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1
Terkhova, Karina. "Capacitance matrix preconditioning." Thesis, University of Oxford, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.244593.
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Garay, Jose. "Asynchronous Optimized Schwarz Methods for Partial Differential Equations in Rectangular Domains." Diss., Temple University Libraries, 2018. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/510451.
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Mathematics<br>Ph.D.<br>Asynchronous iterative algorithms are parallel iterative algorithms in which communications and iterations are not synchronized among processors. Thus, as soon as a processing unit finishes its own calculations, it starts the next cycle with the latest data received during a previous cycle, without waiting for any other processing unit to complete its own calculation. These algorithms increase the number of updates in some processors (as compared to the synchronous case) but suppress most idle times. This usually results in a reduction of the (execution) time to achieve convergence. Optimized Schwarz methods (OSM) are domain decomposition methods in which the transmission conditions between subdomains contain operators of the form \linebreak $\partial/\partial \nu +\Lambda$, where $\partial/\partial \nu$ is the outward normal derivative and $\Lambda$ is an optimized local approximation of the global Steklov-Poincar\'e operator. There is more than one family of transmission conditions that can be used for a given partial differential equation (e.g., the $OO0$ and $OO2$ families), each of these families containing a particular approximation of the Steklov-Poincar\'e operator. These transmission conditions have some parameters that are tuned to obtain a fast convergence rate. Optimized Schwarz methods are fast in terms of iteration count and can be implemented asynchronously. In this thesis we analyze the convergence behavior of the synchronous and asynchronous implementation of OSM applied to solve partial differential equations with a shifted Laplacian operator in bounded rectangular domains. We analyze two cases. In the first case we have a shift that can be either positive, negative or zero, a one-way domain decomposition and transmission conditions of the $OO2$ family. In the second case we have Poisson's equation, a domain decomposition with cross-points and $OO0$ transmission conditions. In both cases we reformulate the equations defining the problem into a fixed point iteration that is suitable for our analysis, then derive convergence proofs and analyze how the convergence rate varies with the number of subdomains, the amount of overlap, and the values of the parameters introduced in the transmission conditions. Additionally, we find the optimal values of the parameters and present some numerical experiments for the second case illustrating our theoretical results. To our knowledge this is the first time that a convergence analysis of optimized Schwarz is presented for bounded subdomains with multiple subdomains and arbitrary overlap. The analysis presented in this thesis also applies to problems with more general domains which can be decomposed as a union of rectangles.<br>Temple University--Theses
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Tang, Wei-pai. "Schwarz splitting and template operators." Stanford, CA : Dept. of Computer Science, Stanford University, 1987. http://doi.library.cmu.edu/10.1184/OCLC/19643650.
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Thesis (Ph. D.)--Stanford University, 1987.<br>"June 1987." "Also numbered Classic-87-03"--Cover. "This research was supported by NASA Ames Consortium Agreement NASA NCA2-150 and Office of Naval Research Contracts N00014-86-K-0565, N00014-82-K-0335, N00014-75-C-1132"--P. vi. Includes bibliographical references (p. 125-129).
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Smith, James A. "“Looking for nothing" : Bayes linear methods for solving equations." Thesis, Durham University, 1993. http://etheses.dur.ac.uk/2207/.
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Here I will describe and implement Bayes linear methods for finding zeros of deterministic functions. We assume that the zero is known to be unique. Initially, the value of the function is modelled simply as the product of two independent factors, the position of the point from the zero and a "slope" which is assumed to vary "smoothly” with position. Additional prior information specifies first and second order properties of the slopes and the position of the zero: in particular, smoothness is specified by modelling the slope process to be stationary with a decreasing correlation function. This research is motivated by problems arising in large scale computer simulation of mathematical models of complex physical phenomena, where a single run of the code can be expensive and the output difficult to assimilate. Scientists are often confident about the structure of their model as a description of a physical process but may be uncertain about the values of certain model "parameters". Such parameters usually refer directly to physical attributes, and so collateral information about their values is usually available. In some applications, the physical process itself has been observed, and several runs of the code are made at different parameter settings in an attempt to match the realisation of the code with the actual realisation. The eventual aim is to aid scientists to search through the "parameter space” efficiently and systematically, using their knowledge of the process. Obviously, there are several respects in which this formulation does not tackle the real problem, as we mainly consider a single-valued function of a real variable. As well as considering this problem I will review the current state of play in the more general field of statistical numerical analysis and its relationship to deterministic computer experiments; and partial belief specification or Bayes linear methods
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Elmikkawy, M. E. A. "Embedded Runge-Kutta-Nystrom methods." Thesis, Teesside University, 1986. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.371400.
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Saravi, Masoud. "Numerical solution of linear ordinary differential equations and differential-algebraic equations by spectral methods." Thesis, Open University, 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.446280.
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This thesis involves the implementation of spectral methods, for numerical solution of linear Ordinary Differential Equations (ODEs) and linear Differential-Algebraic Equations (DAEs). First we consider ODEs with some ordinary problems, and then, focus on those problems in which the solution function or some coefficient functions have singularities. Then, by expressing weak and strong aspects of spectral methods to solve these kinds of problems, a modified pseudospectral method which is more efficient than other spectral methods is suggested and tested on some examples. We extend the pseudo-spectral method to solve a system of linear ODEs and linear DAEs and compare this method with other methods such as Backward Difference Formulae (BDF), and implicit Runge-Kutta (RK) methods using some numerical examples. Furthermore, by using appropriatec hoice of Gauss-Chebyshev-Radapuo ints, we will show that this method can be used to solve a linear DAE whenever some of coefficient functions have singularities by providing some examples. We also used some problems that have already been considered by some authors by finite difference methods, and compare their results with ours. Finally, we present a short survey of properties and numerical methods for solving DAE problems and then we extend the pseudo-spectral method to solve DAE problems with variable coefficient functions. Our numerical experience shows that spectral and pseudo-spectral methods and their modified versions are very promising for linear ODE and linear DAE problems with solution or coefficient functions having singularities. In section 3.2, a modified method for solving an ODE is introduced which is new work. Furthermore, an extension of this method for solving a DAE or system of ODEs which has been explained in section 4.6 of chapter four is also a new idea and has not been done by anyone previously. In all chapters, wherever we talk about ODE or DAE we mean linear.
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Campos, Frederico Ferreira. "Analysis of conjugate gradients-type methods for solving linear equations." Thesis, University of Oxford, 1995. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.282319.
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Blake, Kenneth William. "Moving mesh methods for non-linear parabolic partial differential equations." Thesis, University of Reading, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.369545.
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Shank, Stephen David. "Low-rank solution methods for large-scale linear matrix equations." Diss., Temple University Libraries, 2014. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/273331.
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Mathematics<br>Ph.D.<br>We consider low-rank solution methods for certain classes of large-scale linear matrix equations. Our aim is to adapt existing low-rank solution methods based on standard, extended and rational Krylov subspaces to solve equations which may viewed as extensions of the classical Lyapunov and Sylvester equations. The first class of matrix equations that we consider are constrained Sylvester equations, which essentially consist of Sylvester's equation along with a constraint on the solution matrix. These therefore constitute a system of matrix equations. The second are generalized Lyapunov equations, which are Lyapunov equations with additional terms. Such equations arise as computational bottlenecks in model order reduction.<br>Temple University--Theses
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Fischer, Rainer. "Multigrid methods for anisotropic and indefinite structured linear systems of equations." [S.l.] : [s.n.], 2006. http://mediatum2.ub.tum.de/doc/601806/document.pdf.
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Lopes, Antonio Roldao. "Accelerating iterative methods for solving systems of linear equations using FPGAs." Thesis, Imperial College London, 2010. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.526401.
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Ngounda, Edgard. "Numerical Laplace transformation methods for integrating linear parabolic partial differential equations." Thesis, Stellenbosch : University of Stellenbosch, 2009. http://hdl.handle.net/10019.1/2735.
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Thesis (MSc (Applied Mathematics))--University of Stellenbosch, 2009.<br>ENGLISH ABSTRACT: In recent years the Laplace inversion method has emerged as a viable alternative
method for the numerical solution of PDEs. Effective methods for the
numerical inversion are based on the approximation of the Bromwich integral.
In this thesis, a numerical study is undertaken to compare the efficiency of
the Laplace inversion method with more conventional time integrator methods.
Particularly, we consider the method-of-lines based on MATLAB’s ODE15s
and the Crank-Nicolson method.
Our studies include an introductory chapter on the Laplace inversion method.
Then we proceed with spectral methods for the space discretization where we
introduce the interpolation polynomial and the concept of a differentiation
matrix to approximate derivatives of a function. Next, formulas of the numerical
differentiation formulas (NDFs) implemented in ODE15s, as well as the
well-known second order Crank-Nicolson method, are derived. In the Laplace
method, to compute the Bromwich integral, we use the trapezoidal rule over
a hyperbolic contour. Enhancement to the computational efficiency of these
methods include the LU as well as the Hessenberg decompositions.
In order to compare the three methods, we consider two criteria: The
number of linear system solves per unit of accuracy and the CPU time per
unit of accuracy. The numerical results demonstrate that the new method,
i.e., the Laplace inversion method, is accurate to an exponential order of convergence
compared to the linear convergence rate of the ODE15s and the
Crank-Nicolson methods. This exponential convergence leads to high accuracy
with only a few linear system solves. Similarly, in terms of computational cost, the Laplace inversion method is more efficient than ODE15s and the
Crank-Nicolson method as the results show.
Finally, we apply with satisfactory results the inversion method to the axial
dispersion model and the heat equation in two dimensions.<br>AFRIKAANSE OPSOMMING: In die afgelope paar jaar het die Laplace omkeringsmetode na vore getree
as ’n lewensvatbare alternatiewe metode vir die numeriese oplossing van
PDVs. Effektiewe metodes vir die numeriese omkering word gebasseer op die
benadering van die Bromwich integraal.
In hierdie tesis word ’n numeriese studie onderneem om die effektiwiteit
van die Laplace omkeringsmetode te vergelyk met meer konvensionele tydintegrasie
metodes. Ons ondersoek spesifiek die metode-van-lyne, gebasseer
op MATLAB se ODE15s en die Crank-Nicolson metode.
Ons studies sluit in ’n inleidende hoofstuk oor die Laplace omkeringsmetode.
Dan gaan ons voort met spektraalmetodes vir die ruimtelike diskretisasie,
waar ons die interpolasie polinoom invoer sowel as die konsep van ’n
differensiasie-matriks waarmee afgeleides van ’n funksie benader kan word.
Daarna word formules vir die numeriese differensiasie formules (NDFs) ingebou
in ODE15s herlei, sowel as die welbekende tweede orde Crank-Nicolson
metode. Om die Bromwich integraal te benader in die Laplace metode, gebruik
ons die trapesiumreël oor ’n hiperboliese kontoer. Die berekeningskoste
van al hierdie metodes word verbeter met die LU sowel as die Hessenberg
ontbindings.
Ten einde die drie metodes te vergelyk beskou ons twee kriteria: Die aantal
lineêre stelsels wat moet opgelos word per eenheid van akkuraatheid, en
die sentrale prosesseringstyd per eenheid van akkuraatheid. Die numeriese resultate demonstreer dat die nuwe metode, d.i. die Laplace omkeringsmetode,
akkuraat is tot ’n eksponensiële orde van konvergensie in vergelyking tot
die lineêre konvergensie van ODE15s en die Crank-Nicolson metodes. Die
eksponensiële konvergensie lei na hoë akkuraatheid met slegs ’n klein aantal
oplossings van die lineêre stelsel. Netso, in terme van berekeningskoste is die
Laplace omkeringsmetode meer effektief as ODE15s en die Crank-Nicolson
metode.
Laastens pas ons die omkeringsmetode toe op die aksiale dispersiemodel
sowel as die hittevergelyking in twee dimensies, met bevredigende resultate.
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Shah, A. A. "Iteractive methods for the solution of unsymmetric linear systems of equations." Thesis, University of Kent, 1985. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.333045.
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Nicely, Dywayne A. Morgan Ronald Benjamin. "Restarting the Lanczos algorithm for large eigenvalue problems and linear equations." Waco, Tex. : Baylor University, 2008. http://hdl.handle.net/2104/5219.
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El-Nakla, Jehad A. H. "Finite difference methods for solving mildly nonlinear elliptic partial differential equations." Thesis, Loughborough University, 1987. https://dspace.lboro.ac.uk/2134/10417.
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This thesis is concerned with the solution of large systems of linear algebraic equations in which the matrix of coefficients is sparse. Such systems occur in the numerical solution of elliptic partial differential equations by finite-difference methods. By applying some well-known iterative methods, usually used to solve linear PDE systems, the thesis investigates their applicability to solve a set of four mildly nonlinear test problems. In Chapter 4 we study the basic iterative methods and semiiterative methods for linear systems. In particular, we derive and apply the CS, SOR, SSOR methods and the SSOR method extrapolated by the Chebyshev acceleration strategy. In Chapter 5, three ways of accelerating the SOR method are described together with the applications to the test problems. Also the Newton-SOR method and the SOR-Newton method are derived and applied to the same problems. In Chapter 6, the Alternating Directions Implicit methods are described. Two versions are studied in detail, namely, the Peaceman-Rachford and the Douglas-Rachford methods. They have been applied to the test problems for cycles of 1, 2 and 3 parameters. In Chapter 7, the conjugate gradients method and the conjugate gradient acceleration procedure are described together with some preconditioning techniques. Also an approximate LU-decomposition algorithm (ALUBOT algorithm) is given and then applied in conjunction with the Picard and Newton methods. Chapter 8 contains the final conclusions.
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Choi, Chi-Ho Francis. "Numerical methods for solving some non-linear reaction-diffusion equations in chemistry." Thesis, Brunel University, 2001. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.341555.
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Odland, Tove. "On Methods for Solving Symmetric Systems of Linear Equations Arising in Optimization." Doctoral thesis, KTH, Optimeringslära och systemteori, 2015. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-166675.
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In this thesis we present research on mathematical properties of methods for solv- ing symmetric systems of linear equations that arise in various optimization problem formulations and in methods for solving such problems. In the first and third paper (Paper A and Paper C), we consider the connection be- tween the method of conjugate gradients and quasi-Newton methods on strictly convex quadratic optimization problems or equivalently on a symmetric system of linear equa- tions with a positive definite matrix. We state conditions on the quasi-Newton matrix and the update matrix such that the search directions generated by the corresponding quasi-Newton method and the method of conjugate gradients respectively are parallel. In paper A, we derive such conditions on the update matrix based on a sufficient condition to obtain mutually conjugate search directions. These conditions are shown to be equivalent to the one-parameter Broyden family. Further, we derive a one-to-one correspondence between the Broyden parameter and the scaling between the search directions from the method of conjugate gradients and a quasi-Newton method em- ploying some well-defined update scheme in the one-parameter Broyden family. In paper C, we give necessary and sufficient conditions on the quasi-Newton ma- trix and on the update matrix such that equivalence with the method of conjugate gra- dients hold for the corresponding quasi-Newton method. We show that the set of quasi- Newton schemes admitted by these necessary and sufficient conditions is strictly larger than the one-parameter Broyden family. In addition, we show that this set of quasi- Newton schemes includes an infinite number of symmetric rank-one update schemes. In the second paper (Paper B), we utilize an unnormalized Krylov subspace frame- work for solving symmetric systems of linear equations. These systems may be incom- patible and the matrix may be indefinite/singular. Such systems of symmetric linear equations arise in constrained optimization. In the case of an incompatible symmetric system of linear equations we give a certificate of incompatibility based on a projection on the null space of the symmetric matrix and characterize a minimum-residual solu- tion. Further we derive a minimum-residual method, give explicit recursions for the minimum-residual iterates and characterize a minimum-residual solution of minimum Euclidean norm.<br>I denna avhandling betraktar vi matematiska egenskaper hos metoder för att lösa symmetriska linjära ekvationssystem som uppkommer i formuleringar och metoder för en mängd olika optimeringsproblem. I första och tredje artikeln (Paper A och Paper C), undersöks kopplingen mellan konjugerade gradientmetoden och kvasi-Newtonmetoder när dessa appliceras på strikt konvexa kvadratiska optimeringsproblem utan bivillkor eller ekvivalent på ett symmet- risk linjärt ekvationssystem med en positivt definit symmetrisk matris. Vi ställer upp villkor på kvasi-Newtonmatrisen och uppdateringsmatrisen så att sökriktningen som fås från motsvarande kvasi-Newtonmetod blir parallell med den sökriktning som fås från konjugerade gradientmetoden. I den första artikeln (Paper A), härleds villkor på uppdateringsmatrisen baserade på ett tillräckligt villkor för att få ömsesidigt konjugerade sökriktningar. Dessa villkor på kvasi-Newtonmetoden visas vara ekvivalenta med att uppdateringsstrategin tillhör Broydens enparameterfamilj. Vi tar också fram en ett-till-ett överensstämmelse mellan Broydenparametern och skalningen mellan sökriktningarna från konjugerade gradient- metoden och en kvasi-Newtonmetod som använder någon väldefinierad uppdaterings- strategi från Broydens enparameterfamilj. I den tredje artikeln (Paper C), ger vi tillräckliga och nödvändiga villkor på en kvasi-Newtonmetod så att nämnda ekvivalens med konjugerade gradientmetoden er- hålls. Mängden kvasi-Newtonstrategier som uppfyller dessa villkor är strikt större än Broydens enparameterfamilj. Vi visar också att denna mängd kvasi-Newtonstrategier innehåller ett oändligt antal uppdateringsstrategier där uppdateringsmatrisen är en sym- metrisk matris av rang ett. I den andra artikeln (Paper B), används ett ramverk för icke-normaliserade Krylov- underrumsmetoder för att lösa symmetriska linjära ekvationssystem. Dessa ekvations- system kan sakna lösning och matrisen kan vara indefinit/singulär. Denna typ av sym- metriska linjära ekvationssystem uppkommer i en mängd formuleringar och metoder för optimeringsproblem med bivillkor. I fallet då det symmetriska linjära ekvations- systemet saknar lösning ger vi ett certifikat för detta baserat på en projektion på noll- rummet för den symmetriska matrisen och karaktäriserar en minimum-residuallösning. Vi härleder även en minimum-residualmetod i detta ramverk samt ger explicita rekur- sionsformler för denna metod. I fallet då det symmetriska linjära ekvationssystemet saknar lösning så karaktäriserar vi en minimum-residuallösning av minsta euklidiska norm.<br><p>QC 20150519</p>
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Chow, Tanya L. M., of Western Sydney Macarthur University, and Faculty of Business and Technology. "Systems of partial differential equations and group methods." THESIS_FBT_XXX_Chow_T.xml, 1996. http://handle.uws.edu.au:8081/1959.7/43.
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This thesis is concerned with the derivation of similarity solutions for one-dimensional coupled systems of reaction - diffusion equations, a semi-linear system and a one-dimensional tripled system. The first area of research in this thesis involves a coupled system of diffusion equations for the existence of two distinct families of diffusion paths. Constructing one-parameter transformation groups preserving the invariance of this system of equations enables similarity solutions for this coupled system to be derived via the classical and non-classical procedures. This system of equation is the uncoupled in the hope of recovering further similarity solutions for the system. Once again, one-parameter groups leaving the uncoupled system invariant are obtained, enabling similarity solutions for the system to be elicited. A one-dimensional pattern formation in a model of burning forms the next component of this thesis. The primary focus of this area is the determination of similarity solutions for this reaction - diffusion system by means of one-parameter transformation group methods. Consequently, similarity solutions which are a generalisation of the solutions of the one-dimensional steady equations derived by Forbes are deduced. Attention in this thesis is then directed toward a semi-linear coupled system representing a predator - prey relationship. Two approaches to solving this system are made using the classical procedure, leading to one-parameter transformation groups which are instrumental in elicting the general similarity solution for this system. A triple system of equations representing a one-dimensional case of diffusion in the presence of three diffusion paths constitutes the next theme of this thesis. In association with the classical and non-classical procedures, the derivation of one-parameter transformation groups leaving this system invariant enables similarity solutions for this system to be deduced. The final strand of this thesis involves a one- dimensional case of the general linear system of coupled diffusion equations with cross-effects for which one-parameter transformation group methods are once more employed. The one-parameter groups constructed for this system prove instrumental in enabling the attainment of similarity solutions for this system to be accomplished<br>Faculty of Business and Technology
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Ernst, Oliver G. "Minimal and orthogonal residual methods and their generalizations for solving linear operator equations." Doctoral thesis, Technische Universitaet Bergakademie Freiberg Universitaetsbibliothek "Georgius Agricola", 2009. http://nbn-resolving.de/urn:nbn:de:swb:105-3293998.
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This thesis is concerned with the solution of linear operator equations by projection methods known as minimal residual (MR) and orthogonal residual (OR) methods. We begin with a rather abstract framework of approximation by orthogonal and oblique projection in Hilbert space. When these approximation schemes are applied to sequences of nested spaces, with a simple requirement relating trial and test spaces in case of the OR method, one can derive at this rather general level the basic relations which have been proved for many specific Krylov subspace methods for solving linear systems of equations in the literature. The crucial quantities with which we describe the behavior of these methods are angles between subspaces. By replacing the given inner product with one that is basis-dependent, one can also incorporate methods based on non-orthogonal bases such as those based on the non-Hermitian Lanczos process for solving linear systems. In fact, one can show that any reasonable approximation method based on a nested sequence of approximation spaces can be interpreted as an MR or OR method in this way. When these abstract approximation techniques are applied to the solution of linear operator equations, there are three generic algorithmic formulations, which we identify with some algorithms in the literature. Specializing further to Krylov trial and test spaces, we recover the well known Krylov subspace methods. Moreover, we show that our general framework also covers in a natural way many recent generalizations of Krylov subspace methods, which employ techniques such as augmentation, deflation, restarts and truncation. We conclude with a chapter on error and residual bounds, deriving some old and new results based on the angles framework. This work provides a natural and consistent framework for the sometimes confusing plethora of methods of Krylov subspace type introduced in the last 50 years.
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Johnsen, Pernilla. "Homogenization of Partial Differential Equations using Multiscale Convergence Methods." Licentiate thesis, Mittuniversitetet, Institutionen för matematik och ämnesdidaktik, 2021. http://urn.kb.se/resolve?urn=urn:nbn:se:miun:diva-42036.
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The focus of this thesis is the theory of periodic homogenization of partial differential equations and some applicable concepts of convergence. More precisely, we study parabolic problems exhibiting both spatial and temporal microscopic oscillations and a vanishing volumetric heat capacity type of coefficient. We also consider a hyperbolic-parabolic problem with two spatial microscopic scales. The tools used are evolution settings of multiscale and very weak multiscale convergence, which are extensions of, or closely related to, the classical method of two-scale convergence. The novelty of the research in the thesis is the homogenization results and, for the studied parabolic problems, adapted compactness results of multiscale convergence type.
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Alam, Md Shafiful. "Iterative Methods to Solve Systems of Nonlinear Algebraic Equations." TopSCHOLAR®, 2018. https://digitalcommons.wku.edu/theses/2305.
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Iterative methods have been a very important area of study in numerical analysis since the inception of computational science. Their use ranges from solving algebraic equations to systems of differential equations and many more. In this thesis, we discuss several iterative methods, however our main focus is Newton's method. We present a detailed study of Newton's method, its order of convergence and the asymptotic error constant when solving problems of various types as well as analyze several pitfalls, which can affect convergence. We also pose some necessary and sufficient conditions on the function f for higher order of convergence. Different acceleration techniques are discussed with analysis of the asymptotic behavior of the iterates. Analogies between single variable and multivariable problems are detailed. We also explore some interesting phenomena while analyzing Newton's method for complex variables.
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Eldred, Chris. "Linear and nonlinear properties of numerical methods for the rotating shallow water equations." Thesis, Colorado State University, 2015. http://pqdtopen.proquest.com/#viewpdf?dispub=3720474.
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<p> The shallow water equations provide a useful analogue of the fully compressible Euler equations since they have similar conservation laws, many of the same types of waves and a similar (quasi-) balanced state. It is desirable that numerical models posses similar properties, and the prototypical example of such a scheme is the 1981 Arakawa and Lamb (AL81) staggered (C-grid) total energy and potential enstrophy conserving scheme, based on the vector invariant form of the continuous equations. However, this scheme is restricted to a subset of logically square, orthogonal grids. The current work extends the AL81 scheme to arbitrary non-orthogonal polygonal grids, by combining Hamiltonian methods (work done by Salmon, Gassmann, Dubos and others) and Discrete Exterior Calculus (Thuburn, Cotter, Dubos, Ringler, Skamarock, Klemp and others). </p><p> It is also possible to obtain these properties (along with arguably superior wave dispersion properties) through the use of a collocated (Z-grid) scheme based on the vorticity-divergence form of the continuous equations. Unfortunately, existing examples of these schemes in the literature for general, spherical grids either contain computational modes; or do not conserve total energy and potential enstrophy. This dissertation extends an existing scheme for planar grids to spherical grids, through the use of Nambu brackets (as pioneered by Rick Salmon). </p><p> To compare these two schemes, the linear modes (balanced states, stationary modes and propagating modes; with and without dissipation) are examined on both uniform planar grids (square, hexagonal) and quasi-uniform spherical grids (geodesic, cubed-sphere). In addition to evaluating the linear modes, the results of the two schemes applied to a set of standard shallow water test cases and a recently developed forced-dissipative turbulence test case from John Thuburn (intended to evaluate the ability the suitability of schemes as the basis for a climate model) on both hexagonal-pentagonal icosahedral grids and cubed-sphere grids are presented. Finally, some remarks and thoughts about the suitability of these two schemes as the basis for atmospheric dynamical development are given.</p>
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Al-Shanfari, Fatima. "High-order in time discontinuous Galerkin finite element methods for linear wave equations." Thesis, Brunel University, 2017. http://bura.brunel.ac.uk/handle/2438/15332.
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In this thesis we analyse the high-order in time discontinuous Galerkin nite element method (DGFEM) for second-order in time linear abstract wave equations. Our abstract approximation analysis is a generalisation of the approach introduced by Claes Johnson (in Comput. Methods Appl. Mech. Engrg., 107:117-129, 1993), writing the second order problem as a system of fi rst order problems. We consider abstract spatial (time independent) operators, highorder in time basis functions when discretising in time; we also prove approximation results in case of linear constraints, e.g. non-homogeneous boundary data. We take the two steps approximation approach i.e. using high-order in time DGFEM; the discretisation approach in time introduced by D Schötzau (PhD thesis, Swiss Federal institute of technology, Zürich, 1999) to fi rst obtain the semidiscrete scheme and then conformal spatial discretisation to obtain the fully-discrete formulation. We have shown solvability, unconditional stability and conditional a priori error estimates within our abstract framework for the fully discretized problem. The skew-symmetric spatial forms arising in our abstract framework for the semi- and fully-discrete schemes do not full ll the underlying assumptions in D. Schötzau's work. But the semi-discrete and fully discrete forms satisfy an Inf-sup condition, essential for our proofs; in this sense our approach is also a generalisation of D. Schötzau's work. All estimates are given in a norm in space and time which is weaker than the Hilbert norm belonging to our abstract function spaces, a typical complication in evolution problems. To the best of the author's knowledge, with the approximation approach we used, these stability and a priori error estimates with their abstract structure have not been shown before for the abstract variational formulation used in this thesis. Finally we apply our abstract framework to the acoustic and an elasto-dynamic linear equations with non-homogeneous Dirichlet boundary data.
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Ahmad, Ab Rahman bin. "The AGE iterative methods for solving large linear systems occurring in differential equations." Thesis, Loughborough University, 1993. https://dspace.lboro.ac.uk/2134/32905.
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The work presented in this thesis is wholly concerned with the Alternating Group Explicit (AGE) iterative methods for solving large linear systems occurring in solving Ordinary and Partial Differential Equations (ODEs and PDEs) using finite difference approximations.
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Ernst, Oliver G. "Minimal and orthogonal residual methods and their generalizations for solving linear operator equations." Doctoral thesis, [S.l. : s.n.], 2000. https://tubaf.qucosa.de/id/qucosa%3A22355.
Full textAbstract:
This thesis is concerned with the solution of linear operator equations by projection methods known as minimal residual (MR) and orthogonal residual (OR) methods. We begin with a rather abstract framework of approximation by orthogonal and oblique projection in Hilbert space. When these approximation schemes are applied to sequences of nested spaces, with a simple requirement relating trial and test spaces in case of the OR method, one can derive at this rather general level the basic relations which have been proved for many specific Krylov subspace methods for solving linear systems of equations in the literature. The crucial quantities with which we describe the behavior of these methods are angles between subspaces. By replacing the given inner product with one that is basis-dependent, one can also incorporate methods based on non-orthogonal bases such as those based on the non-Hermitian Lanczos process for solving linear systems. In fact, one can show that any reasonable approximation method based on a nested sequence of approximation spaces can be interpreted as an MR or OR method in this way. When these abstract approximation techniques are applied to the solution of linear operator equations, there are three generic algorithmic formulations, which we identify with some algorithms in the literature. Specializing further to Krylov trial and test spaces, we recover the well known Krylov subspace methods. Moreover, we show that our general framework also covers in a natural way many recent generalizations of Krylov subspace methods, which employ techniques such as augmentation, deflation, restarts and truncation. We conclude with a chapter on error and residual bounds, deriving some old and new results based on the angles framework. This work provides a natural and consistent framework for the sometimes confusing plethora of methods of Krylov subspace type introduced in the last 50 years.
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Donfack, Simplice. "Methods and algorithms for solving linear systems of equations on massively parallel computers." Thesis, Paris 11, 2012. http://www.theses.fr/2012PA112042.
Full textAbstract:
Les processeurs multi-cœurs sont considérés de nos jours comme l'avenir des calculateurs et auront un impact important dans le calcul scientifique. Cette thèse présente une nouvelle approche de résolution des grands systèmes linéaires creux et denses, qui soit adaptée à l'exécution sur les futurs machines pétaflopiques et en particulier celles ayant un nombre important de cœurs. Compte tenu du coût croissant des communications comparé au temps dont les processeurs mettent pour effectuer les opérations arithmétiques, notre approche adopte le principe de minimisation des communications au prix de quelques calculs redondants et utilise plusieurs adaptations pour atteindre de meilleures performances sur les machines multi-cœurs. Nous décomposons le problème à résoudre en plusieurs phases qui sont ensuite mises en œuvre séparément. Dans la première partie, nous présentons un algorithme basé sur le partitionnement d'hypergraphe qui réduit considérablement le remplissage ("fill-in") induit lors de la factorisation LU des matrices creuses non symétriques. Dans la deuxième partie, nous présentons deux algorithmes de réduction de communication pour les factorisations LU et QR qui sont adaptés aux environnements multi-cœurs. La principale contribution de cette partie est de réorganiser les opérations de la factorisation de manière à réduire la sollicitation du bus tout en utilisant de façon optimale les ressources. Nous étendons ensuite ce travail aux clusters de processeurs multi-cœurs. Dans la troisième partie, nous présentons une nouvelle approche d'ordonnancement et d'optimisation. La localité des données et l'équilibrage des charges représentent un sérieux compromis pour le choix des méthodes d'ordonnancement. Sur les machines NUMA par exemple où la localité des données n'est pas une option, nous avons observé qu'en présence de perturbations systèmes (" OS noise"), les performances pouvaient rapidement se dégrader et devenir difficiles à prédire. Pour cela, nous présentons une approche combinant un ordonnancement statique et dynamique pour ordonnancer les tâches de nos algorithmes. Nos résultats obtenues sur plusieurs architectures montrent que tous nos algorithmes sont efficaces et conduisent à des gains de performances significatifs. Nous pouvons atteindre des améliorations de l'ordre de 30 à 110% par rapport aux correspondants de nos algorithmes dans les bibliothèques numériques bien connues de la littérature<br>Multicore processors are considered to be nowadays the future of computing, and they will have an important impact in scientific computing. In this thesis, we study methods and algorithms for solving efficiently sparse and dense large linear systems on future petascale machines and in particular these having a significant number of cores. Due to the increasing communication cost compared to the time the processors take to perform arithmetic operations, our approach embrace the communication avoiding algorithm principle by doing some redundant computations and uses several adaptations to achieve better performance on multicore machines.We decompose the problem to solve into several phases that would be then designed or optimized separately. In the first part, we present an algorithm based on hypergraph partitioning and which considerably reduces the fill-in incurred in the LU factorization of sparse unsymmetric matrices. In the second part, we present two communication avoiding algorithms that are adapted to multicore environments. The main contribution of this part is to reorganize the computations such as to reduce bus contention and using efficiently resources. Then, we extend this work for clusters of multi-core processors. In the third part, we present a new scheduling and optimization approach. Data locality and load balancing are a serious trade-off in the choice of the scheduling strategy. On NUMA machines for example, where the data locality is not an option, we have observed that in the presence of noise, performance could quickly deteriorate and become difficult to predict. To overcome this bottleneck, we present an approach that combines a static and a dynamic scheduling approach to schedule the tasks of our algorithms.Our results obtained on several architectures show that all our algorithms are efficient and lead to significant performance gains. We can achieve from 30 up to 110% improvement over the corresponding routines of our algorithms in well known libraries
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Tang, Shaowu [Verfasser]. "Multiscale and geometric methods for linear elliptic and parabolic partial differential equations / Shaowu Tang." Bremen : IRC-Library, Information Resource Center der Jacobs University Bremen, 2008. http://d-nb.info/1034787411/34.
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Liesen, Jörg. "Construction and analysis of polynomial iterative methods for non-hermitian systems of linear equations." [S.l. : s.n.], 1998. http://deposit.ddb.de/cgi-bin/dokserv?idn=955877776.
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Fu, Hongbo. "Differential methods for intuitive 3D shape modeling /." View abstract or full-text, 2007. http://library.ust.hk/cgi/db/thesis.pl?CSED%202007%20FU.
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Krueger, Denise A. "Stabilized Finite Element Methods for Feedback Control of Convection Diffusion Equations." Diss., Virginia Tech, 2004. http://hdl.handle.net/10919/11214.
Full textAbstract:
We study the behavior of numerical stabilization schemes in the context of linear quadratic regulator (LQR) control problems for convection diffusion equations. The motivation for this effort comes from the observation that when linearization is applied to fluid flow control problems the resulting equations have the form of a convection diffusion equation. This effort is focused on the specific problem of computing the feedback functional gains that are the kernels of the feedback operators defined by solutions of operator Riccati equations. We develop a stabilization scheme based on the Galerkin Least Squares (GLS) method and compare this scheme to the standard Galerkin finite element method. We use cubic B-splines in order to keep the higher order terms that occur in GLS formulation. We conduct a careful numerical investigation into the convergence and accuracy of the functional gains computed using stabilization. We also conduct numerical studies of the role that the stabilization parameter plays in this convergence. Overall, we discovered that stabilization produces much better approximations to the functional gains on coarse meshes than the unstabilized method and that adjustments in the stabilization parameter greatly effects the accuracy and convergence rates. We discovered that the optimal stabilization parameter for simulation and steady state analysis is not necessarily optimal for solving the Riccati equation that defines the functional gains. Finally, we suggest that the stabilized GLS method might provide good initial values for iterative schemes on coarse meshes.<br>Ph. D.
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Wise, Steven M. "POLSYS_PLP: A Partitioned Linear Product Homotopy Code for Solving Polynomial Systems of Equations." Thesis, Virginia Tech, 1998. http://hdl.handle.net/10919/36933.
Full textAbstract:
Globally convergent, probability-one homotopy methods have proven to be very effective for finding all the isolated solutions to polynomial systems of equations. After many years of development, homotopy path trackers based on probability-one homotopy methods are reliable and fast. Now, theoretical advances reducing the number of homotopy paths that must be tracked, and in the handling of singular solutions, have made probability-one homotopy methods even more practical. This thesis describes the theory behind and performance of the new code POLSYS_PLP, which consists of Fortran 90 modules for finding all isolated solutions of a complex coefficient polynomial system of equations by a probability-one homotopy method. The package is intended to be used in conjunction with HOMPACK90, and makes extensive use of Fortran 90 derived data types to support a partitioned linear product (PLP) polynomial system structure. PLP structure is a generalization of m-homogeneous structure, whereby each component of the system can have a different m-homogeneous structure. POLSYS_PLP employs a sophisticated power series end game for handling singular solutions, and provides support for problem definition both at a high level and via hand-crafted code. Different PLP structures and their corresponding Bezout numbers can be systematically explored before committing to root finding.<br>Master of Science
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Indratno, Sapto Wahyu. "Numerical methods for solving linear ill-posed problems." Diss., Kansas State University, 2011. http://hdl.handle.net/2097/8109.
Full textAbstract:
Doctor of Philosophy<br>Department of Mathematics<br>Alexander G. Ramm<br>A new method, the Dynamical Systems Method (DSM), justified
recently, is applied to solving ill-conditioned linear algebraic
system (ICLAS). The DSM gives a new approach to solving a wide class
of ill-posed problems. In Chapter 1 a new iterative scheme for
solving ICLAS is proposed. This iterative scheme is based on the DSM
solution. An a posteriori stopping rules for the proposed method is
justified. We also gives an a posteriori stopping rule for a
modified iterative scheme developed in A.G.Ramm, JMAA,330
(2007),1338-1346, and proves convergence of the solution obtained by
the iterative scheme. In Chapter 2 we give a convergence analysis of
the following iterative scheme:
u[subscript]n[superscript]delta=q u[subscript](n-1)[superscript]delta+(1-q)T[subscript](a[subscript]n)[superscript](-1) K[superscript]*f[subscript]delta, u[subscript]0[superscript]delta=0,
where T:=K[superscript]* K, T[subscript]a :=T+aI, q in the interval (0,1),\quad
a[subscript]n := alpha[subscript]0 q[superscript]n, alpha_0>0, with finite-dimensional
approximations of T and K[superscript]* for solving stably Fredholm integral
equations of the first kind with noisy data. In Chapter 3 a new
method for inverting the Laplace transform from the real axis is
formulated. This method is based on a quadrature formula. We assume
that the unknown function f(t) is continuous with (known) compact
support. An adaptive iterative method and an adaptive stopping rule,
which yield the convergence of the approximate solution to f(t),
are proposed in this chapter.
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Voigtmann, Steffen. "General linear methods for integrated circuit design." Doctoral thesis, Berlin Logos-Verl, 2006. http://deposit.d-nb.de/cgi-bin/dokserv?id=2850248&prov=M&dok_var=1&dok_ext=htm.
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Solodukhov, Yuri Olegovich. "Reduced-basis methods applied to locally non-affine and locally non-linear partial differential equations." Thesis, Massachusetts Institute of Technology, 2005. http://hdl.handle.net/1721.1/32390.
Full textAbstract:
Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mechanical Engineering, 2005.<br>Includes bibliographical references (p. 191-196).<br>In modern engineering and scientific applications there is a huge demand for solutions of parameter-based partial differential equations and associated outputs of interest expressed as functionals of these solutions. Areas that require solving partial differential equations include - but are not restricted to heat transfer, elasticity, and fluid dynamics. Since in most cases it is not feasible to obtain an analytic solution, many numerical approaches to obtain approximate numerical solutions - such as finite elements. finite differences, finite volumes have been developed. For applications like optimization. design, and inverse problems, where it is crucial to evaluate the field solution/output repeatedly, it might be overly computationally expensive to apply conventional numerical methods. To address this issue we present and compare two new reduced basis techniques for the rapid and reliable prediction of linear functional outputs of linear elliptic partial differential equations with locally non-affine parameter dependence: the partition of unity method (PUM) and the minimax coefficient approximation method (MCAM). We also describe the minimax coefficient approximation method (MCAM) in application to locally non-linear elliptic partial differential equations.<br>(cont.) The essential components for both the PUM and the MCAM are (i) (provably) rapidly convergent global reduced basis approximations Galerkin projection onto a low-dimensional space spanned by the solutions of the governing partial differential equation at N selected points in the parameter space; (ii) a posteriori error estimation - relaxations of the error-residual equation that provide inexpensive yet sharp and rigorous bounds for the error in the outputs of interest; and (iii) off- line/on-line computational procedures - methods which decouple the generation and projection stages of the approximation process. The operation count for the on-line stage - in which, given a new parameter value, we calculate the output of interest and associated error bound depends only on N (typically very small), the affine parametric complexity of the problem ad the number of points in the region where the non-affine/non-linear dependence is observed. The ratio of the error bound to the real error (which we call effectivity or the sharpness of our error estimate) typically does not exceed 100. The partition of unity approach relies on domain decomposition with respect to the separation of the affine part from the non-affine part and estimation of contributions to the error bound from these two parts. The minimax coefficient approximation approach is based on approximating the non-affine/non-linear dependence with an affine-like approximation and the subsequent treatment of the problem based on the ideas previously developed for affine problems. As a test for these new methods we consider several model problems involving steady heat transfer.<br>(cont.) Numerical results are provided with respect to the accuracy and computational savings provided by the described reduced basis methods.<br>by Yuri Olegovich Solodukhov.<br>Ph.D.
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Ramirez, Edgardo II. "Finite element methods for parameter identification problem of linear and nonlinear steady-state diffusion equations." Diss., Virginia Tech, 1997. http://hdl.handle.net/10919/28173.
Full textAbstract:
We study a parameter identification problem for the steady state
diffusion equations. In this thesis, we transform this
identification problem into a minimization problem by considering an
appropriate cost functional and propose a finite element method for the
identification of the parameter for the linear and nonlinear partial
differential equation.
The cost functional involves the classical
output least square term, a term approximating the
derivative of the piezometric head <I>u(x)</I>, an equation error term
plus some regularization terms,
which happen to be a norm or a semi-norm of the variables in the cost functional in an
appropriate Sobolev space. The existence and uniqueness of the
minimizer for the cost functional is proved.
Error estimates in a weighted
<I>H</I><sup>-1</sup>-norm, <I>L</I><sup>2</sup>-norm and <I>L</I><sup>1</sup>-norm for the numerical solution are derived.
Numerical examples
will be given to show features of this numerical method.<br>Ph. D.
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Zhang, Hong. "Efficient Time Stepping Methods and Sensitivity Analysis for Large Scale Systems of Differential Equations." Diss., Virginia Tech, 2014. http://hdl.handle.net/10919/50492.
Full textAbstract:
Many fields in science and engineering require large-scale numerical simulations of complex systems described by differential equations. These systems are typically multi-physics (they are driven by multiple interacting physical processes) and multiscale (the dynamics takes place on vastly different spatial and temporal scales). Numerical solution of such systems is highly challenging due to the dimension of the resulting discrete problem, and to the complexity that comes from incorporating multiple interacting components with different characteristics. The main contributions of this dissertation are the creation of new families of time integration methods for multiscale and multiphysics simulations, and the development of industrial-strengh tools for sensitivity analysis.
This work develops novel implicit-explicit (IMEX) general linear time integration methods for multiphysics and multiscale simulations typically involving both stiff and non-stiff components. In an IMEX approach, one uses an implicit scheme for the stiff components and an explicit scheme for the non-stiff components such that the combined method has the desired stability and accuracy properties. Practical schemes with favorable properties, such as maximized stability, high efficiency, and no order reduction, are constructed and applied in extensive numerical experiments to validate the theoretical findings and to demonstrate their advantages. Approximate matrix factorization (AMF) technique exploits the structure of the Jacobian of the implicit parts, which may lead to further efficiency improvement of IMEX schemes. We have explored the application of AMF within some high order IMEX Runge-Kutta schemes in order to achieve high efficiency.
Sensitivity analysis gives quantitative information about the changes in a dynamical model outputs caused by caused by small changes in the model inputs. This information is crucial for data assimilation, model-constrained optimization, inverse problems, and uncertainty quantification. We develop a high performance software package for sensitivity analysis in the context of stiff and nonstiff ordinary differential equations. Efficiency is demonstrated by direct comparisons against existing state-of-art software on a variety of test problems.<br>Ph. D.
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Cunha, Rudnei Dias da. "A study of iterative methods for the solution of systems of linear equations on transputer networks." Thesis, University of Kent, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.314642.
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Eneyew, Eyaya Birara. "Efficient computation of shifted linear systems of equations with application to PDEs." Thesis, Stellenbosch : Stellenbosch University, 2011. http://hdl.handle.net/10019.1/17827.
Full textAbstract:
Thesis (MSc)--Stellenbosch University, 2011.<br>ENGLISH ABSTRACT: In several numerical approaches to PDEs shifted linear systems of the form (zI - A)x = b, need to be solved for several values of the complex scalar z. Often, these linear systems
are large and sparse. This thesis investigates efficient numerical methods for these systems
that arise from a contour integral approximation to PDEs and compares these methods
with direct solvers.
In the first part, we present three model PDEs and discuss numerical approaches to solve
them. We use the first problem to demonstrate computations with a dense matrix, the
second problem to demonstrate computations with a sparse symmetric matrix and the
third problem for a sparse but nonsymmetric matrix. To solve the model PDEs numerically
we apply two space discrerization methods, namely the finite difference method and the
Chebyshev collocation method. The contour integral method mentioned above is used to
integrate with respect to the time variable.
In the second part, we study a Hessenberg reduction method for solving shifted linear
systems with a dense matrix and present numerical comparison of it with the built-in
direct linear system solver in SciPy. Since both are direct methods, in the absence of
roundoff errors, they give the same result. However, we find that the Hessenberg reduction
method is more efficient in CPU-time than the direct solver. As application we solve a
one-dimensional version of the heat equation.
In the third part, we present efficient techniques for solving shifted systems with a sparse
matrix by Krylov subspace methods. Because of their shift-invariance property, the Krylov
methods allow one to obtain approximate solutions for all values of the parameter, by generating a single approximation space. Krylov methods applied to the linear systems are
generally slowly convergent and hence preconditioning is necessary to improve the convergence.
The use of shift-invert preconditioning is discussed and numerical comparisons with
a direct sparse solver are presented. As an application we solve a two-dimensional version
of the heat equation with and without a convection term. Our numerical experiments
show that the preconditioned Krylov methods are efficient in both computational time and
memory space as compared to the direct sparse solver.<br>AFRIKAANSE OPSOMMING: In verskeie numeriese metodes vir PDVs moet geskuifde lineêre stelsels van die vorm (zI − A)x = b, opgelos word vir verskeie waardes van die komplekse skalaar z. Hierdie stelsels is dikwels
groot en yl. Hierdie tesis ondersoek numeriese metodes vir sulke stelsels wat voorkom in
kontoerintegraalbenaderings vir PDVs en vergelyk hierdie metodes met direkte metodes
vir oplossing.
In die eerste gedeelte beskou ons drie model PDVs en bespreek numeriese benaderings
om hulle op te los. Die eerste probleem word gebruik om berekenings met ’n vol matriks
te demonstreer, die tweede probleem word gebruik om berekenings met yl, simmetriese
matrikse te demonstreer en die derde probleem vir yl, onsimmetriese matrikse. Om die
model PDVs numeries op te los beskou ons twee ruimte-diskretisasie metodes, naamlik
die eindige-verskilmetode en die Chebyshev kollokasie-metode. Die kontoerintegraalmetode
waarna hierbo verwys is word gebruik om met betrekking tot die tydveranderlike te
integreer.
In die tweede gedeelte bestudeer ons ’n Hessenberg ontbindingsmetode om geskuifde lineêre
stelsels met ’n vol matriks op te los, en ons rapporteer numeriese vergelykings daarvan met
die ingeboude direkte oplosser vir lineêre stelsels in SciPy. Aangesien beide metodes direk
is lewer hulle dieselfde resultate in die afwesigheid van afrondingsfoute. Ons het egter
bevind dat die Hessenberg ontbindingsmetode meer effektief is in terme van rekenaartyd in
vergelyking met die direkte oplosser. As toepassing los ons ’n een-dimensionele weergawe
van die hittevergelyking op.
In die derde gedeelte beskou ons effektiewe tegnieke om geskuifde stelsels met ’n yl matriks op te los, met Krylov subruimte-metodes. As gevolg van hul skuifinvariansie eienskap,
laat die Krylov metodes mens toe om benaderde oplossings te verkry vir alle waardes van
die parameter, deur slegs een benaderingsruimte voort te bring. Krylov metodes toegepas
op lineêre stelsels is in die algemeen stadig konvergerend, en gevolglik is prekondisionering
nodig om die konvergensie te verbeter. Die gebruik van prekondisionering gebasseer
op skuif-en-omkeer word bespreek en numeriese vergelykings met direkte oplossers word
aangebied. As toepassing los ons ’n twee-dimensionele weergawe van die hittevergelyking
op, met ’n konveksie term en daarsonder. Ons numeriese eksperimente dui aan dat die
Krylov metodes met prekondisionering effektief is, beide in terme van berekeningstyd en
rekenaargeheue, in vergelyking met die direkte metodes.
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Dolean, Victorita. "Algorithmes par decomposition de domaine et méthodes de discrétisation d'ordre elevé pour la résolution des systèmes d'équations aux dérivées partielles. Application aux problèmes issus de la mécanique des fluides et de l'électromagnétisme." Habilitation à diriger des recherches, Université de Nice Sophia-Antipolis, 2009. http://tel.archives-ouvertes.fr/tel-00413574.
Full textAbstract:
My main research topic is about developing new domain decomposition algorithms for the solution of systems of partial differential equations. This was mainly applied to fluid dynamics problems (as compressible Euler or Stokes equations) and electromagnetics (time-harmonic and time-domain first order system of Maxwell's equations). Since the solution of large linear systems is strongly related to the application of a discretization method, I was also interested in developing and analyzing the application of high order methods (such as Discontinuos Galerkin methods) to Maxwell's equations (sometimes in conjuction with time-discretization schemes in the case of time-domain problems). As an active member of NACHOS pro ject (besides my main afiliation as an assistant professor at University of Nice), I had the opportunity to develop certain directions in my research, by interacting with permanent et non-permanent members (Post-doctoral researchers) or participating to supervision of PhD Students. This is strongly refflected in a part of my scientific contributions so far. This memoir is composed of three parts: the first is about the application of Schwarz methods to fluid dynamics problems; the second about the high order methods for the Maxwell's equations and the last about the domain decomposition algorithms for wave propagation problems.
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Hoskins, Jeremy G. "The application of symmetry methods and conservation laws to ordinary differential equations and a linear wave equation." Thesis, University of British Columbia, 2012. http://hdl.handle.net/2429/43321.
Full textAbstract:
Symmetry analysis and conservation laws are widely used in analyzing and solving differential equations. Conservation laws are also called first integrals when dealing with ordinary differential equations (ODEs). In this thesis, the complementary nature of these two approaches is explored; specifically, the use of symmetries to find integrating factors and, conversely, the use of conservation laws to seek new symmetries. In Part 1, building upon results in [3] and [10], it is shown that a higher-order symmetries of an ODE induces a point symmetry of the corresponding integrating factor determining equations (IFDE), and an explicit expression for this induced symmetry is obtained. Secondly, it is shown that the converse also holds for a special class of Lie point symmetries of the IFDE; namely, all Lie point symmetries of the IFDE which are of this form project onto point symmetries of the original scalar ODE.
In Part 2, the use of conservation laws to find non-local symmetries is shown for a linear one-dimensional wave equation in a two-layered medium with a smooth transition layer. The resulting analytic solutions are then studied in order to investigate the effect of the transmission and reflection of energy between the two media. It is found that the reflection and transmission coefficients depend on the ratio of the wave speeds in the two media as well as the ratio of the characteristic length of the incoming signal to the width of the transition layer. Approximations of the dependence of the reflection and transmission coefficients on these two parameters are also presented, obtained via numerical experiments performed using both the analytic solution and a finite element method.
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Magruder, Robin L. "SOLVING LINEAR EQUATIONS: A COMPARISON OF CONCRETE AND VIRTUAL MANIPULATIVES IN MIDDLE SCHOOL MATHEMATICS." UKnowledge, 2012. http://uknowledge.uky.edu/edc_etds/2.
Full textAbstract:
The purpose of this embedded quasi-experimental mixed methods research was to use solving simple linear equations as the lens for looking at the effectiveness of concrete and virtual manipulatives as compared to a control group using learning methods without manipulatives. Further, the researcher wanted to investigate unique benefits and drawbacks associated with each manipulative.
Qualitative research methods such as observation, teacher interviews, and student focus group interviews were employed. Quantitative data analysis techniques were used to analyze pretest and posttest data of middle school students (n=76). ANCOVA, analysis of covariance, uncovered statistically significant differences in favor of the control group. Differences in posttest scores, triangulated with qualitative data, suggested that concrete and virtual manipulatives require more classroom time because of administrative issues and because of time needed to learn how to operate the manipulative in addition to necessary time to learn mathematics content. Teachers must allow students enough time to develop conceptual understanding linking the manipulatives to the mathematics represented. Additionally, a discussion of unique benefits and drawbacks of each manipulative sheds light on the use of manipulatives in middle school mathematics.
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Pipkins, Daniel Scott. "Non-linear analysis of (i) wave propagation using transform methods and (ii) plates and shells using integral equations." Diss., Georgia Institute of Technology, 1992. http://hdl.handle.net/1853/20052.
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Gärtner, Christian [Verfasser], and Christian [Akademischer Betreuer] Bender. "Primal-dual methods for dynamic programming equations arising in non-linear option pricing / Christian Gärtner ; Betreuer: Christian Bender." Saarbrücken : Saarländische Universitäts- und Landesbibliothek, 2017. http://d-nb.info/1154438392/34.
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Arnold, Andrea. "Sequential Monte Carlo Parameter Estimation for Differential Equations." Case Western Reserve University School of Graduate Studies / OhioLINK, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=case1396617699.
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Jung, Michael, and Todor D. Todorov. "On the Convergence Factor in Multilevel Methods for Solving 3D Elasticity Problems." Universitätsbibliothek Chemnitz, 2006. http://nbn-resolving.de/urn:nbn:de:swb:ch1-200601510.
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The constant gamma in the strengthened Cauchy-Bunyakowskii-Schwarz inequality is a basic tool for constructing of two-level and multilevel preconditioning matrices. Therefore many authors consider estimates or computations of this quantity. In this paper the bilinear form arising from 3D linear elasticity problems is considered on a polyhedron. The cosine of the abstract angle between multilevel finite element subspaces is computed by a spectral analysis of a general eigenvalue problem. Octasection and bisection approaches are used for refining the triangulations. Tetrahedron, pentahedron and hexahedron meshes are considered. The dependence of the constant $\gamma$ on the Poisson ratio is presented graphically.
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Murphy, Steven. "Methods for solving discontinuous-Galerkin finite element equations with application to neutron transport." Phd thesis, Toulouse, INPT, 2015. http://oatao.univ-toulouse.fr/14650/1/murphy.pdf.
Full textAbstract:
We consider high order discontinuous-Galerkin finite element methods for partial differential equations, with a focus on the neutron transport equation. We begin by examining a method for preprocessing block-sparse matrices, of the type that arise from discontinuous-Galerkin methods, prior to factorisation by a multifrontal solver. Numerical experiments on large two and three dimensional matrices show that this pre-processing method achieves a significant reduction in fill-in, when compared to methods that fail to exploit block structures. A discontinuous-Galerkin finite element method for the neutron transport equation is derived that employs high order finite elements in both space and angle. Parallel Krylov subspace based solvers are considered for both source problems and $k_{eff}$-eigenvalue problems. An a-posteriori error estimator is derived and implemented as part of an h-adaptive mesh refinement algorithm for neutron transport $k_{eff}$-eigenvalue problems. This algorithm employs a projection-based error splitting in order to balance the computational requirements between the spatial and angular parts of the computational domain. An hp-adaptive algorithm is presented and results are collected that demonstrate greatly improved efficiency compared to the h-adaptive algorithm, both in terms of reduced computational expense and enhanced accuracy. Computed eigenvalues and effectivities are presented for a variety of challenging industrial benchmarks. Accurate error estimation (with effectivities of 1) is demonstrated for a collection of problems with inhomogeneous, irregularly shaped spatial domains as well as multiple energy groups. Numerical results are presented showing that the hp-refinement algorithm can achieve exponential convergence with respect to the number of degrees of freedom in the finite element space
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Iwamura, Chihiro, and chihiro_iwamura@ybb ne jp. "A fast solver for large systems of linear equations for finite element analysis on unstructured meshes." Swinburne University of Technology, 2004. http://adt.lib.swin.edu.au./public/adt-VSWT20051020.091538.
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The objective of this thesis is to develop a more efficient solver for a large system of linear equations arising from finite element discretization on unstructured tetrahedral meshes for a scalar elliptic partial differential equation of second order for pressure in a commercial computational fluid dynamics (CFD) simulation.
Segregated solution methods (or pressure correction type methods) are a widely used approach to obtain solutions of Navier-Stokes equations during numerical simulation by many commercial CFD codes. At each time step, these simulations usually require the approximate solution of a series of scalar equations for velocity, pressure and temperature. Even if the simulation does not require high-accuracy approximations, the large systems of linear equations for pressure may not be efficiently solved. The matrices of these systems of linear equations of real-life industry problems often strongly violate weak diagonal dominance and the numerical simulation often requires solutions of very large systems with over a few hundred thousands degrees of freedom. These conditions produce very ill-conditioned systems of linear equations. Therefore, it is very difficult to solve such systems of linear equations efficiently using most currently available common iterative solvers.
A survey of solvers for systems of linear equations was undertaken to determine the preferred solution methodology. An algebraic multigrid preconditioned conjugate gradient (AMGPCG) method solver was chosen for these problems. This solver uses the algebraic multigrid (AMG) cycle as a preconditioner for the conjugate gradient (CG) method. The disadvantages of the conventional AMG method are an expensive setup time and large memory requirements, particularly for three dimensional problems. The disadvantage of an expensive setup time needs to be overcome because the simulation usually requires only low-accuracy approximations for pressure. Also it is important to overcome the disadvantage of the large memory requirements for use in commercial software.
In this work, an efficient AMGPCG solver is developed by overcoming the disadvantages of the conventional AMG method. The robustness of AMGPCG is shown theoretically so that the solver is always convergent. Optimum or close to optimum rates of convergence behavior for the solver are shown numerically so that the number of necessary iterations to obtain the estimated solution is approximately independent of mesh resolution. Furthermore, numerical experiments of solving pressure for some industry problems were carried out and compared with other efficient solvers including a fast commercial AMGPCG solver (SAMG, release 20b1). It was found that the developed AMGPCG solver was the fastest among these solvers for solving these problems and its algorithm has been numerically proven to be efficient. In addition, the memory requirement is at an acceptable level for commercial CFD codes.
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Schneider, Olaf. "Krylov subspace methods and their generalizations for solving singular linear operator equations with applications to continuous time Markov chains." Doctoral thesis, Technische Universitaet Bergakademie Freiberg Universitaetsbibliothek "Georgius Agricola", 2009. http://nbn-resolving.de/urn:nbn:de:bsz:105-1148840.
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Viele Resultate über MR- und OR-Verfahren zur Lösung linearer Gleichungssysteme bleiben (in leicht modifizierter Form) gültig, wenn der betrachtete Operator nicht invertierbar ist. Neben dem für reguläre Probleme charakteristischen Abbruchverhalten, kann bei einem singulären Gleichungssystem auch ein so genannter singulärer Zusammenbruch auftreten. Für beide Fälle werden verschiedene Charakterisierungen angegeben. Die Unterrauminverse, eine spezielle verallgemeinerte Inverse, beschreibt die Näherungen eines MR-Unterraumkorrektur-Verfahrens. Für Krylov-Unterräume spielt die Drazin-Inverse eine Schlüsselrolle. Bei Krylov-Unterraum-Verfahren kann a-priori entschieden werden, ob ein regulärer oder ein singulärer Abbruch auftritt. Wir können zeigen, dass ein Krylov-Verfahren genau dann für beliebige Startwerte eine Lösung des linearen Gleichungssystems liefert, wenn der Index der Matrix nicht größer als eins und das Gleichungssystem konsistent ist. Die Berechnung stationärer Zustandsverteilungen zeitstetiger Markov-Ketten mit endlichem Zustandsraum stellt eine praktische Aufgabe dar, welche die Lösung eines singulären linearen Gleichungssystems erfordert. Die Eigenschaften der Übergangs-Halbgruppe folgen aus einfachen Annahmen auf rein analytischem und matrixalgebrischen Wege. Insbesondere ist die erzeugende Matrix eine singuläre M-Matrix mit Index 1. Ist die Markov-Kette irreduzibel, so ist die stationäre Zustandsverteilung eindeutig bestimmt.
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Anderson, Curtis James. "Estimating the Optimal Extrapolation Parameter for Extrapolated Iterative Methods When Solving Sequences of Linear Systems." University of Akron / OhioLINK, 2013. http://rave.ohiolink.edu/etdc/view?acc_num=akron1383826559.
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Campos, Fabio Antonio Araujo de 1984. "Métodos matemáticos para o problema de acústica linear estocástica." [s.n.], 2015. http://repositorio.unicamp.br/jspui/handle/REPOSIP/306070.
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Orientador: Maria Cristina de Castro Cunha<br>Tese (doutorado) - Universidade Estadual de Campinas, Instituto de Matemática Estatística e Computação Científica<br>Made available in DSpace on 2018-08-26T19:33:01Z (GMT). No. of bitstreams: 1
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Previous issue date: 2015<br>Resumo: Neste trabalho estudamos o sistema de equações diferenciais estocásticas obtido na linearização do modelo de propagação de ondas acústicas. Mais especificamente, analisamos métodos para solução do sistema de equações diferenciais usado na acústica linear, onde a matriz com dados aleatórios e um vetor de funções aleatórias que define as condições iniciais. Além do tradicional Método de Monte Carlo aplicamos o Método de Transformações de Variáveis Aleatórias e o Método de Galerkin Estocástico. Apresentamos resultados obtidos usando diferentes distribuições de probabilidades dos dados do problema. Também comparamos os métodos através da distribuição de probabilidade e momentos estatísticos da solução<br>Abstract: On the present work we study the system of stochastic differential equations obtained from the linearization of the propagation model of acoustic waves. More specifically we analyze methods for the solution of the system of differential equations used in the linear acoustics, where the matrix with random data and a vector of random functions defining initial conditions. In addition to the traditional Monte Carlo Method we apply the Variable Transformations of Random Method and the Galerkin Stochastic Method. We present results obtained using different probability distributions of problem data. We also compared the methods through the distribution of probabilities and statistical moments of the solution<br>Doutorado<br>Matematica Aplicada<br>Doutor em Matemática Aplicada