Дисертації з теми "Subspaces methods"
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1
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.
Анотація:
MathematicsPh.D.
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.
Temple University--Theses
2
UGWU, UGOCHUKWU OBINNA. "Iterative tensor factorization based on Krylov subspace-type methods with applications to image processing." Kent State University / OhioLINK, 2021. http://rave.ohiolink.edu/etdc/view?acc_num=kent1633531487559183.
3
Hossain, Mohammad Sahadet. "Numerical Methods for Model Reduction of Time-Varying Descriptor Systems." Doctoral thesis, Universitätsbibliothek Chemnitz, 2011. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-74776.
Анотація:
This dissertation concerns the model reduction of linear periodic descriptor systems both in continuous and discrete-time case. In this dissertation, mainly the projection based approaches are considered for model order reduction of linear periodic time varying descriptor systems. Krylov based projection method is used for large continuous-time periodic descriptor systems and balancing based projection technique is applied to large sparse discrete-time periodic descriptor systems to generate the reduce systems.
For very large dimensional state space systems, both the techniques produce large dimensional solutions. Hence, a recycling technique is used in Krylov based projection methods which helps to compute low rank solutions of the state space systems and also accelerate the computational convergence. The outline of the proposed model order reduction procedure is given with more details. The accuracy and suitability of the proposed method is demonstrated through different examples of different orders.
Model reduction techniques based on balance truncation require to solve matrix equations. For periodic time-varying descriptor systems, these matrix equations are projected generalized periodic Lyapunov equations and the solutions are also time-varying. The cyclic lifted representation of the periodic time-varying descriptor systems is considered in this dissertation and the resulting lifted projected Lyapunov equations are solved to achieve the periodic reachability and observability Gramians of the original periodic systems. The main advantage of this solution technique is that the cyclic structures of projected Lyapunov equations can handle the time-varying dimensions as well as the singularity of the period matrix pairs very easily. One can also exploit the theory of time-invariant systems for the control of periodic ones, provided that the results achieved can be easily re-interpreted in the periodic framework.
Since the dimension of cyclic lifted system becomes very high for large dimensional periodic systems, one needs to solve the very large scale periodic Lyapunov equations which also generate very large dimensional solutions. Hence iterative techniques, which are the generalization and modification of alternating directions implicit (ADI) method and generalized Smith method, are implemented to obtain low rank Cholesky factors of the solutions of the periodic Lyapunov equations. Also the application of the solvers in balancing-based model reduction of discrete-time periodic descriptor systems is discussed. Numerical results are given to illustrate the effciency and accuracy of the proposed methods.
4
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.
5
Shatnawi, Heba Awad Addad. "Frequency estimation using subspace methods." Thesis, Wichita State University, 2009. http://hdl.handle.net/10057/2419.
Анотація:
Complex frequency estimation problem plays a significant role in many engineering
applications. The estimation process was traditionally achieved by the Eigenvalue
Decomposition (EVD) of the spatial correlation matrix of observations. Frequency estimation
has fundamental significant and wide relevance for many reasons. First, any arbitrary signal may
be modeled as a sum of frequencies. Hence, any signal estimation problem may be expressed in
terms of frequency estimation problems. Second, many parameter estimation applications may
be mathematically expressed as a frequency estimation problem.
In this thesis an improved frequency estimation technique is presented based on the
unitary transformation, which was basically applied in the direction of arrival problem. The key
idea of the proposed technique is to convert the complex valued autocorrelation, cumulant, or the
direct data matrix in Hankel like shape into a real valued data matrix with the same dimension.
The resultant real valued matrix will be used to extract the noise and/or the signal subspace
instead of the original complex one. It is well known that real manipulations are easier and faster
than the complex ones.Thesis (M.S.)--Wichita State University, College of Engineering, Dept. of Electrical and Computer Engineering
6
Ensor, Jonathan Edward. "Subspace methods for eigenstructure assignment." Thesis, University of York, 2000. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.341821.
7
Mestrah, Ali. "Identification de modèles sous forme de représentation d'état pour les systèmes à sortie binaire." Electronic Thesis or Diss., Normandie, 2023. http://www.theses.fr/2023NORMC255.
Анотація:
Cette thèse porte sur la modélisation paramétrique des systèmes linéaires invariants à partir de mesures binaires de la sortie. Ce problème demodélisation est abordée via l’usage des méthodes des sous-espaces. Ces méthodes permettent l’estimation de modèles sous forme de représentation d’état,un des avantages de ces méthodes étant que leur mise en œuvre ne nécessite pas la connaissance préalable de l’ordre du système. Ces méthodes ne sontinitialement pas adaptées au traitement de données binaires, l’objectif de cette thèse est ainsi leur adaptation à ce contexte d’identification. Dans cette thèse nousproposons trois méthodes des sous-espaces. Les propriétés de convergence de deux d’entre elles sont établies. Des résultats de simulations de Monte-Carlo sontprésentés afin de montrer les bonnes performances, mais aussi les limites, de ces méthodesThis thesis focuses on parametric modeling of invariant linear systems from binary output measurements. This identification problem is addressed via the use ofsubspace methods. These methods allow the estimation of state-space models, an added benefit of these methods being the fact that their implementation doesnot require the prior knowledge of the order of the system. These methods are initially adapted to high resolution data processing, the objective of this thesis istherefore their adaptation to the identification using binary measurements. In this thesis we propose three subspace methods. Convergence properties of two ofthem are established. Monte Carlo simulation results are presented to show the good performance, but also limits, of these methods
8
Nguyen, Hieu. "Linear subspace methods in face recognition." Thesis, University of Nottingham, 2011. http://eprints.nottingham.ac.uk/12330/.
Анотація:
Despite over 30 years of research, face recognition is still one of the most difficult problems in the field of Computer Vision. The challenge comes from many factors affecting the performance of a face recognition system: noisy input, training data collection, speed-accuracy trade-off, variations in expression, illumination, pose, or ageing. Although relatively successful attempts have been made for special cases, such as frontal faces, no satisfactory methods exist that work under completely unconstrained conditions. This thesis proposes solutions to three important problems: lack of training data, speed-accuracy requirement, and unconstrained environments. The problem of lacking training data has been solved in the worst case: single sample per person. Whitened Principal Component Analysis is proposed as a simple but effective solution. Whitened PCA performs consistently well on multiple face datasets. Speed-accuracy trade-off problem is the second focus of this thesis. Two solutions are proposed to tackle this problem. The first solution is a new feature extraction method called Compact Binary Patterns which is about three times faster than Local Binary Patterns. The second solution is a multi-patch classifier which performs much better than a single classifier without compromising speed. Two metric learning methods are introduced to solve the problem of unconstrained face recognition. The first method called Indirect Neighourhood Component Analysis combines the best ideas from Neighourhood Component Analysis and One-shot learning. The second method, Cosine Similarity Metric Learning, uses Cosine Similarity instead of the more popular Euclidean distance to form the objective function in the learning process. This Cosine Similarity Metric Learning method produces the best result in the literature on the state-of-the-art face dataset: the Labelled Faces in the Wild dataset. Finally, a full face verification system based on our real experience taking part in ICPR 2010 Face Verification contest is described. Many practical points are discussed.
9
Tao, Dacheng. "Discriminative linear and multilinear subspace methods." Thesis, Birkbeck (University of London), 2007. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.438996.
10
Yu, Xuebo. "Generalized Krylov subspace methods with applications." Kent State University / OhioLINK, 2014. http://rave.ohiolink.edu/etdc/view?acc_num=kent1401937618.
11
Shi, Ruijie. "Subspace identification methods for process dynamic modeling /." *McMaster only, 2001.
12
Lam, Xuan-Binh. "Uncertainty quantification for stochastic subspace indentification methods." Rennes 1, 2011. http://www.theses.fr/2011REN1S133.
Анотація:
In Operational Modal Analysis, the modal parameters (natural frequencies, damping ratios, and mode shapes) obtained from Stochastic Subspace Identification (SSI) of a structure, are afflicted with statistical uncertainty. For evaluating the quality of the obtained results it is essential to know the appropriate uncertainty bounds of these terms. In this thesis, the algorithms, that automatically compute the uncertainty bounds of modal parameters obtained from SSI of a structure based on vibration measurements, are presented. With these new algorithms, the uncertainty bounds of the modal parameters of some relevant industrial examples are computed. To quantify the statistical uncertainty of the obtained modal parameters, the statistical uncertainty in the data can be evaluated and propagated to the system matrices and, thus, to the modal parameters. In the uncertainty quantification algorithm, which is a perturbation-based method, it has been shown how uncertainty bounds of modal parameters can be determined from the covariances of the system matrices, which are obtained from some covariance of the data and the covariances of subspace matrices. In this thesis, several results are derived. Firstly, a novel and more realistic scheme for the uncertainty calculation of the mode shape is presented, the mode shape is normalized by the phase angle of the component having the maximal absolute value instead of by one of its components. Secondly, the uncertainty quantification is derived and developed for several identification methods, first few of them are covariance- and data-driven SSI. The thesis also mentions about Eigensystem Realization Algorithm (ERA), a class of identification methods, and its uncertainty quantification scheme. This ERA approach is introduced in conjunction with the singular value decomposition to derive the basic formulation of minimum order realization. Besides, the thesis supposes efficient algorithms to estimate the system matrices at multiple model orders, the uncertainty quantification is also derived for this new multi-order SSI method. Two last interesting sections of the thesis are discovering the uncertainty of multi-setups SSI algorithm and recursive algorithms. In summary, subspace algorithms are efficient tools for vibration analysis, fitting a model to input/output or output-only measurements taken from a system. However, uncertainty quantification for SSI was missing for a long time. The uncertainty quantification is very important feature for credibility of modal analysis exploitationEn analyse modale operationelle, les paramètres modaux (fréquence, amortissement, déforméees) peuvent être obtenus par des méthodes d'identification de type sous espaces et sont définis à une incertitude stochastique près. Pour évaluer la qualité des résultats obtenus, il est essentiel de connaître les bornes de confiance sur ces résultats. Dans cette thèse sont développés des algorithmes qui calcule automatiquement de telles bornes de confiance pour des paramètres modaux caractèristiques d'une structure mécanique. Ces algorithmes sont validés sur des exemples industriels significatifs. L'incertitude est tout d'abord calculé sur les données puis propagée sur les matrices du système par calcul de sensibilité, puis finalement sur les paramètres modaux. Les algorithmes existants sur lesquels se basent cette thèse dérivent l'incertitude des matrices du système de l'incertitude sur les covariances des entrées mesurées. Dans cette thèse, plusieurs résultats ont été obtenus. Tout d'abord, l'incertitude sur les déformées modales est obtenue par un schema de calcul plus réaliste que précédemment, utilisant une normalisation par l'angle de phase de la composante de valeur maximale. Ensuite, plusieurs méthodes de sous espaces et non seulement les méthodes à base de covariance sont considérées, telles que la méthode de réalisation stochastique ERA ainsi que la méthode UPC, à base des données. Pour ces méthodes, le calcul d'incertitude est explicité. Deu autres problèmatiques sont adressés : tout d'abord l'estimation multi ordre par méthode de sous espace et l'estimation à partir de jeux de données mesurées séparément. Pour ces deux problèmes, les schemas d'incertitude sont développés. En conclusion, cette thèse s'est attaché à développer des schemas de calcul d'incertitude pour une famille de méthodes sous espaces ainsi que pour un certain nombre de problèmes pratiques. La thèse finit avec le calcul d'incertitudes pour les méthodes récursives. Les méthodes sous espaces sont considérées comme une approche d'estimation robuste et consistante pour l'extraction des paramètres modaux à partir de données temporelles. Le calcul des incertitudes pour ces méthodes est maintenant possible, rendant ces méthodes encore plus crédible dans le cadre de l'exploitation de l'analyse modale
13
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.
14
Neuman, Arthur James III. "Regularization Methods for Ill-posed Problems." Kent State University / OhioLINK, 2010. http://rave.ohiolink.edu/etdc/view?acc_num=kent1273611079.
15
Soodhalter, Kirk McLane. "Krylov Subspace Methods with Fixed Memory Requirements: Nearly Hermitian Linear Systems and Subspace Recycling." Diss., Temple University Libraries, 2012. http://cdm16002.contentdm.oclc.org/cdm/ref/collection/p245801coll10/id/192337.
Анотація:
MathematicsPh.D.
Krylov subspace iterative methods provide an effective tool for reducing the solution of large linear systems to a size for which a direct solver may be applied. However, the problems of limited storage and speed are still a concern. Therefore, in this dissertation work, we present iterative Krylov subspace algorithms for non-Hermitian systems which do have fixed memory requirements and have favorable convergence characteristics. This dissertation describes three projects. The first project concerns short-term recurrence Krylov subspace methods for nearly-Hermitian linear systems. In 2008, Beckermann and Reichel introduced a short-term recurrence progressive GMRES algorithm for nearly-Hermitian linear systems. However, we have found this method to be unstable. We document the instabilities and introduce a different fixed-memory algorithm to treat nearly-Hermitian problems. We present numerical experiments demonstrating that the performance of this algorithm is competitive. The other two projects involve extending a strategy called Krylov subspace recycling, introduced by Parks and colleagues in 2005. This method requires more overhead than other subspace augmentation methods but offers the ability to recycle subspace information between cycles for a single linear system and recycle information between related linear systems. In the first project, we extend subspace recycling to the block Krylov subspace setting. A block Krylov subspace is a generalization of Krylov subspace where a single starting vector is replaced with a block of linearly independent starting vectors. We then apply our method to a sequence of matrices arising in a Newton iteration applied to fluid density functional theory and present some numerical experiments. In the second project, we extend the methods of subspace recycling to a family of linear systems differing only by multiples of the identity. These problems arise in the theory of quantum chromodynamics, a theory of the behavior of subatomic particles. We wish to build on the class of Krylov methods which allow the simultaneous solution of all shifted linear systems while generating only one subspace. However, the mechanics of subspace recycling complicates this situation and interferes with our ability to simultaneously solve all systems using these techniques. Therefore, we introduce an algorithm which avoids this complication and present some numerical experiments demonstrating its effectiveness.
Temple University--Theses
16
Wilkins, Bryce Daniel. "The E² Bathe subspace iteration method." Thesis, Massachusetts Institute of Technology, 2019. https://hdl.handle.net/1721.1/122238.
Анотація:
Thesis: S.M., Massachusetts Institute of Technology, Department of Mechanical Engineering, 2019Cataloged from PDF version of thesis.
Includes bibliographical references (pages 91-93).
Since its development in 1971, the Bathe subspace iteration method has been widely-used to solve the generalized symmetric-definite eigenvalue problem. The method is particularly useful for solving large eigenvalue problems when only a few of the least dominant eigenpairs are sought. In reference [18], an enriched subspace iteration method was proposed that accelerated the convergence of the basic method by replacing some of the iteration vectors with more effective turning vectors. In this thesis, we build upon this recent acceleration effort and further enrich the subspace of each iteration by replacing additional iteration vectors with our new turning-of-turning vectors. We begin by reviewing the underpinnings of the subspace iteration methodology. Then, we present the steps of our new algorithm, which we refer to as the Enriched- Enriched (E2 ) Bathe subspace iteration method. This is followed by a tabulation of the number of floating point operations incurred during a general iteration of the E2 algorithm. Additionally, we perform a simplified convergence analysis showing that the E2 method converges asymptotically at a faster rate than the enriched method. Finally, we examine the results from several test problems that were used to illustrate the E2 method and to assess its potential computational savings compared to the enriched method. The sample results for the E2 method are consistent with the theoretical asymptotic convergence rate that was obtained in our convergence analysis. Further, the results from the CPU time tests suggest that the E2 method can often provide a useful reduction in computational effort compared to the enriched method, particularly when relatively few iteration vectors are used in comparison with the number of eigenpairs that are sought.
by Bryce Daniel Wilkins.
S.M.
S.M. Massachusetts Institute of Technology, Department of Mechanical Engineering
17
Gonder, Ozkan. "A Comparison Of Subspace Based Face Recognition Methods." Master's thesis, METU, 2005. http://etd.lib.metu.edu.tr/upload/2/12605291/index.pdf.
Анотація:
Different approaches to the face recognition are studied in this thesis. These approaches are PCA (Eigenface), Kernel Eigenface and Fisher LDA. Principal component analysis extracts the most important information contained in the face to construct a computational model that best describes the face. In Eigenface approach, variation between the face images are described by using a set of characteristic face images in order to find out the eigenvectors (Eigenfaces) of the covariance matrix of the distribution, spanned by a training set of face images. Then, every face image is represented by a linear combination of these eigenvectors. Recognition is implemented by projecting a new image into the face subspace spanned by the Eigenfaces and then classifying the face by comparing its position in face space with the positions of known individuals. In Kernel Eigenface method, non-linear mapping of input space is implemented before PCA in order to handle non-linearly embedded properties of images (i.e. background differences, illumination changes, and facial expressions etc.). In Fisher LDA, LDA is applied after PCA to increase the discrimination between classes.
These methods are implemented on three databases that are: Yale face database, AT&T (formerly Olivetti Research Laboratory) face database, and METU Vision Lab face database. Experiment results are compared with respect to the effects of changes in illumination, pose and expression. Kernel Eigenface and Fisher LDA show slightly better performance with respect to Eigenfaces method under changes in illumination. Expression differences did not affect the performance of Eigenfaces method. From test results, it can be observed that Eigenfaces approach is an adequate method that can be used in face recognition systems due to its simplicity, speed and learning capability. By this way, it can easily be used in real time systems.
18
Zhao, Yong. "Identification of ankle joint stiffness using subspace methods." Thesis, McGill University, 2010. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=86800.
Анотація:
Studying joint stiffness against a compliant load is a difficult problem because the intrinsic and reflex torques cannot be measured separately experimentally. Moreover, the joint stiffness is operated within a closed loop because the ankle torque is fed back through the load to change the ankle position. In this thesis, a state space model for ankle joint stiffness is developed. Then, a discrete-time, subspace-based method is used to estimate this state space model for overall stiffness. By using appropriate instrumental variables, the subspace method can estimate the state space model for joint stiffness in both open-loop and in closed-loop conditions. This thesis also presents a subspace method to identify state space models for biomedical systems with short transients or systems with time-varying behaviors, from ensembles of short transients. The simulation and experimental results demonstrate that those algorithms provide accurate estimates under their respective conditions.L'étude de la rigidité articulaire en réponse à une charge est un problème difficile car les couples réflexes et intrinsèques ne peuvent pas être mesurés séparément expérimentalement. En outre, la rigidité articulaire opère en boucle fermée car le couple de la cheville est réinjectée à travers la charge pour modifier la position de la cheville. Dans cette thèse, un modèle d'espace d'état pour la rigidité articulaire de la cheville est développé. Une méthode sous-espace à temps discret est ensuite utilisée pour estimer ce modèle d'espace d'état pour la rigidité globale. En considérant les variables instrumentales appropriées, la méthode sous-espace permet d'estimer le modèle espace d'état pour la rigidité articulaire en boucles ouverte et fermée. Cette thèse présente également une méthode sous-espace pour identifier les modèles d'espace d'état pour les systèmes biomédicauxou les systèmes variant dans le temps caractérisés par des phénomènes transitoires de courte durée. Les simulations et les résultats expérimentaux démontrent que ces algorithmes fournissent des estimations précises en fonction de leurs conditions propres.
19
Chui, Nelson Loong Chik. "Subspace methods and informative experiments for system identification." Thesis, University of Cambridge, 1997. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.298794.
20
Ahuja, Kapil. "Recycling Krylov Subspaces and Preconditioners." Diss., Virginia Tech, 2011. http://hdl.handle.net/10919/29539.
Анотація:
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 error formulation in the model reduction framework. Using BiCG to evaluate bilinear forms -- for example, in the variational Monte Carlo (VMC) algorithm for electronic structure calculations -- leads to a quadratic error bound. Since one of our focus areas is sequences of dual linear systems, we introduce recycling BiCG, a BiCG method that recycles two Krylov subspaces from one pair of dual linear systems to the next pair. The derivation of recycling BiCG also builds the foundation for developing recycling variants of other bi-Lanczos based methods like CGS, BiCGSTAB, BiCGSTAB2, BiCGSTAB(l), QMR, and TFQMR.
We develop a generalized bi-Lanczos algorithm, where the two matrices of the bi-Lanczos procedure are not each other's conjugate transpose but satisfy this relation over the generated Krylov subspaces. This is sufficient for a short term recurrence. Next, we derive an augmented bi-Lanczos algorithm with recycling and show that this algorithm is a special case of generalized bi-Lanczos. The Petrov-Galerkin approximation that includes recycling in the iteration leads to modified two-term recurrences for the solution and residual updates.
We generalize and extend the framework of our recycling BiCG to CGS, BiCGSTAB and BiCGSTAB2. We perform extensive numerical experiments and analyze the generated recycle space. We test all of our recycling algorithms on a discretized partial differential equation (PDE) of convection-diffusion type. This PDE problem provides well-known test cases that are easy to analyze further. We use recycling BiCG in the Iterative Rational Krylov Algorithm (IRKA) for interpolatory model reduction and in the VMC algorithm. For a model reduction problem, we show up to 70% savings in iterations, and we also demonstrate that solving the problem without recycling leads to (about) a 50% increase in runtime. Experiments with recycling BiCG for VMC gives promising results.
We also present an algorithm that recycles preconditioners, leading to a dramatic reduction in the cost of VMC for large(r) systems. The main cost of the VMC method is in constructing a sequence of Slater matrices and computing the ratios of determinants for successive Slater matrices. Recent work has improved the scaling of constructing Slater matrices for insulators, so that the cost of constructing Slater matrices in these systems is now linear in the number of particles. However, the cost of computing determinant ratios remains cubic in the number of particles. With the long term aim of simulating much larger systems, we improve the scaling of computing determinant ratios in the VMC method for simulating insulators by using preconditioned iterative solvers.
The main contribution here is the development of a method to efficiently compute for the Slater matrices a sequence of preconditioners that make the iterative solver converge rapidly. This involves cheap preconditioner updates, an effective reordering strategy, and a cheap method to monitor instability of ILUTP preconditioners. Using the resulting preconditioned iterative solvers to compute determinant ratios of consecutive Slater matrices reduces the scaling of the VMC algorithm from O(n^3) per sweep to roughly O(n^2), where n is the number of particles, and a sweep is a sequence of n steps, each attempting to move a distinct particle. We demonstrate experimentally that we can achieve the improved scaling without increasing statistical errors.Ph. D.
21
Dahlen, Anders. "Identification of stochastic systems : Subspace methods and covariance extension." Doctoral thesis, KTH, Mathematics, 2001. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3178.
22
Benner, Peter. "Structured Krylov Subspace Methods for Eigenproblems with Spectral Symmetries." Universitätsbibliothek Chemnitz, 2010. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-201000852.
Анотація:
We consider large and sparse eigenproblems where the spectrum exhibits
special symmetries. Here we focus on Hamiltonian symmetry, that is,
the spectrum is symmetric with respect to the real and imaginary
axes. After briefly discussing quadratic eigenproblems with
Hamiltonian spectra we review structured Krylov subspace methods to
aprroximate parts of the spectrum of Hamiltonian operators. We will
discuss the optimization of the free parameters in the resulting
symplectic Lanczos process in order to minimize the conditioning of
the (non-orthonormal) Lanczos basis. The effects of our findings are
demonstrated for several numerical examples.
23
Zhou, Ning. "Subspace methods of system identification applied to power systems." Laramie, Wyo. : University of Wyoming, 2005. http://proquest.umi.com/pqdweb?did=1095432761&sid=1&Fmt=2&clientId=18949&RQT=309&VName=PQD.
24
Dahlén, Anders. "Identification of stochastic systems : subspace methods and covariance extension /." Stockholm : Tekniska högsk, 2001. http://urn.kb.se/resolve?urn=urn:nbn:se:kth:diva-3178.
25
Anderson, Penelope L. "Matrix based derivations and representations of Krylov subspace methods." Thesis, McGill University, 1995. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=23766.
Анотація:
This thesis is based on recent work by Paige which gave a formalism for presenting and analyzing the class of algorithms which manipulate an appropriate Krylov subspace in solving large sparse systems of linear equations. This formalism--a way of dividing a method of solution into a Krylov process and an associated subproblem--is described and then applied to several of the more popular algorithms in use today including the methods of Conjugate Gradients and BiConjugate Gradients. The aim is to clarify these algorithms to make them easier to understand, analyze and use. Several of the methods presented in this thesis were developed in exactly this way--notably the Symmetric LQ method and the Generalized Minimum Residual method--and required little or no effort to characterize using the formalism. It was successfully applied to Conjugate Gradients and BiConjugate Gradients, already recognized as being closely related to the symmetric and unsymmetric Lanczos processes respectively. The newer algorithms such as Conjugate Gradients Squared and BiConjugate Gradients Stabilized, with less obvious relation to a specific Krylov process, provided more difficulty in their clarification.
26
Embree, Mark. "Convergence of Krylov subspace methods for non-normal matrices." Thesis, University of Oxford, 1999. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.326004.
27
Moufawad, Sophie. "Enlarged Krylov Subspace Methods and Preconditioners for Avoiding Communication." Thesis, Paris 6, 2014. http://www.theses.fr/2014PA066438/document.
Анотація:
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 méthodes, ce qui constitue une importante limitation. Dans cette thèse, nous présentons le préconditionneur nommé ''Communication-Avoiding ILU0'', pour la résolution des systèmes d’équations linéaires (Ax=b) de très grandes tailles. Nous proposons une nouvelle renumérotation de la matrice A ('alternating min-max layers'), avec laquelle nous montrons que le préconditionneur en question réduit la communication. Il est ainsi possible d’effectuer « s » itérations d’une méthode itérative préconditionnée sans communication. Nous présentons aussi deux nouvelles méthodes itératives, que nous nommons 'multiple search direction with orthogonalization CG' (MSDO-CG) et 'long recurrence enlarged CG' (LRE-CG). Ces dernières servent à la résolution des systèmes linéaires d’équations de très grandes tailles, et sont basées sur l’enrichissement de l’espace de Krylov par la décomposition du domaine de la matrice AThe performance of an algorithm on any architecture is dependent on the processing unit’s speed for performing floating point operations (flops) and the speed of accessing memory and disk. As the cost of communication is much higher than arithmetic operations, and since this gap is expected to continue to increase exponentially, communication is often the bottleneck in numerical algorithms. In a quest to address the communication problem, recent research has focused on communication avoiding Krylov subspace methods based on the so called s-step methods. However there are very few communication avoiding preconditioners, and this represents a serious limitation of these methods. In this thesis, we present a communication avoiding ILU0 preconditioner for solving large systems of linear equations (Ax=b) by using iterative Krylov subspace methods. Our preconditioner allows to perform s iterations of the iterative method with no communication, by applying a heuristic alternating min-max layers reordering to the input matrix A, and through ghosting some of the input data and performing redundant computation. We also introduce a new approach for reducing communication in the Krylov subspace methods, that consists of enlarging the Krylov subspace by a maximum of t vectors per iteration, based on the domain decomposition of the graph of A. The enlarged Krylov projection subspace methods lead to faster convergence in terms of iterations and to parallelizable algorithms with less communication, with respect to Krylov methods. We discuss two new versions of Conjugate Gradient, multiple search direction with orthogonalization CG (MSDO-CG) and long recurrence enlarged CG (LRE-CG)
28
Patterson, Andrew David. "A subspace reduction method for microwave device characterisation." Thesis, Queen's University Belfast, 1992. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.333831.
29
Driver, Maria Sosonkina Jr. "Parallel Sparse Linear Algebra for Homotopy Methods." Diss., Virginia Tech, 1997. http://hdl.handle.net/10919/30718.
Анотація:
Globally convergent homotopy methods are used to solve difficult
nonlinear systems of equations by tracking the zero curve of a homotopy
map. Homotopy curve tracking involves solving a sequence of linear
systems, which often vary greatly in difficulty. In this research, a
popular iterative solution tool, GMRES(k), is adapted to deal with the
sequence of such systems. The proposed adaptive strategy of GMRES(k)
allows tuning of the restart parameter k based on the GMRES
convergence rate for the given problem. Adaptive GMRES(k) is shown
to be superior to several other iterative techniques on analog
circuit simulation problems and on postbuckling structural analysis
problems.
Developing parallel techniques for robust but expensive sequential
computations, such as globally convergent homotopy methods, is
important. The design of these techniques encompasses the functionality
of the iterative method (adaptive GMRES(k)) implemented sequentially
and is based on the results of a parallel
performance analysis of several implementations. An implementation of
adaptive GMRES(k) with Householder reflections in its
orthogonalization phase is developed. It is shown that
the efficiency of linear system solution by the adaptive GMRES(k)
algorithm depends on the change in problem difficulty when the problem
is scaled.
In contrast, a standard GMRES(k) implementation using Householder
reflections maintains a
constant efficiency with increase in problem size and number of
processors, as concluded analytically and experimentally. The supporting
numerical results are obtained on three distributed memory homogeneous
parallel architectures: CRAY T3E, Intel Paragon, and IBM SP2.Ph. D.
30
Lei, Siu Long. "Some applications of Krylov subspace methods with circulant-type preconditioners." Thesis, University of Macau, 2000. http://umaclib3.umac.mo/record=b1446687.
31
Ewen, Tracy Leanne. "Mixed product Krylov subspace methods for solving nonsymmetric linear systems." Thesis, National Library of Canada = Bibliothèque nationale du Canada, 2000. http://www.collectionscanada.ca/obj/s4/f2/dsk2/ftp03/MQ51707.pdf.
32
Swirydowicz, Katarzyna. "Strategies For Recycling Krylov Subspace Methods and Bilinear Form Estimation." Diss., Virginia Tech, 2017. http://hdl.handle.net/10919/78695.
Анотація:
The main theme of this work is effectiveness and efficiency of Krylov subspace methods and Krylov subspace recycling. While solving long, slowly changing sequences of large linear systems, such as the ones that arise in engineering, there are many issues we need to consider if we want to make the process reliable (converging to a correct solution) and as fast as possible. This thesis is built on three main components. At first, we target bilinear and quadratic form estimation. Bilinear form $c^TA^{-1}b$ is often associated with long sequences of linear systems, especially in optimization problems. Thus, we devise algorithms that adapt cheap bilinear and quadratic form estimates for Krylov subspace recycling. In the second part, we develop a hybrid recycling method that is inspired by a complex CFD application. We aim to make the method robust and cheap at the same time. In the third part of the thesis, we optimize the implementation of Krylov subspace methods on Graphic Processing Units (GPUs). Since preconditioners based on incomplete matrix factorization (ILU, Cholesky) are very slow on the GPUs, we develop a preconditioner that is effective but well suited for GPU implementation.Ph. D.
33
Zounon, Mawussi. "On numerical resilience in linear algebra." Thesis, Bordeaux, 2015. http://www.theses.fr/2015BORD0038/document.
Анотація:
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 pas de ressources supplémentaires, à savoir, des unités de calcul ou du temps de calcul additionnel, en l’absence de pannes. En supposant qu’un mécanisme distinct assure la détection des pannes, nous proposons des algorithmes numériques pour extraire des informations pertinentes à partir des données disponibles après une pannes. Après l’extraction de données, les données critiques manquantes sont régénérées grâce à des stratégies d’interpolation pour constituer des informations pertinentes pour redémarrer numériquement l’algorithme. Nous avons conçu ces méthodes appelées techniques d’Interpolation-restart pour des problèmes d’algèbre linéaire numérique tels que la résolution de systèmes linéaires ou des problèmes aux valeurs propres qui sont indispensables dans de nombreux noyaux scientifiques et applications d’ingénierie. La résolution de ces problèmes est souvent la partie dominante; en termes de temps de calcul, des applications scientifiques. Dans le cadre solveurs linéaires du sous-espace de Krylov, les entrées perdues de l’itération sont interpolées en utilisant les entrées disponibles sur les nœuds encore disponibles pour définir une nouvelle estimation de la solution initiale avant de redémarrer la méthode de Krylov. En particulier, nous considérons deux politiques d’interpolation qui préservent les propriétés numériques clés de solveurs linéaires bien connus, à savoir la décroissance monotone de la norme-A de l’erreur du gradient conjugué ou la décroissance monotone de la norme résiduelle de GMRES. Nous avons évalué l’impact du taux de pannes et l’impact de la quantité de données perdues sur la robustesse des stratégies de résilience conçues. Les expériences ont montré que nos stratégies numériques sont robustes même en présence de grandes fréquences de pannes, et de perte de grand volume de données. Dans le but de concevoir des solveurs résilients de résolution de problèmes aux valeurs propres, nous avons modifié les stratégies d’interpolation conçues pour les systèmes linéaires. Nous avons revisité les méthodes itératives de l’état de l’art pour la résolution des problèmes de valeurs propres creux à la lumière des stratégies d’Interpolation-restart. Pour chaque méthode considérée, nous avons adapté les stratégies d’Interpolation-restart pour régénérer autant d’informations spectrale que possible. Afin d’évaluer la performance de nos stratégies numériques, nous avons considéré un solveur parallèle hybride (direct/itérative) pleinement fonctionnel nommé MaPHyS pour la résolution des systèmes linéaires creux, et nous proposons des solutions numériques pour concevoir une version tolérante aux pannes du solveur. Le solveur étant hybride, nous nous concentrons dans cette étude sur l’étape de résolution itérative, qui est souvent l’étape dominante dans la pratique. Les solutions numériques proposées comportent deux volets. A chaque fois que cela est possible, nous exploitons la redondance de données entre les processus du solveur pour effectuer une régénération exacte des données en faisant des copies astucieuses dans les processus. D’autre part, les données perdues qui ne sont plus disponibles sur aucun processus sont régénérées grâce à un mécanisme d’interpolationAs the computational power of high performance computing (HPC) systems continues to increase by using huge number of cores or specialized processing units, HPC applications are increasingly prone to faults. This study covers a new class of numerical fault tolerance algorithms at application level that does not require extra resources, i.e., computational unit or computing time, when no fault occurs. Assuming that a separate mechanism ensures fault detection, we propose numerical algorithms to extract relevant information from available data after a fault. After data extraction, well chosen part of missing data is regenerated through interpolation strategies to constitute meaningful inputs to numerically restart the algorithm. We have designed these methods called Interpolation-restart techniques for numerical linear algebra problems such as the solution of linear systems or eigen-problems that are the inner most numerical kernels in many scientific and engineering applications and also often ones of the most time consuming parts. In the framework of Krylov subspace linear solvers the lost entries of the iterate are interpolated using the available entries on the still alive nodes to define a new initial guess before restarting the Krylov method. In particular, we consider two interpolation policies that preserve key numerical properties of well-known linear solvers, namely the monotony decrease of the A-norm of the error of the conjugate gradient or the residual norm decrease of GMRES. We assess the impact of the fault rate and the amount of lost data on the robustness of the resulting linear solvers.For eigensolvers, we revisited state-of-the-art methods for solving large sparse eigenvalue problems namely the Arnoldi methods, subspace iteration methods and the Jacobi-Davidson method, in the light of Interpolation-restart strategies. For each considered eigensolver, we adapted the Interpolation-restart strategies to regenerate as much spectral information as possible. Through intensive experiments, we illustrate the qualitative numerical behavior of the resulting schemes when the number of faults and the amount of lost data are varied; and we demonstrate that they exhibit a numerical robustness close to that of fault-free calculations. In order to assess the efficiency of our numerical strategies, we have consideredan actual fully-featured parallel sparse hybrid (direct/iterative) linear solver, MaPHyS, and we proposed numerical remedies to design a resilient version of the solver. The solver being hybrid, we focus in this study on the iterative solution step, which is often the dominant step in practice. The numerical remedies we propose are twofold. Whenever possible, we exploit the natural data redundancy between processes from the solver toperform an exact recovery through clever copies over processes. Otherwise, data that has been lost and is not available anymore on any process is recovered through Interpolationrestart strategies. These numerical remedies have been implemented in the MaPHyS parallel solver so that we can assess their efficiency on a large number of processing units (up to 12; 288 CPU cores) for solving large-scale real-life problems
34
Byers, R., C. He, and V. Mehrmann. "The Matrix Sign Function Method and the Computation of Invariant Subspaces." Universitätsbibliothek Chemnitz, 1998. http://nbn-resolving.de/urn:nbn:de:bsz:ch1-199800619.
Анотація:
A perturbation analysis shows that if a numerically stable
procedure is used to compute the matrix sign function, then it is competitive
with conventional methods for computing invariant subspaces.
Stability analysis of the Newton iteration improves an earlier result of Byers
and confirms that ill-conditioned iterates may cause numerical
instability. Numerical examples demonstrate the theoretical results.
35
Frangos, Michalis. "Adaptive Krylov subspace methods for model reduction of large scale systems." Thesis, Imperial College London, 2008. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.485411.
Анотація:
The ultimate goal of every theory according to Albert Einstein, is as follows: "It is the grand object of all theory to make these irreducible elements (axioms/assumptions) as simple and as few in number as possible, without having to renounce the adequate representation of any empirical content whatever" (Albert Einstein, 1954). The main goal of this dissertation falls in the above definition which is focused on model reduction of large-scale linear systems, to derive small and accurate linear systems based on efficient Krylov subspace projection techniques. The rational Arnoldi algorithm which belongs to the class of Krylov subspace projection methods has been applied for deriving reduced order models that are rational interpolating approximations of the original system. The rational Arnoldi algorithm is known in the literature and it is used extensively for the approximation of large scale linear systems due to its numerical stability and efficiency. However there are some outstanding issues which can affect and improve its performance which are investigated in this thesis. The first issue is in the development of a set of simple equations, the Arnoldi-like equations, to describe the rational Arnoldi-algorithm. This set of equations is of the same form as in the case of the well-known standard Arnoldi algorithm, an algorithm based on which many techniques for model reduction have emerged. The reduced order models developed by the rational Arnoldi algorithm interpolate the original system at multiple interpolation points, while the standard Arnoldi algorithm interpolates the original system around infinity. The second issue is in the development of adaptive schemes for the selection of the interpolation points which result in significantly improved approximations, without a priori knowledge of the system's transfer function characteristics. The information about the interpolation points rises from simple error expressions and error approximation expressions derived posing the Arnoldi-like equations. The third issue concerned in this work is in the development of a simple and easy to understand modified version of the rational Arnoldi algorithm which is suitable for adaptive interpolation. A breakdown analysis and an error analysis, essential for the adaptive schemes, are provided. Based on the modified Arnoldi algorithm an efficient restart technique for the algorithm is also developed to improve the approximation further while the order of the approximation remains fixed. The performance of the reduced order approximations is based on updates of some of the interpolation points of the approximations. A drawback of the rational interpolating methods is that they do not guarantee stability for the reduced order models. The fourth issue addressed in the thesis is the parameterisation of a set of interpolating approximations in terms of a free parameter. As a post-processing step of the rational Arnoldi algorithm any unstable reduced order models can be stabilised by a proper selection of the free parameter. Future research directions are provided at the conclusions of the thesis.
36
Maciver, Mark Alasdair. "Electromagnetic characterisation of structures using Krylov subspace model order reduction methods." Thesis, University of Glasgow, 2003. http://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.433619.
37
Ho, Michael T. "A Comparision of Wideband Subspace Methods for Direction-of-Arrival Estimation." The Ohio State University, 2002. http://rave.ohiolink.edu/etdc/view?acc_num=osu1419265813.
38
Božović, Nemanja [Verfasser]. "Deflation Based Krylov Subspace Methods for Sequences of Linear Systems / Nemanja Božović." Wuppertal : Universitätsbibliothek Wuppertal, 2017. http://d-nb.info/1141193531/34.
39
Lu, James 1977. "Krylov subspace methods for simultaneous primal-dual solutions and superconvergent functional estimates." Thesis, Massachusetts Institute of Technology, 2002. http://hdl.handle.net/1721.1/82253.
40
Gazzola, Silvia. "Regularization techniques based on Krylov subspace methods for ill-posed linear systems." Doctoral thesis, Università degli studi di Padova, 2014. http://hdl.handle.net/11577/3423528.
Анотація:
This thesis is focussed on the regularization of large-scale linear discrete ill-posed problems. Problems of this kind arise in a variety of applications, and, in a continuous setting, they are often formulated as Fredholm integral equations of the first kind, with smooth kernel, modeling an inverse problem (i.e., the unknown of these equations is the cause of an observed effect). Upon discretization, linear systems whose coefficient matrix is ill-conditioned and whose right-hand side vector is affected by some perturbations (noise) must be solved. %Because of the ill-conditioning of the system matrix and the errors in the data,
In this setting, a straightforward solution of the available linear system is meaningless because the computed solution would be dominated by errors; moreover, for large-scale problems, solving directly the available system could be computationally infeasible. Therefore, in order to recover a meaningful approximation of the original solution, some regularization must be employed, i.e., the original linear system must be replaced by a nearby problem having better numerical properties.
The first part of this thesis (Chapter 1) gives an overview on inverse problems and briefly describes their properties in the continuous setting; then, in a discrete setting, the most well-known regularization techniques relying on some factorization of the system matrix are surveyed.
The remaining part of the thesis is concerned with iterative regularization strategies based on some Krylov subspaces methods, which are well-suited for large-scale problems. More precisely, in Chapter 2, an extensive overview of the Krylov subspace methods most successfully employed with regularizing purposes is presented: historically, the first methods to be used were related to the normal equations and many issues linked to the analysis of their behavior have already been addressed. The situation is different for the methods based on the Arnoldi algorithm, whose regularizing properties are not well understood or widely accepted, yet. Therefore, still in Chapter 2, a novel analysis of the approximation properties of the Arnoldi algorithm when employed to solve linear discrete ill-posed problems is presented, in order to provide some insight on the use of Arnoldi-based methods for regularization purposes.
The core results of this thesis are related to class of the Arnoldi-Tikhonov methods, first introduced about ten years ago, and described in Chapter 3. The Arnoldi-Tikhonov approach to regularization consists in solving a Tikhonov-regularized problem by means of an iterative strategy based on the Arnoldi algorithm. With respect to a purely iterative approach to regularization, Arnoldi-Tikhonov methods can deliver more accurate approximations by easily incorporating some information about the behavior of the solution within the reconstruction process. In connection with Arnoldi-Tikhonov methods, many open questions still remain, the most significant ones being the choice of the regularization parameters and the choice of the regularization matrices. The first issues are addressed in Chapter 4, where two new efficient and original parameter selection strategies to be employed with the Arnoldi-Tikhonov methods are derived and extensively tested; still in Chapter 4, a novel extension of the Arnoldi-Tikhonov method to the multi-parameter Tikhonov regularization case is described. Finally, in Chapter 5, two efficient and innovative schemes to approximate the solution of nonlinear regularized problems are presented: more precisely, the regularization terms originally defined by the 1-norm or by the Total Variation functional are approximated by adaptively updating suitable regularization matrices within the Arnoldi-Tikhonov iterations.
Along this thesis, the results of many numerical experiments are presented in order to show the performance of the newly proposed methods, and to compare them with the already existing strategies.Questa tesi è incentrata sulle tecniche di regolarizzazione per problemi lineari discreti malposti e di grandi dimensioni. Molteplici applicazioni fisiche ed ingegneristiche sono modellate da questo genere di problemi che, in ambito continuo, sono spesso formulati mediante equazioni integrali di Fredholm di prima specie con nucleo regolare. Più precisamente, queste equazioni modellano i cosiddetti problemi inversi, cioè problemi in cui la causa di un effetto osservato deve essere ricostruita. Una volta discretizzati, questi problemi si presentano come sistemi lineari, la cui matrice dei coefficienti è fortemente malcondizionata e il cui vettore dei termini noti è affetto da qualche perturbazione (spesso chiamata rumore). In questo contesto, risolvere direttamente il sistema lineare discretizzato produrrebbe una soluzione priva di significato, in quanto pesantemente dominata da errori; inoltre, a causa delle grandi dimensioni del sistema, tale procedimento potrebbe risultare infattibile, perchè computazionalmente troppo costoso. Pertanto, qualche forma di regolarizzazione deve essere applicata in modo da poter calcolare una approssimazione fisicamente significativa della soluzione esatta del problema trattato: regolarizzare significa appunto sostituire il sistema lineare con un problema ad esso collegato ma avente migliori proprietà numeriche. La prima parte di questa tesi (Capitolo 1) offre una panoramica sui problemi inversi e descrive brevemente le loro proprietà nel continuo. Quindi, nel discreto, vengono esaminati i più comuni metodi di regolarizzazione basati su una qualche fattorizzazione della matrice del sistema. La restante parte della tesi riguarda le tecniche di regolarizzazione iterative che consistono nell'applicazione di metodi di Krylov: questo tipo di regolarizzazione è particolarmente appropriato quando devono essere risolti sistemi lineari di grandi dimensioni. Più precisamente, nel Capitolo 2, viene proposta un'accurata descrizione dei metodi di Krylov più popolari nell'ambito della regolarizzazione: storicamente, i primi metodi ad essere utilizzati a tale scopo sono stati quelli legati alle equazioni normali e le proprietà regolarizzanti di molti di essi sono già state analizzate. Per quanto riguarda i metodi basati sull'algoritmo di Arnoldi, la situazione è differente: nella maggior parte dei casi, le loro proprietà regolarizzanti non sono ancora state rigorosamente studiate. Pertanto, sempre nel Capitolo 2, viene proposta un'analisi originale delle proprietà approssimanti dell'algoritmo di Arnoldi nel caso in cui esso venga impiegato per la risoluzione di sistemi lineari malposti: l'obbiettivo di questa analisi è di fornire maggiori spiegazioni riguardo all'utilizzo dei metodi basati sull'algoritmo di Arnoldi per la regolarizzazione. I risultati più significativi presentati nella tesi riguardano la classe dei metodi di tipo Arnoldi-Tikhonov, introdotti per la prima volta una decina di anni fa e descritti nel Capitolo 3. L'approccio di tipo Arnoldi-Tikhonov consiste nel risolvere, mediante un metodo iterativo basato sull'algoritmo di Arnoldi, un problema regolarizzato tramite il metodo di Tikhonov. Rispetto ad un approccio regolarizzante puramente iterativo, i metodi di tipo Arnoldi-Tikhonov sono in grado di calcolare soluzioni approssimate più accurate, in quanto all'interno del procedimento iterativo di tipo Arnoldi-Tikhonov possono essere facilmente incorporate alcune informazioni sul comportamento e sulla regolarità della soluzione. Fra i maggiori problemi aperti legati all'utilizzo dei metodi di tipo Arnoldi-Tikhonov figurano la ricerca di metodi efficienti per la scelta dei parametri di regolarizzazione e la scelta di opportune matrici di regolarizzazione. Le problematiche relative alla scelta dei parametri sono trattate nel Capitolo 4, dove vengono derivate due nuove tecniche che possono essere utilizzate congiuntamente ai metodi di tipo Arnoldi-Tikhonov; sempre nel Capitolo 4 viene descritta una nuova estensione del metodo di Arnoldi-Tikhonov al caso della regolarizzazione di Tikhonov a più parametri. Infine, nel Capitolo 5, vengono presentate due innovative ed efficienti strategie per approssimare la soluzione di problemi regolarizzati nonlineari: più precisamente, i termini di regolarizzazione inizialmente definiti utilizzando la norma 1 o il funzionale di Variazione Totale (TV) sono approssimati mediante opportune matrici di regolarizzazione che vengono aggiornate adattivamente durante le iterazioni del metodo di Arnoldi-Tikhonov. In generale, nel corso della trattazione, vengono illustrati i risultati di molteplici esperimenti numerici, con l'obbiettivo di mostrare il comportamento dei nuovi metodi proposti e di confrontarli con quelli già esistenti.
41
Gazzola, Silvia. "Regularization techniques based on Krylov subspace methods for ill-posed linear systems." Thesis, University of Bath, 2014. https://ethos.bl.uk/OrderDetails.do?uin=uk.bl.ethos.707564.
42
Zhang, Ping. "Iterative Methods for Computing Eigenvalues and Exponentials of Large Matrices." UKnowledge, 2009. http://uknowledge.uky.edu/gradschool_diss/789.
Анотація:
In this dissertation, we study iterative methods for computing eigenvalues and exponentials of large matrices. These types of computational problems arise in a large number of applications, including mathematical models in economics, physical and biological processes. Although numerical methods for computing eigenvalues and matrix exponentials have been well studied in the literature, there is a lack of analysis in inexact iterative methods for eigenvalue computation and certain variants of the Krylov subspace methods for approximating the matrix exponentials. In this work, we proposed an inexact inverse subspace iteration method that generalizes the inexact inverse iteration for computing multiple and clustered eigenvalues of a generalized eigenvalue problem. Compared with other methods, the inexact inverse subspace iteration method is generally more robust. Convergence analysis showed that the linear convergence rate of the exact case is preserved. The second part of the work is to present an inverse Lanczos method to approximate the product of a matrix exponential and a vector. This is proposed to allow use of larger time step in a time-propagation scheme for solving linear initial value problems. Error analysis is given for the inverse Lanczos method, the standard Lanczos method as well as the shift-and-invert Lanczos method. The analysis demonstrates different behaviors of these variants and helps in choosing which variant to use in practice.
43
Nilsen, Geir Werner. "Topics in open and closed loop subspace system identification : finite data-based methods." Doctoral thesis, Norwegian University of Science and Technology, Faculty of Information Technology, Mathematics and Electrical Engineering, 2005. http://urn.kb.se/resolve?urn=urn:nbn:no:ntnu:diva-1752.
44
Birk, Sebastian [Verfasser]. "Deflated Shifted Block Krylov Subspace Methods for Hermitian Positive Definite Matrices / Sebastian Birk." Wuppertal : Universitätsbibliothek Wuppertal, 2015. http://d-nb.info/1073127559/34.
45
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.
Анотація:
MathematicsPh.D.
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 subdomain. These are called local problems. Second, if a coarse problem is introduced, they are optimal in the sense that bounds on the convergence rate of the preconditioned iterative method are independent (or slowly dependent) on the finite element mesh size and the number of subproblems. We study certain cases where the same optimality can be obtained without a coarse grid correction. In another part of this thesis we consider inexact GMRES when applied to singular (or nearly singular) linear systems. This applies when instead of matrix-vector products with A, one uses A = A+E for some error matrix E which usually changes from one iteration to the next. Following a similar study by Simoncini and Szyld (2003) for nonsingular systems, we prescribe how to relax the exactness of the matrixvector product and still achieve the desired convergence. In addition, similar criteria is given to guarantee that the computed residual with the inexact method is close to the true residual. Furthermore, we give a new computable criteria to determine the inexactness of matrix-vector product using in inexact CG, and applied it onto control problems governed by parabolic partial differential equations.
Temple University--Theses
46
Katyal, Bhavana. "Multiple current dipole estimation in a realistic head model using signal subspace methods." Online access for everyone, 2004. http://www.dissertations.wsu.edu/Thesis/Summer2004/b%5Fkatyal%5F072904.pdf.
47
Guan, Yu. "Covariate-invariant gait recognition using random subspace method and its extensions." Thesis, University of Warwick, 2015. http://wrap.warwick.ac.uk/67147/.
Анотація:
Compared with other biometric traits like fingerprint or iris, the most significant advantage of gait is that it can be used for remote human identification without cooperation from the subjects. The technology of gait recognition may play an important role in crime prevention, law enforcement, etc. Yet the performance of automatic gait recognition may be affected by covariate factors such as speed, carrying condition, elapsed time, shoe, walking surface, clothing, camera viewpoint, video quality, etc. In this thesis, we propose a random subspace method (RSM) based classifier ensemble framework and its extensions for robust gait recognition. Covariates change the human gait appearance in different ways. For example, speed may change the appearance of human arms or legs; camera viewpoint alters the human visual appearance in a global manner; carrying condition and clothing may change the appearance of any parts of the human body (depending on what is being carried/wore). Due to the unpredictable nature of covariates, it is difficult to collect all the representative training data. We claim overfitting may be the main problem that hampers the performance of gait recognition algorithms (that rely on learning). First, for speed-invariant gait recognition, we employ a basic RSM model, which can reduce the generalisation errors by combining a large number of weak classifiers in the decision level (i.e., by using majority voting). We find that the performance of RSM decreases when the intra-class variations are large. In RSM, although weak classifiers with lower dimensionality tend to have better generalisation ability, they may have to contend with the underfitting problem if the dimensionality is too low. We thus enhance the RSM-based weak classifiers by extending RSM to multimodal-RSM. In tackling the elapsed time covariate, we use face information to enhance the RSM-based gait classifiers before the decision-level fusion. We find significant performance gain can be achieved when lower weight is assigned to the face information. We also employ a weak form of multimodal-RSM for gait recognition from low quality videos (with low resolution and low frame-rate) when other modalities are unavailable. In this case, model-based information is used to enhance the RSM-based weak classifiers. Then we point out the relationship of base classifier accuracy, classifier ensemble accuracy, and diversity among the base classifiers. By incorporating the model-based information (with lower weight) into the RSM-based weak classifiers, the diversity of the classifiers, which is positively correlated to the ensemble accuracy, can be enhanced. In contrast to multimodal systems, large intra-class variations may have a significant impact on unimodal systems. We model the effect of various unknown covariates as a partial feature corruption problem with unknown locations in the spatial domain. By making some assumptions in ideal cases analysis, we provide the theoretical basis of RSM-based classifier ensemble in the application of covariate-invariant gait recognition. However, in real cases, these assumptions may not hold precisely, and the performance may be affected when the intra-class variations are large. We propose a criterion to address this issue. That is, in the decision-level fusion stage, for a query gait with unknown covariates, we need to dynamically suppress the ratio of the false votes and the true votes before the majority voting. Two strategies are employed, i.e., local enhancing (LE) which can increase true votes, and the proposed hybrid decision-level fusion (HDF) which can decrease false votes. Based on this criterion, the proposed RSM-based HDF (RSM-HDF) framework achieves very competitive performance in tackling the covariates such as walking surface, clothing, and elapsed time, which were deemed as the open questions. The factor of camera viewpoint is different from other covariates. It alters the human appearance in a global manner. By employing unitary projection (UP), we form a new space, where the same subjects are closer from different views. However, it may also give rise to a large amount of feature distortions. We deem these distortions as the corrupted features with unknown locations in the new space (after UP), and use the RSM-HDF framework to address this issue. Robust view-invariant gait recognition can be achieved by using the UP-RSM-HDF framework. In this thesis, we propose a RSM-based classifier ensemble framework and its extensions to realise the covariate-invariant gait recognition. It is less sensitive to most of the covariate factors such as speed, shoe, carrying condition, walking surface, video quality, clothing, elapsed time, camera viewpoint, etc., and it outperforms other state-of-the-art algorithms significantly on all the major public gait databases. Specifically, our method can achieve very competitive performance against (large changes in) view, clothing, walking surface, elapsed time, etc., which were deemed as the most difficult covariate factors.
48
Tian, Yunhao. "Subspace method for blind equalization of multiple time-varying FIR channels." Thesis, McGill University, 2012. http://digitool.Library.McGill.CA:80/R/?func=dbin-jump-full&object_id=107827.
Анотація:
Wireless communications is the fastest growing segment of communication technologies. In a wireless communication system, the inter-symbol interference (ISI) is a linear distortion which causes decision errors at the receiver. The equalizer is required to remove the ISI. In the past decade, the blind channel equalization has been a popular research topic in the area of wireless communication. A particular class of blind equalization approaches is based on the second order statistics (SOS) of the received signals. Within this framework, subspace methods exploit the orthogonality between the signal and noise subspaces in order to identify the channel characteristics so that the equalizer can be constructed.This thesis investigates a new equalization algorithm for the time-varying (TV) channel under the single-input multiple-output (SIMO) framework. The TV channel is decomposed using arbitrary basis functions associated with time variable properties of the channels, and with expansion coefficients associated with multi-path delays. An equivalent time-invariant (TI) multiple-input multiple-output (MIMO) system is built for the TV SIMO system. The equivalent TI MIMO system is assumed to match the necessary and sufficient conditions of the SOS identification framework. The blind subspace method is exploited to identify the expansion coefficients when considered as channel characteristics of the MIMO system. The associated ambiguity matrix is identified by using the least square (LS) method. The zero forcing equalizer is realized based on the result of the subspace channel equalization and the ambiguity matrix. The simulation results indicate that the proposed equalizer can effectively recover the source signal in TV SIMO channel applications.La communication sans fil est le segment de croissance le plus dynamique parmi les techniques de la communication. Dans un système de communication sans fil, l'interférence inter-symboles (ISI) est une distorsion linéaire qui provoque des erreurs de décisions au niveau du récepteur. L'égaliseur est nécessaire pour éliminer l'ISI. Récemment, l'égalisation aveugle du canal est devenue un sujet de recherche populaire dans les domaines de la communication sans fil. Un des jalons de la technologie aveugle est fondé sur le cadre des statistiques du second ordre (SOS) du signal reçu. Tout particulièrement, la méthode du sous-espace exploite l'orthogonalité entre le sous-espace signal et le sous-espace bruit afin d'identifier les caractéristiques du canal de telle sorte que l'égaliseur puisse être construit. Dans cette thèse, j'ai proposé un algorithme de péréquation pour le canal à variation temporelle (TV) des systèmes à entrée unique et sorties multiples (SIMO). Le canal TV est décomposé en fonctions arbitraires associées aux propriétés de TV du cacal, et avec les coefficients d'expansion associés à chacun des retards multi-trajet. Un système équivalent invariant dans le temps (TI), à entrée multiples et sorties multiples (MIMO) est conçu pour le TV SIMO. Le systèmeéquivalent TI MIMO est supposé correspondre aux conditions nécessaires et suffisantes dans le cadre de la théorie SOS. La méthode sous-espace aveugle est exploitée pour identifier les coefficients d'expansion quand ils sont considérés comme caractéristiques du canal du système MIMO. La matrice d'ambiguïté est déterminée par la méthode des moindres carrés (LS). La remise à zéro forcée de l'égaliseur est réalisée sur la base des résultats de l'égalisation des canaux de sous-espace et de la matrice d'ambiguïté. Des expériences de simulations numériques sont utilisées afin de démontrer le potential d'application de la nouvelle méthode.
49
Liang, Qiao. "Singular Value Computation and Subspace Clustering." UKnowledge, 2015. http://uknowledge.uky.edu/math_etds/30.
Анотація:
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 inverse-free preconditioned Krylov subspace method, which uses preconditioning instead of shift-and-invert to accelerate the convergence. To compute several eigenvalues, Wielandt is used in a straightforward manner. However, the Wielandt deflation alters the structure of the problem and may cause some difficulties in certain applications such as the singular value computations. So we first propose to consider a deflation by restriction method for the inverse-free Krylov subspace method. We generalize the original convergence theory for the inverse-free preconditioned Krylov subspace method to justify this deflation scheme. We next extend the inverse-free Krylov subspace method with deflation by restriction to the singular value problem. We consider preconditioning based on robust incomplete factorization to accelerate the convergence. Numerical examples are provided to demonstrate efficiency and robustness of the new algorithm.
In the second part of this thesis, we consider the so-called subspace clustering problem, which aims for extracting a multi-subspace structure from a collection of points lying in a high-dimensional space. Recently, methods based on self expressiveness property (SEP) such as Sparse Subspace Clustering and Low Rank Representations have been shown to enjoy superior performances than other methods. However, methods with SEP may result in representations that are not amenable to clustering through graph partitioning. We propose a method where the points are expressed in terms of an orthonormal basis. The orthonormal basis is optimally chosen in the sense that the representation of all points is sparsest. Numerical results are given to illustrate the effectiveness and efficiency of this method.
50
Farina, Sofia. "Barycentric Subspace Analysis on the Sphere and Image Manifolds." Master's thesis, Alma Mater Studiorum - Università di Bologna, 2018. http://amslaurea.unibo.it/15797/.
Анотація:
In this dissertation we present a generalization of Principal Component Analysis (PCA) to Riemannian manifolds called Barycentric Subspace Analysis and show some applications. The notion of barycentric subspaces has been first introduced first by X. Pennec. Since they lead to hierarchy of properly embedded linear subspaces of increasing dimension, they define a generalization of PCA on manifolds called Barycentric Subspace Analysis (BSA). We present a detailed study of the method on the sphere since it can be considered as the finite dimensional projection of a set of probability densities that have many practical applications. We also show an application of the barycentric subspace method for the study of cardiac motion in the problem of image registration, following the work of M.M. Rohé.