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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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Maticové funkce a jejich numerické aproximace
Suchá, Darja ; Hnětynková, Iveta (advisor) ; Strakoš, Zdeněk (referee)
In the presented work, we study numerical methods for approximation of a function f of a matrix A. First, we give theoretical background - definitions of matrix functions, and their properties. Further, we summarize basic numerical methods for computation of an approximation of matrix functions f(A). In many applications, we need to approximate the matrix function f(A) applied on an apriory given vector b, i.e. f(A)b. Especially, when A is large and sparse, the computation of approximation to f(A) and subsequent multiplication by the vector b can be computationaly expensive. Therefore we study methods, which compute the approximation of f(A)b directly. Main emphasis is placed on the polynomial approximation in the least squares sense, and several modifications of Krylov subspace methods. Numerical experiments compare convergence and computa- tional time required to obtain reasonable approximation to f(A)b. 1
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Approximate Polynomial Greatest Common Divisor
Eliaš, Ján ; Zítko, Jan (advisor) ; Hnětynková, Iveta (referee)
Title: Approximate Polynomial Greatest Common Divisor Author: Ján Eliaš Department: Department of Numerical Mathematics, MFF UK Supervisor: Doc. RNDr. Jan Zítko, CSc., Department of Numerical Mathematics, MFF UK Abstract: The computation of polynomial greatest common divisor (GCD) ranks among basic algebraic problems with many applications. The Euclidean algorithm is the oldest and usual technique for computing GCD. However, the GCD computation problem is ill-posed, particularly when some unknown noise is applied to the polyno- mial coefficients. Since the Euclidean algorithm is unstable, new methods have been extensively studied in recent years. Methods based on the numerical rank estimation represent one group of current meth- ods. Their disadvantage is that the numerical rank cannot be computed reliably due to the sensitivity of singular values on noise. The aim of the work is to overcome the ill-posed sensitivity of GCD computation in the presence of noise. Keywords: AGCD, Sylvester matrix, numerical rank, TLS
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Numerické řešení inverzních integrálních rovnic matematického modelování ve výzkumu biopaliv
Bílková, Zuzana ; Hnětynková, Iveta (advisor) ; Kofroň, Josef (referee)
Název práce: Numerické řešení inverzních integrálních rovnic matema- tického modelování ve výzkumu biopaliv Autor: Zuzana Bílková Katedra / Ústav: Katedra numerické matematiky Vedoucí bakalářské práce: RNDr. Iveta Hnětynková, Ph.D., Katedra nu- merické matematiky MFF UK Abstrakt: Cílem této bakalářské práce je numerické řešení Fredholmových integrálních rovnic prvního řádu, které se vyskytují ve výzkumu biopaliv. Práce se zaměřuje na studium Lagrangových interpolačních kvadraturních formulí. Uvažujeme lichoběžníkové a Simpsonovo pravidlo s využitím ekvidistantního a logaritmického dělení. Cílem práce je srovnání těchto pravidel a nalezení nej- vhodnější metody. Práce se dále zabývá určením minimálního počtu naměře- ných dat tak, abychom dosáhli dané přesnosti. Poznatky jsou demonstrovány na numerických experimentech se simulovanými daty. Klíčová slova: inverzní integrální rovnice, kvadratura, konvergence, odhad chyby 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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