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Dissertations / Theses on the topic 'Preconditioned Krylov subspace method'

Author: Grafiati

Published: 4 June 2021

Last updated: 16 February 2022

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1

Du, Xiuhong. "Additive Schwarz Preconditioned GMRES, Inexact Krylov Subspace Methods, and Applications of Inexact CG." Diss., Temple University Libraries, 2008. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/6474.

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Mathematics<br>Ph.D.<br>The GMRES method is a widely used iterative method for solving the linear systems, of the form Ax = b, especially for the solution of discretized partial differential equations. With an appropriate preconditioner, the solution of the linear system Ax = b can be achieved with less computational effort. Additive Schwarz Preconditioners have two good properties. First, they are easily parallelizable, since several smaller linear systems need to be solved: one system for each of the sub-domains, usually corresponding to the restriction of the differential operator to that s

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2

Ahuja, Kapil. "Recycling Krylov Subspaces and Preconditioners." Diss., Virginia Tech, 2011. http://hdl.handle.net/10919/29539.

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Science and engineering problems frequently require solving a sequence of single linear systems or a sequence of dual linear systems. We develop algorithms that recycle Krylov subspaces and preconditioners from one system (or pair of systems) in the sequence to the next, leading to efficient solutions. Besides the benefit of only having to store few Lanczos vectors, using BiConjugate Gradients (BiCG) to solve dual linear systems may have application-specific advantages. For example, using BiCG to solve the dual linear systems arising in interpolatory model reduction provides a backward err

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3

Pestana, Jennifer. "Nonstandard inner products and preconditioned iterative methods." Thesis, University of Oxford, 2011. http://ora.ox.ac.uk/objects/uuid:2e5b636b-1145-461e-80fa-ea2041ec476f.

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By considering Krylov subspace methods in nonstandard inner products, we develop in this thesis new methods for solving large sparse linear systems and examine the effectiveness of existing preconditioners. We focus on saddle point systems and systems with a nonsymmetric, diagonalizable coefficient matrix. For symmetric saddle point systems, we present a preconditioner that renders the preconditioned saddle point matrix nonsymmetric but self-adjoint with respect to an inner product and for which scaling is not required to apply a short-term recurrence method. The robustness and effectiveness o

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4

Liang, Qiao. "Singular Value Computation and Subspace Clustering." UKnowledge, 2015. http://uknowledge.uky.edu/math_etds/30.

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In this dissertation we discuss two problems. In the first part, we consider the problem of computing a few extreme eigenvalues of a symmetric definite generalized eigenvalue problem or a few extreme singular values of a large and sparse matrix. The standard method of choice of computing a few extreme eigenvalues of a large symmetric matrix is the Lanczos or the implicitly restarted Lanczos method. These methods usually employ a shift-and-invert transformation to accelerate the speed of convergence, which is not practical for truly large problems. With this in mind, Golub and Ye proposes an in

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5

Liu, Jun. "NEW COMPUTATIONAL METHODS FOR OPTIMAL CONTROL OF PARTIAL DIFFERENTIAL EQUATIONS." OpenSIUC, 2015. https://opensiuc.lib.siu.edu/dissertations/1076.

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Partial differential equations are the chief means of providing mathematical models in science, engineering and other fields. Optimal control of partial differential equations (PDEs) has tremendous applications in engineering and science, such as shape optimization, image processing, fluid dynamics, and chemical processes. In this thesis, we develop and analyze several efficient numerical methods for the optimal control problems governed by elliptic PDE, parabolic PDE, and wave PDE, respectively. The thesis consists of six chapters. In Chapter 1, we briefly introduce a few motivating applicati

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6

Moufawad, Sophie. "Enlarged Krylov Subspace Methods and Preconditioners for Avoiding Communication." Thesis, Paris 6, 2014. http://www.theses.fr/2014PA066438/document.

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La performance d'un algorithme sur une architecture donnée dépend à la fois de la vitesse à laquelle le processeur effectue des opérations à virgule flottante (flops) et de la vitesse d'accès à la mémoire et au disque. Etant donné que le coût de la communication est beaucoup plus élevé que celui des opérations arithmétiques, celle-là forme un goulot d'étranglement dans les algorithmes numériques. Récemment, des méthodes de sous-espace de Krylov basées sur les méthodes 's-step' ont été développées pour réduire les communications. En effet, très peu de préconditionneurs existent pour ces méthode

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7

Lei, Siu Long. "Some applications of Krylov subspace methods with circulant-type preconditioners." Thesis, University of Macau, 2000. http://umaclib3.umac.mo/record=b1446687.

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8

Badahmane, Achraf. "Méthodes de sous espaces de Krylov préconditionnées pour les problèmes de point-selle avec plusieurs seconds membres." Thesis, Littoral, 2019. http://www.theses.fr/2019DUNK0543.

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La résolution numérique des problèmes de point-selle a eu une attention particulière ces dernières années. À titre d'exemple, la mécanique des fluides et solides conduit souvent à des problèmes de point-selle. Ces problèmes se présentent généralement par des équations aux dérivées partielles que nous linéarisons et discrétisons. Le problème linéaire obtenu est souvent mal conditionné. Le résoudre par des méthodes itératives standard n'est donc pas approprié. En plus, lorsque la taille du problème est grande, il est nécessaire de procéder par des méthodes de projections. Nous nous intéressons d

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9

Moulin, Johann. "On the flutter bifurcation in laminar flows : linear and nonlinear modal methods." Thesis, Institut polytechnique de Paris, 2020. http://www.theses.fr/2020IPPAX093.

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L'instabilité de flottement a été le sujet de nombreuses études depuis le milieu du vingtième siècle à cause de ses applications critiques en aéronautique. Elle est classiquement décrite comme un instabilité linéaire en écoulement potentiel, mais les effets visqueux et nonlinéaires du fluide peuvent avoir un impact crucial.La première partie de cette thèse est consacrée au développement de méthodes théoriques et numériques pour l'analyse linéaire et nonlinéaire de la dynamique d'une ``section typique aéroélastique'' --- une plaque montée sur des ressorts de flexion et torsion --- plongée dans

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10

Li, Ming. "Recycling Preconditioners for Sequences of Linear Systems and Matrix Reordering." Diss., Virginia Tech, 2015. http://hdl.handle.net/10919/64382.

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In science and engineering, many applications require the solution of a sequence of linear systems. There are many ways to solve linear systems and we always look for methods that are faster and/or require less storage. In this dissertation, we focus on solving these systems with Krylov subspace methods and how to obtain effective preconditioners inexpensively. We first present an application for electronic structure calculation. A sequence of slowly changing linear systems is produced in the simulation. The linear systems change by rank-one updates. Properties of the system matrix are analyz

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11

Yu, Xuebo. "Generalized Krylov subspace methods with applications." Kent State University / OhioLINK, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=kent1401937618.

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12

Carr, Arielle Katherine Grim. "Recycling Techniques for Sequences of Linear Systems and Eigenproblems." Diss., Virginia Tech, 2021. http://hdl.handle.net/10919/104143.

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Sequences of matrices arise in many applications in science and engineering. In this thesis we consider matrices that are closely related (or closely related in groups), and we take advantage of the small differences between them to efficiently solve sequences of linear systems and eigenproblems. Recycling techniques, such as recycling preconditioners or subspaces, are popular approaches for reducing computational cost. In this thesis, we introduce two novel approaches for recycling previously computed information for a subsequent system or eigenproblem, and demonstrate good results for seq

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13

Ahmed, Nisar. "Implicit restart schemes for Krylov subspace model reduction methods." Thesis, Imperial College London, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.340535.

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14

Birge, Jonathan R. (Jonathan Richards). "A preconditioned Newton-Krylov method for computing steady-state pulse solutions of mode-locked lasers." Thesis, Massachusetts Institute of Technology, 2008. http://hdl.handle.net/1721.1/42454.

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Thesis (S.M.)--Massachusetts Institute of Technology, Computation for Design and Optimization Program, 2008.<br>Includes bibliographical references (p. 47-48).<br>We solve the periodic boundary value problem for a mode-locked laser cavity using a specially preconditioned matrix-implicit Newton-Krylov solver. Solutions are obtained at least an order of magnitude faster than with dynamic simulation, the standard method. Our method is demonstrated experimentally on a one-dimensional temporal model of an eight femtosecond mode-locked laser operating in the dispersion-managed soliton regime. Our so

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15

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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16

Quillen, Patrick D. "GENERALIZATIONS OF AN INVERSE FREE KRYLOV SUBSPACE METHOD FOR THE SYMMETRIC GENERALIZED EIGENVALUE PROBLEM." UKnowledge, 2005. http://uknowledge.uky.edu/gradschool_diss/380.

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Symmetric generalized eigenvalue problems arise in many physical applications and frequently only a few of the eigenpairs are of interest. Typically, the problems are large and sparse, and therefore traditional methods such as the QZ algorithm may not be considered. Moreover, it may be impractical to apply shift-and-invert Lanczos, a favored method for problems of this type, due to difficulties in applying the inverse of the shifted matrix. With these difficulties in mind, Golub and Ye developed an inverse free Krylov subspace algorithm for the symmetric generalized eigenvalue problem. This me

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17

Ahuja, Kapil. "Recycling Bi-Lanczos Algorithms: BiCG, CGS, and BiCGSTAB." Thesis, Virginia Tech, 2009. http://hdl.handle.net/10919/34765.

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Engineering problems frequently require solving a sequence of dual linear systems. This paper introduces recycling BiCG, that recycles the Krylov subspace from one pair of linear systems to the next pair. Augmented bi-Lanczos algorithm and modified two-term recurrence are developed for using the recycle space. Recycle space is built from the approximate invariant subspace corresponding to eigenvalues close to the origin. Recycling approach is extended to the CGS and the BiCGSTAB algorithms. Experiments on a convection-diffusion problem give promising results.<br>Master of Science

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18

Newman, Christopher K. "Exponential Integrators for the Incompressible Navier-Stokes Equations." Diss., Virginia Tech, 2003. http://hdl.handle.net/10919/29340.

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We provide an algorithm and analysis of a high order projection scheme for time integration of the incompressible Navier-Stokes equations (NSE). The method is based on a projection onto the subspace of divergence-free (incompressible) functions interleaved with a Krylov-based exponential time integration (KBEI). These time integration methods provide a high order accurate, stable approach with many of the advantages of explicit methods, and can reduce the computational resources over conventional methods. The method is scalable in the sense that the computational costs grow linearly with probl

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19

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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20

Benner, Peter, and Cedric Effenberger. "A rational SHIRA method for the Hamiltonian eigenvalue problem." Universitätsbibliothek Chemnitz, 2009. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200900026.

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The SHIRA method of Mehrmann and Watkins belongs among the structure preserving Krylov subspace methods for solving skew-Hamiltonian eigenvalue problems. It can also be applied to Hamiltonian eigenproblems by considering a suitable transformation. Structure induced shift-and-invert techniques are employed to steer the algorithm towards the interesting region of the spectrum. However, the shift cannot be altered in the middle of the computation without discarding the information that has been accumulated so far. This paper shows how SHIRA can be combined with ideas from Ruhe's Rational Krylov a

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21

Sacchi, Giulia. "A new convergence model for the GMRES method." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2017. http://amslaurea.unibo.it/13501/.

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Tra i metodi sugli spazi di Krylov, il metodo gmres (Generalized Minimal RESidual) è particolarmente adatto per risolvere sistemi non simmetrici di grandi dimensioni. Esso tuttavia diventa inutilizzabile quando presenta una convergenza troppo lenta. Analizzarne la convergenza è dunque molto importante per rendere il suo utilizzo più agevole, ma anche al fine di comprendere il comportamento di altri algoritmi non necessariamente ottimali. In questa tesi viene analizzato il particolare caso in cui la matrice del sistema, A, è diagonalizzabile e quasi unitaria, e tuttavia presenta mal condiziona

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22

Benner, Peter, Thomas Bonin, Heike Faßbender, Jens Saak, Andreas Soppa, and Michael Zaeh. "Novel Model Reduction Techniques for Control of Machine Tools." Universitätsbibliothek Chemnitz, 2009. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-200901837.

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Computational methods for reducing the complexity of Finite Element (FE) models in structural dynamics are usually based on modal analysis. Classical approaches such as modal truncation, static condensation (Craig-Bampton, Guyan), and component mode synthesis (CMS) are available in many CAE tools such as ANSYS. In other disciplines, different techniques for Model Order Reduction (MOR) have been developed in the previous 2 decades. Krylov subspace methods are one possible choice and often lead to much smaller models than modal truncation methods given the same prescribed tolerance threshold.

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23

Zounon, Mawussi. "On numerical resilience in linear algebra." Thesis, Bordeaux, 2015. http://www.theses.fr/2015BORD0038/document.

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Comme la puissance de calcul des systèmes de calcul haute performance continue de croître, en utilisant un grand nombre de cœurs CPU ou d’unités de calcul spécialisées, les applications hautes performances destinées à la résolution des problèmes de très grande échelle sont de plus en plus sujettes à des pannes. En conséquence, la communauté de calcul haute performance a proposé de nombreuses contributions pour concevoir des applications tolérantes aux pannes. Cette étude porte sur une nouvelle classe d’algorithmes numériques de tolérance aux pannes au niveau de l’application qui ne nécessite p

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24

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 se

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25

Archid, Atika. "Méthodes par blocs adaptées aux matrices structurées et au calcul du pseudo-inverse." Thesis, Littoral, 2013. http://www.theses.fr/2013DUNK0394/document.

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Nous nous intéressons dans cette thèse, à l'étude de certaines méthodes numériques de type krylov dans le cas symplectique, en utilisant la technique de blocs. Ces méthodes, contrairement aux méthodes classiques, permettent à la matrice réduite de conserver la structure Hamiltonienne ou anti-Hamiltonienne ou encore symplectique d'une matrice donnée. Parmi ces méthodes, nous nous sommes intéressés à la méthodes d'Arnoldi symplectique par bloc que nous appelons aussi bloc J-Arnoldi. Notre but essentiel est d’étudier cette méthode de façon théorique et numérique, sur la nouvelle structure du K-mo

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26

Duminil, Sébastien. "Extrapolation vectorielle et applications aux équations aux dérivées partielles." Phd thesis, Université du Littoral Côte d'Opale, 2012. http://tel.archives-ouvertes.fr/tel-00790115.

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Nous nous intéressons, dans cette thèse, à l'étude des méthodes d'extrapolation polynômiales et à l'application de ces méthodes dans l'accélération de méthodes de points fixes pour des problèmes donnés. L'avantage de ces méthodes d'extrapolation est qu'elles utilisent uniquement une suite de vecteurs qui n'est pas forcément convergente, ou qui converge très lentement pour créer une nouvelle suite pouvant admettreune convergence quadratique. Le développement de méthodes cycliques permet, deplus, de limiter le coût de calculs et de stockage. Nous appliquons ces méthodes à la résolution des équat

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27

Chen, Wei-quan, and 陳威全. "Krylov Subspace Method for different Term-Document Matrices." Thesis, 2006. http://ndltd.ncl.edu.tw/handle/74139689981784585878.

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碩士<br>國立中正大學<br>應用數學研究所<br>95<br>An information retrieval system is to search documents in a large database for those which are relevant to a user query. A well-known method called Krylov subspace method is one of subspace methods for IR. It applies the Golub-Kahan bidiagonalization algorithm to a term-document matrix, starting from the query vector. Here we will study the famous and powerful method. And term weighting is common for improving the accuracy of an IR system. After introducing several term weighting schemes, we test two noted databases, Medline and Cranfield, for different term we

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28

Rother, Stephan. "Modellreduktion thermischer Felder unter Berücksichtigung der Wärmestrahlung." 2019. https://tud.qucosa.de/id/qucosa%3A36164.

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Transiente Simulationen im Rahmen von Parameterstudien oder Optimierungsprozessen erfor-dern die Anwendung der Modellordnungsreduktion zur Minimierung der Berechnungs¬zeiten. Die aus der Wärmestrahlung resultierende Nichtlinearität bei der Analyse thermischer Felder wird hier als äußere Last betrachtet, wodurch die entkoppelte Ermittlung der strahlungs-beding¬ten Wärmeströme gelingt. Darüber hinaus ermöglichen die infolgedessen konstanten System¬matrizen die Reduktion des Temperaturvektors mit etablierten Verfahren für lineare Systeme, wie beispielsweise den Krylov-Unterraummethoden. Die aus d

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29

Καλαντζής, Βασίλειος. "Επιτάχυνση της οικογένειας αλγορίθμων Spike μέσω τεχνικών επίλυσης γραμμικών συστημάτων με πολλά δεξιά μέλη". Thesis, 2014. http://hdl.handle.net/10889/8331.

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Στη παρούσα διπλωματική εργασία ασχολούμαστε με την αποδοτική επίλυση ταινιακών και γενικών, αραιών γραμμικών συστημάτων σε παράλληλες αρχιτεκτονικές μέσω της οικογένειας αλγορίθμων Spike. Ζητούμενο είναι η βελτίωση (μείωση) του χρόνου επίλυσης μέσω τεχνικών επίλυσης γραμμικών συστημάτων με πολλά δεξιά μέλη. Πιο συγκεκριμένα, επικεντρωνόμαστε στην επίλυση της εξίσωσης μητρώου $AX=F$ (1) όπου $A\in \mathbb{R}^{n\times n}$ είναι το μητρώο συντελεστών και το οποίο είναι αραιό ή/και ταινιακό, $F\in \mathbb{R}^{n\times s}$ είναι ένα μητρώο με $s$ στήλες το οποίο ονομάζεται μητρώο δεξιών μελών

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