolution kernel at each iteration. This admits again a quasi-linear numerical algorithm.
4) The same idea is applicable to wave processes with obstacles which represent an arbitrary set of linear rigid screen of finite length. The iteration process is proposed, when at each iteration step one needs only solution of the problem for every isolated single screen. All equations in this case are of convolution type, and they are reduced again to Toepliz-like matrix equations in a discrete form.
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Garšvaitė, Skaistė. "Dvimatės elipsinės lygties su nelokaliąja sąlyga sprendimas baigtinių skirtumų metodu." Master's thesis, Lithuanian Academic Libraries Network (LABT), 2008. http://vddb.library.lt/obj/LT-eLABa-0001:E.02~2008~D_20080619_122640-56101.
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Šiame darbe nagrinėjame elipsinės lygties stačiakampėje srityje su nelokaliąja sąlyga sprendimą baigtinių skirtumų metodu. Sprendžiame dvimates skirtumines lygčių sistemas, jas gavome pakeitę diferencialinę lygtį skirtumine. Trumpai apžvelgtas maksimumo principas ir sprendinio radimas iteraciniais metodais bei tikrinių reikšmių radimas dvimačiu atveju. Įvertinta skirtuminės lygčių sistemos paklaida, kuri gaunama sprendžiant elipsinę lygtį skirtuminiu metodu. Darbo pabaigoje išspręstas konkretus uždavinys.In this work we consider two dimensional elliptic equation on the rectangle with non local condition by finite difference method. We solve two dimensional equations instead one intricate differential equation. A short review of maximum principle and solution finding with iteration method, and the proper account finding with two dimensional case. Estimated differential equationerror, this making calculate elliptic equation difference method. Finally we solve particilar example with different steps.
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Albertyn, Martin. "Generic simulation modelling of stochastic continuous systems." Thesis, Pretoria : [s.n.], 2004. http://upetd.up.ac.za/thesis/available/etd-05242005-112442.
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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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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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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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Aydogdu, Oktay. "Pseudospin Symmetry And Its Applications." Phd thesis, METU, 2009. http://etd.lib.metu.edu.tr/upload/12611298/index.pdf.
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The pseudospin symmetry concept is investigated by solving the Dirac equation for the exactly solvable potentials such as pseudoharmonic potential, Mie-type potential, Woods-Saxon potential and Hulthén plus ring-shaped potential with any spin-orbit coupling term $kappa$. Nikiforov-Uvarov Method, Asymptotic Iteration Method and functional analysis method are used in the calculations. The energy eigenvalue equations of the Dirac particles are found and the corresponding radial wave functions are presented in terms of special functions. We look for the contribution of the ring-shaped potential to the energy spectra of the Dirac particles. Particular cases of the potentials are also discussed. By considering some particular cases, our results are reduced to the well-known ones presented in the literature. In addition, by taking equal mixture of scalar and vector potentials together with tensor potential, solutions of the Dirac equation are found and then the energy splitting between the two states in the pseudospin doublets is investigated. We indicate that degeneracy between members of pseudospin doublet is removed by tensor interactions. Effects of the potential parameters on the pseudospin doublet splitting are also studied. Radial nodes structure of the Dirac spinor are presented.
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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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Demetriou, Demetris. "An investigation into nonlinear random vibrations based on Wiener series theory." Thesis, University of Cambridge, 2019. https://www.repository.cam.ac.uk/handle/1810/287637.
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In support of society's technological evolution, the study of nonlinear systems in engineering and sciences has become a vital research area. Aiming to contribute in this field, this thesis investigates the behaviour of nonlinear systems using the 'Wiener theories'. As a useful example the Duffing oscillator is investigated in this work. In many real-life applications, nonlinear systems are excited randomly so this work examines systems under white-noise excitation using the Wiener series. Equivalent Linearisation (EL) is a well-known and simple method that approximates a nonlinear system by an equivalent linear system. However, it has deficiencies which this thesis attempts to improve. Initially, the performance of EL for different types of nonlinearities will be assessed and an alternative method to enhance it is suggested. This requires the calculation of the first Wiener kernel of various system defined quantities. The first Wiener kernel, as it will be shown, is the foundation of this research and a central element of the Wiener theory. In this thesis, an analytical proof to explain the interesting behaviour of the first Wiener kernel for a system with nonlinear stiffness is included using an energy transfer approach. Furthermore, the method mentioned above to enhance EL known as the Single-Pole Fit method (SPF) is to be tested for different kinds of systems to prove its robustness and validity. Its direct application to systems with nonlinear stiffness and nonlinear damping is shown as well as its ability to perform for systems with two degrees of freedom where an extension of the SPF method is required to achieve the desired solution. Finally, an investigation to understand and replicate the complex behaviour observed by the first Wiener kernel in the early chapters is carried out. The groundwork for this investigation is done by modelling an isolated nonlinear spring with a series of linear filters and certain nonlinear operations. Subsequently, an attempt is made to relate the principles governing the successful spring model presented to the original nonlinear system. An iterative procedure is used to demonstrate the application of this method, which also enables this new modelling approach to be related to the SPF method.
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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.
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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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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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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.
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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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Galbraith, Steven Douglas. "Iterations of elliptic curves." Thesis, Georgia Institute of Technology, 1991. http://hdl.handle.net/1853/28620.
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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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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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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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Khavanin, Mohammad. "The Method of Mixed Monotony and First Order Delay Differential Equations." Pontificia Universidad Católica del Perú, 2014. http://repositorio.pucp.edu.pe/index/handle/123456789/96643.
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In this paper I extend the method of mixed monotony, to construct monotone sequences that converge to the unique solution of an initial value delay differential equation.En este artículo se prueba una generalización del método de monotonía mixta, para construir sucesiones monótonas que convergen a la solución única de una ecuación diferencial de retraso con valor inicial.
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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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McKay, Melanie. "Iterative methods for incompressible flow." Thesis, University of Ottawa (Canada), 2009. http://hdl.handle.net/10393/28063.
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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.
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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.
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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.
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MacHardy, William R. "Iterative methods for parameter estimation." Thesis, Monterey, California : Naval Postgraduate School, 1990. http://handle.dtic.mil/100.2/ADA246174.
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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.
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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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Solov'ëv, Sergey I. "Preconditioned iterative methods for a class of nonlinear eigenvalue problems." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601389.
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In this paper we develop new preconditioned
iterative methods for solving monotone nonlinear
eigenvalue problems. We investigate the convergence
and derive grid-independent error estimates for
these methods. Numerical experiments demonstrate
the practical effectiveness of the proposed methods
for a model problem.
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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.
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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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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
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Helou, Neto Elias Salomão. "Algoritmos incrementais com aplicações em tomografia computadorizada." [s.n.], 2009. http://repositorio.unicamp.br/jspui/handle/REPOSIP/307603.
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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
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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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Berti, Lilian Ferreira 1988. "Iteração continuada aplicada ao método de pontos interiores." [s.n.], 2012. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306753.
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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
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Figueiredo, Patric. "Iterative method for solving inverse heat conduction problems." Master's thesis, Universidade de Aveiro, 2014. http://hdl.handle.net/10773/14400.
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