
Regularization properties of Krylov subspace methods
Kučerová, Andrea ; Hnětynková, Iveta (advisor) ; Kučera, Václav (referee)
The aim of this thesis is to study and describe regularizing properties of iterative Krylov subspace methods for finding a solution of linear algebraic ill posed problems contaminated by white noise. First we explain properties of this kind of problems, especially their sensitivity to small perturbations in data. It is shown that classical methods for solving approximation problems (such as the least squares method) fail here. Thus we turn to explanation of regularizing pro perties of projections onto Krylov subspaces. Basic Krylov regularizing methods are considered, namely RRGMRES, CGLS, and LSQR. The results are illustrated on model problems from Regularization toolbox in MATLAB. 1


Comparison of least squares methods for problems with errors in the model
Dvořák, Jan ; Hnětynková, Iveta (advisor) ; Kopal, Jiří (referee)
In this work we study the least squares and the total least squares problem for the solution of linear aproximation problems. We introduce both formulations and we discuss the existence and uniqueness of their solutions. We show selected computational methods based on the singular value decomposition. Further we focus on algebraic relations between both solutions and associated residuals. We present several estimates for the norm of the difference between the least squares and total least squares solution. We discuss their dependance on singular values of A and ( A b ) . Finally we illustrate the methods on selected test problems in the Matlab enviroment. 1


Global krylov methods for solving linear algebraic problems with matrix observations
Rapavý, Martin ; Hnětynková, Iveta (advisor) ; Tichý, Petr (referee)
In this thesis we study methods for solving systems of linear algebraic equati ons with multiple right hand sides. Specifically we focus on block Krylov subspace methods and global Krylov subspace methods, which can be derived by various approaches to generalization of methods GMRES and LSQR for solving systems of linear equations with single right hand side. We describe the difference in construction of orthonormal basis in block methods and Forthonormal basis in global methods, in detail. Finally, we provide numerical experiments for the deri ved algorithms in MATLAB enviroment. On carefully selected test problems we compare convergence properties of the methods. 1


The Lanczos method in finite precision arithmetic
Šimonová, Dorota ; Tichý, Petr (advisor) ; Hnětynková, Iveta (referee)
In this thesis we consider the Lanczos algoritm and its behaviour in finite precision. Having summarized theoretical properties of the algorithm and its connection to orthogonal polynomials, we recall the idea of the Lanczos method for approximating the matrix eigenvalues. As the behaviour of the algorithm is strongly influenced by finite precision arithmetic, the linear independence of the Lanczos vectors is usually lost after a few iterations. We use the most im portant results from analysis of the finite precision Lanczos algorithm according to Paige, Greenbaum, Strakos and others. Based on that, we study formulation and properties of the mathematical model of finite presicion Lanczos computati ons suggested by Greenbaum. We carry out numerical experiments in Matlab, which support the theoretical results.


Application of computational methods in classification of glass stones
Lébl, Matěj ; Hnětynková, Iveta (advisor)
Application of computational methods in classification of glass stones Bc. Matěj Lébl Abstrakt: The goal of this thesis is to employ mathematical image processing methods in automatic quality control of glass jewellery stones. The main math ematical subject is a matrix of specific attributes representing digital image of the studied products. First, the thesis summarizes mathematical definition of digital image and some standard image processing methods. Then, a complete solution to the considered problem is presented. The solution consists of stone localization within the image followed by analysis of the localized area. Two lo calization approaches are presented. The first is based on the matrix convolution and optimized through the Fourier transform. The second uses mathematical methods of thresholding and median filtering, and data projection into one di mension. The localized area is analyzed based on statistical distribution of the stone brightness. All methods are implemented in the MATLAB environment. 1


Regularization methods for discrete inverse problems in single particle analysis
Havelková, Eva ; Hnětynková, Iveta (advisor) ; Plešinger, Martin (referee)
The aim of this thesis is to investigate applicability of regulariza tion by Krylov subspace methods to discrete inverse problems arising in single particle analysis (SPA). We start with a smooth model formulation and describe its discretization, yielding an illposed inverse problem Ax ≈ b, where A is a lin ear operator and b represents the measured noisy data. We provide theoretical background and overview of selected methods for the solution of general linear inverse problems. Then we focus on specific properties of inverse problems from SPA, and provide experimental analysis based on synthetically generated SPA datasets (experiments are performed in the Matlab enviroment). Turning to the solution of our inverse problem, we investigate in particular an approach based on iterative Hybrid LSQR with inner Tikhonov regularization. A reliable stopping criterion for the iterative part as well as parameterchoice method for the inner regularization are discussed. Providing a complete implementation of the proposed solver (in Matlab and in C++), its performance is evaluated on various SPA model datasets, considering high levels of noise and realistic distri bution of orientations of scanning angles. Comparison to other regularization methods, including the ART method traditionally used in SPA,...


Numerical Methods in Discrete Inverse Problems
Kubínová, Marie ; Hnětynková, Iveta (advisor) ; Gazzola, Silvia (referee) ; Meurant, Gerard (referee)
Title: Numerical Methods in Discrete Inverse Problems Author: Marie Kubínová Department: Department of Numerical Mathematics Supervisor: RNDr. Iveta Hnětynková, Ph.D., Department of Numerical Mathe matics Abstract: Inverse problems represent a broad class of problems of reconstruct ing unknown quantities from measured data. A common characteristic of these problems is high sensitivity of the solution to perturbations in the data. The aim of numerical methods is to approximate the solution in a computationally efficient way while suppressing the influence of inaccuracies in the data, referred to as noise, that are always present. Properties of noise and its behavior in reg ularization methods play crucial role in the design and analysis of the methods. The thesis focuses on several aspects of solution of discrete inverse problems, in particular: on propagation of noise in iterative methods and its representation in the corresponding residuals, including the study of influence of finiteprecision computation, on estimating the noise level, and on solving problems with data polluted with noise coming from various sources. Keywords: discrete inverse problems, iterative solvers, noise estimation, mixed noise, finiteprecision arithmetic  iii 


Exploiting numerical linear algebra to accelerate the computation of the MCD estimator
Sommerová, Kristýna ; Duintjer Tebbens, Erik Jurjen (advisor) ; Hnětynková, Iveta (referee)
This work is dealing with speeding up the algorithmization of the MCD es timator for detection of the mean and the covariance matrix of a normally dis tributed multivariate data contaminated with outliers. First, the main idea of the estimator and its wellknown aproximation by the FastMCD algorithm is discussed. The main focus was to be placed on possibilities of a speedup of the iteration step known as Cstep while maintaining the quality of the estimations. This proved to be problematic, if not impossible. The work is, therefore, aiming at creating a new implementation based on the Cstep and Jacobi method for eigenvalues. The proposed JacobiMCD algorithm is compared to the FastMCD in terms of floating operation count and results. In conclusion, JacobiMCD is not found to be fully equivalent to FastMCD but hints at a possibility of its usage on larger problems. The numerical experiments suggest that the computation can indeed be quicker by an order of magnitude, while the quality of results is close to those from FastMCD in some settings. 1


Reorthogonalization strategies in GolubKahan iterative bidiagonalization
Šmelík, Martin ; Hnětynková, Iveta (advisor) ; Kučera, Václav (referee)
The main goal of this thesis is to describe GolubKahan iterative bidiagonalization and its connection with Lanczos tridiagonalization and Krylov space theory. The GolubKahan iterative bidiagonalization is based on short recurrencies and when computing in finite precision arithmetics, the loss of orthogonality often occurs. Consequently, with the aim to reduce the loss of orthogonality, we focus on various reorthogonalization strategies. We compare them in numerical experiments on testing matrices available in the MATLAB environment. We study the dependency of the loss of orthogonalization and computational time on the choice of the method or the attributes of the matrix.

 