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Mixed Precision in Uncertainty Quantification Methods
Martínek, Josef ; Carson, Erin Claire (advisor) ; Hnětynková, Iveta (referee)
This work is concerned with analysing and exploiting mixed precision arithmetic in un- certainty quantification methods with emphasis on the multilevel Monte Carlo (MLMC) method. Although mixed precision can improve performance, it should be used carefully to avoid unwanted effects on the solution accuracy. We provide a rigorous analysis of uncertainty quantification methods in finite precision arithmetic. Based on this analysis, we exploit mixed precision arithmetic in uncertainty quantification methods to improve runtime while preserving the overall error. We begin by stating the model problem, an elliptic PDE with random coefficients and a random right-hand side. Such a problem arises, for example, in uncertainty quantification for groundwater flow. Our focus is on approximating a quantity of interest given as the expected value of a functional of the solution of the PDE problem. To this end, we use the conforming finite element method for approximation in the spatial variable and the MLMC method for approximation of the expected value. We provide a novel rigorous analysis of the MLMC method in finite precision arithmetic and based on this we formulate an adaptive algorithm which determines the optimal precision value on each level of discretisation. To our knowledge, this is a new approach. Our...
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Iterative methods for Tichonov regularization with generalized regularization terms
Kučerová, Andrea ; Hnětynková, Iveta (advisor) ; Carson, Erin Claire (referee)
The aim of this thesis is to study hybrid methods for solving ill-posed linear inverse problems corrupted by white noise. These approaches are based on the combination of iterative Krylov subspace methods and the Tichonov regularization with a general regularization term. We explain the basic properties of ill-posed problems, the idea of regularization, the role of the regularization term to enforce desirable properties to the solution and the theoretical background of Standard and General Tichonov minimization. Then we explain shift invariance of Krylov subspaces. This allows us to describe a hybrid approach where the full size problem is first projected onto a Krylov subspace of a smaller dimension and then the Tichonov minimization is applied to the small projected problem. We focus on the regularization based on the finite difference approximation of derivatives of the solution. The well known regularization terms constructed from forward differences for the first and the second derivative are summarized, then we use the Taylor expansion to construct finite differences of higher orders of precision. We incorporate different variants of boundary conditions. Then the impact of the order of precision of the finite difference schemes on the quality of the solution is studied. In the experiments we use the...
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Algebraic view on the PCA method in selected applications
Hammerbauer, Tomáš ; Hnětynková, Iveta (advisor) ; Tichý, Petr (referee)
This thesis deals with describing algebraic and statistic view on Principal component analysis and the way of exporting important variables. Basic properties of the singular value decomposition are introduced and the best rank k aproximation of a matrix is de- rived. Then, a conection between PCA and singular value decomposition is described. At the end, PCA is ilustrated on two numerical experiments on image databases. It is shown, how we can aproximate images simillar to the elements of the database. Theo- retical foundations for the experiments are presented and then they are implemented in the Matlab enviroment. 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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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 finite-precision 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, finite-precision arithmetic - iii -
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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 ill-posed 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 parameter-choice 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,...
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Reorthogonalization strategies in Golub-Kahan iterative bidiagonalization
Šmelík, Martin ; Hnětynková, Iveta (advisor) ; Kučera, Václav (referee)
The main goal of this thesis is to describe Golub-Kahan iterative bidiagonalization and its connection with Lanczos tridiagonalization and Krylov space theory. The Golub-Kahan 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.
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