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Numerical computation with functions using Chebfun
Lébl, Matěj ; Tichý, Petr (advisor) ; Hnětynková, Iveta (referee)
Goal of this work is to introduce Chebfun software and show ideas behind it. In the first chapter we summarize the theory of polynomial interpolation with focus on the Chebyshev interpolants. In the second chapter we introduce Chebfun software, its basic commands and principles of constructing interpolants. The third chapter is devoted to demonstrate theorems from the first chapter and to show practical applications of Chebfun when finding roots of a function and solving differential equations. Powered by TCPDF (www.tcpdf.org)
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Numerical methods in image processing for applications in jewellery industry
Petrla, Martin ; Hnětynková, Iveta (advisor) ; Plešinger, Martin (referee)
Presented thesis deals with a problem from the field of image processing for application in multiple scanning of jewelery stones. The aim is to develop a method for preprocessing and subsequent mathematical registration of images in order to increase the effectivity and reliability of the output quality control. For these purposes the thesis summerizes mathematical definition of digital image as well as theoretical base of image registration. It proposes a method adjusting every single image to increase effectivity of its subsequent processing. One image for every evaluated gemstone is generated using image registration. The method is implementated in the MATLAB environment. Powered by TCPDF (www.tcpdf.org)
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Efficient implementation of dimension reduction methods for high-dimensional statistics
Pekař, Vojtěch ; Duintjer Tebbens, Erik Jurjen (advisor) ; Hnětynková, Iveta (referee)
The main goal of our thesis is to make the implementation of a classification method called linear discriminant analysis more efficient. It is a model of multivariate statistics which, given samples and their membership to given groups, attempts to determine the group of a new sample. We focus especially on the high-dimensional case, meaning that the number of variables is higher than number of samples and the problem leads to a singular covariance matrix. If the number of variables is too high, it can be practically impossible to use the common methods because of the high computational cost. Therefore, we look at the topic from the perspective of numerical linear algebra and we rearrange the obtained tasks to their equivalent formulation with much lower dimension. We offer new ways of solution, provide examples of particular algorithms and discuss their efficiency. Powered by TCPDF (www.tcpdf.org)
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Computation and applications of the MCD estimator for robust statistical analysis
Sommerová, Kristýna ; Duintjer Tebbens, Erik Jurjen (advisor) ; Hnětynková, Iveta (referee)
This work describes one of the basic problems of robust statistics con- cerning outlier detection and its possible solution by using the Minimum covariance determinant estimator for estimates of the mean value and the covariance matrix with multivariate data. It explains how the estimator works and analyses its properties. The work concentrates on its approximation based on the fastMCD algorithm and specifies its numerical properties with emphasis on computational costs and stability of the standard implementation in MATLAB. It also discusses possible modifications of the algorithm and its effects on numerical properties. Lastly the work shows the usage of the fastMCD algorithm on a few real data experiments. Powered by TCPDF (www.tcpdf.org)
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Regularization techniques based on the least squares method
Kubínová, Marie ; Hnětynková, Iveta (advisor)
Title: Regularization Techniques Based on the Least Squares Method Author: Marie Michenková Department: Department of Numerical Mathematics Supervisor: RNDr. Iveta Hnětynková, Ph.D. Abstract: In this thesis we consider a linear inverse problem Ax ≈ b, where A is a linear operator with smoothing property and b represents an observation vector polluted by unknown noise. It was shown in [Hnětynková, Plešinger, Strakoš, 2009] that high-frequency noise reveals during the Golub-Kahan iterative bidiagonalization in the left bidiagonalization vectors. We propose a method that identifies the iteration with maximal noise revealing and reduces a portion of high-frequency noise in the data by subtracting the corresponding (properly scaled) left bidiagonalization vector from b. This method is tested for different types of noise. Further, Hnětynková, Plešinger, and Strakoš provided an estimator of the noise level in the data. We propose a modification of this estimator based on the knowledge of the point of noise revealing. Keywords: ill-posed problems, regularization, Golub-Kahan iterative bidiagonalization, noise revealing, noise estimate, denoising 1
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Medical image registration
Lacmanová, Zdeňka ; Soukup, Jindřich (advisor) ; Hnětynková, Iveta (referee)
This thesis deals with the registration of images, which differ only in the shift and rotation, taken from CT angiography. The term of registration and transformation between the images are explained here. Three methods are used for registration. The first method is based on Fourier transform and it is called the phase correlation. The other two methods use the measured data by phase correlation and then the data is processed using the least squares method or singular value decomposition. There is given detailed theoretical foundation, the methods are mathematically derived and then implemented and tested in Matlab. Powered by TCPDF (www.tcpdf.org)
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Regularizační metody založené na metodách nejmenších čtverců
Michenková, Marie ; Hnětynková, Iveta (advisor) ; Zítko, Jan (referee)
Title: Regularization Techniques Based on the Least Squares Method Author: Marie Michenková Department: Department of Numerical Mathematics Supervisor: RNDr. Iveta Hnětynková, Ph.D. Abstract: In this thesis we consider a linear inverse problem Ax ≈ b, where A is a linear operator with smoothing property and b represents an observation vector polluted by unknown noise. It was shown in [Hnětynková, Plešinger, Strakoš, 2009] that high-frequency noise reveals during the Golub-Kahan iterative bidiagonalization in the left bidiagonalization vectors. We propose a method that identifies the iteration with maximal noise revealing and reduces a portion of high-frequency noise in the data by subtracting the corresponding (properly scaled) left bidiagonalization vector from b. This method is tested for different types of noise. Further, Hnětynková, Plešinger, and Strakoš provided an estimator of the noise level in the data. We propose a modification of this estimator based on the knowledge of the point of noise revealing. Keywords: ill-posed problems, regularization, Golub-Kahan iterative bidiagonalization, noise revealing, noise estimate, denoising 1
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Analysis of Krylov subspace methods
Gergelits, Tomáš ; Strakoš, Zdeněk (advisor) ; Hnětynková, Iveta (referee)
Title: Analysis of Krylov subspace methods Author: Tomáš Gergelits Department: Department of Numerical Mathematics Supervisor: prof. Ing. Zdeněk Strakoš, DrSc. Abstract: After the derivation of the Conjugate Gradient method (CG) and the short review of its relationship with other fields of mathematics, this thesis is focused on its convergence behaviour both in exact and finite precision arith- metic. Fundamental difference between the CG and the Chebyshev semi-iterative method is described in detail. Then we investigate the use of the widespread lin- ear convergence bound based on Chebyshev polynomials. Through the example of the composite polynomial convergence bounds it is showed that the effects of rounding errors must be included in any consideration concerning the CG rate of convergence relevant to practical computations. Furthermore, the close corre- spondence between the trajectories of the CG approximations generated in finite precision and exact arithmetic is studied. The thesis is concluded with the discus- sion concerning the sensitivity of the closely related Gauss-Christoffel quadrature. The last two topics may motivate our further research. Keywords: Conjugate Gradient Method, Chebyshev semi-iterative method, fi- nite precision computations, delay of convergence, composite polynomial conver-...
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Teorie a aplikace krylovovských metod v souvislostech
Gergelits, Tomáš ; Strakoš, Zdeněk (advisor) ; Hnětynková, Iveta (referee)
Title: Krylov subspace methods: Theory, applications and interconnections Author: Tomáš Gergelits Department: Department of Numerical Mathematics Supervisor: prof. Ing. Zdeněk Strakoš, DrSc. Abstract: After recalling of properties of Chebyshev polynomials and of sta- tionary iterative methods, this thesis is focused on the description of Conjugate Gradient Method (CG), the Krylov method of the choice for symmetric positive definite matrices. Fundamental difference between stationary iterative methods and Krylov subspace methods is emphasized. CG is derived using the minimiza- tion of the quadratic functional and the relationship with several other fields of mathematics (Lanczos method, orthogonal polynomials, quadratic rules, moment problem) is pointed out. Effects of finite precision arithmetic are emphasized. In compliance with the theoretical part, the numerical experiments examine a bound derived assuming exact arithmetic which is often presented in literature. It is shown that this bound inevitably fails in practical computations. The thesis is concluded with description of two open problems which can motivate further research. Keywords: Krylov subspace methods, convergence behaviour, numerical stabil- ity, spectral information, convergence rate bounds
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