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Lineární algebraické modelování úloh s nepřesnými daty
Vasilík, Kamil ; Hnětynková, Iveta (advisor) ; Janovský, Vladimír (referee)
In this thesis we consider problems Ax b arising from the discretization of ill-posed problems, where the right-hand side b is polluted by (unknown) noise. It was shown in [29] that under some natural assumptions, using the Golub-Kahan iterative bidiagonalization the noise level in the data can be estimated at a negligible cost. Such information can be further used in solving ill-posed problems. Here we suggest criteria for detecting the noise revealing iteration in the Golub-Kahan iterative bidiagonalization. We discuss the presence of noise of different colors. We study how the loss of orthogonality affects the noise revealing property of the bidiagonalization.
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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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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
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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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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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Numerické metody zpracování obrazu
Tóthová, Katarína ; Hnětynková, Iveta (advisor) ; Zítko, Jan (referee)
The aim of this thesis is to provide a concise overview of the numerical techniques in digital image processing, specifically to discuss the construction, properties and methods of solving of the image deblurring problems modelled by a linear system Ax = b. Often, these problems fall within a group of the ill-posed problems with severely ill-conditioned matrix A and hence require special numerical treatment. We provide a brief overview of selected regularization methods that can be used in this situation, including direct (TSVD, Tikhonov regularization) and iterative ones (CGLS, LSQR), together with the pertinent parameter-choice methods - L-curve, GCV and the discrepancy principle. The theoretical discussion is supplemented by the numerical experiments with real-life image data.
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Lineární algebraické modelování úloh s nepřesnými daty
Vasilík, Kamil ; Hnětynková, Iveta (advisor) ; Janovský, Vladimír (referee)
In this thesis we consider problems Ax b arising from the discretization of ill-posed problems, where the right-hand side b is polluted by (unknown) noise. It was shown in [29] that under some natural assumptions, using the Golub-Kahan iterative bidiagonalization the noise level in the data can be estimated at a negligible cost. Such information can be further used in solving ill-posed problems. Here we suggest criteria for detecting the noise revealing iteration in the Golub-Kahan iterative bidiagonalization. We discuss the presence of noise of different colors. We study how the loss of orthogonality affects the noise revealing property of the bidiagonalization.
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