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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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Numerical solution of equations describing the dynamics of flocking
Živčáková, Andrea ; Kučera, Václav (advisor)
This work is devoted to the numerical solution of equations describing the dynamics of flocks of birds. Specifically, we pay attention to the Euler equati- ons for compressible flow with a right-hand side correction. This model is based on the work Fornasier et al. (2010). Due to the complexity of the model, we focus only on the one-dimensional case. For the numerical solution we use a semi- implicit discontinuous Galerkin method. Discretization of the right-hand side is chosen so that we preserve the structure of the semi-implicit scheme for the Euler equations presented in the work Feistauer, Kučera (2007). The proposed numeri- cal scheme was implemented and numerical experiments showing the robustness of the scheme were carried out. 1
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On the Dijkstra's algorithm in the pedestrian flow problem
Petrášová, Tereza ; Felcman, Jiří (advisor) ; Kučera, Václav (referee)
Title: On the Dijkstra's algorithm in the Pedestrian Flow Problem Author: Tereza Petrášová Department: Department of Numerical Mathematics Supervisor: doc. RNDr. Jiří Felcman, CSc., Department of Numerical Mathe- matics Abstract: The pedestrian flow problem is described by a coupled system of the first order hyperbolic partial differential equations with the source term and by the functional minimization problem for the desired direction of motion. The functional minimization is based on the modified Dijkstra's algorithm used to find the minimal path to the exit. The original modification of the Dijkstra's algorithm is proposed to increase its efficiency in the pedestrian flow problem. This approach is compared with the algorithm of Bornemann and Rasch for determination of the desired direction of motion based on the solution of the so- called Eikonal equation. Both approaches are numerically tested in the framework of two splitting algorithms for solution of the coupled problem. The former splitting algorithm is based on the finite volume method yielding for the given time instant the piecewise constant approximation of the solution. The latter one uses the implicit discretization by a space-time discontinuous Galerkin method based on the discontinuous piecewise polynomial approximation. The numerical examples...
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The Gibbs phenomenon in the discontinuous Galerkin method
Stará, Lenka ; Kučera, Václav (advisor) ; Sváček, Petr (referee)
The solution of the Burgers' equation computed by the standard finite element method is degraded by oscillations, which are the manifestation of the Gibbs phenomenon. In this work we study the following numerical me- thods: Discontinuous Galerkin method, stable low order schemes and the flux corrected technique method in order to prevent the undesired Gibbs phenomenon. The focus is on the reduction of severe overshoots and under- shoots and the preservation of the smoothness of the solution. We consider a simple 1D problem on the interval (0, 1) with different initial conditions to demonstrate the properties of the presented methods. The numerical results of individual methods are provided. 1
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Numerical solution of traffic flow models
Vacek, Lukáš ; Kučera, Václav (advisor) ; Janovský, Vladimír (referee)
Our work describes the simulation of traffic flows on networks. These are described by partial differential equations. For the numerical solution of our models, we use the discontinuous Galerkin method in space and a multistep method in time. This combination of the two methods on networks is unique and leads to a robust numerical scheme. We use several different approaches to model the traffic flow. Thus, our program must solve both scalar problems as well as systems of equations described by first and second order partial differential equations. The output of our programs is, among other things, the evolution of traffic density in time and 1D space. Since this is a physical quantity, we introduce limiters which keep the density in an admissible interval. Moreover, limiters prevent spurious oscillations in the numerical solution. All the above is performed on networks. Thus, we must deal with the situation at the junctions, which is not standard. The main task is to ensure that the law of conservation of the total amount of cars passing through the junction is still satisfied. This is achieved by modifying the numerical flux for junctions. The result of this work is the comparison of all the models, the demonstration of the benefits of the discontinuous Galerkin method and the influence of limiters.
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Legal regulation of employment termination
Kučera, Václav ; Vysokajová, Margerita (advisor) ; Štefko, Martin (referee)
Presented rigorous thesis deals with legislation of employment termination and also institutions included in the employment termination. The termination notice is very frequented way of ending the employment although this unilateral juridical proceedings may be find as complications free, questions are arising during its application in place, and this thesis is trying to find satisfactory answers. It can be considered, that application of the institute has been carrying on contra legem, especially in terms of application by the employer, as we can find in rich court juridical. Employers do not hesitate to terminate employment by using termination notice even though, they did not meet legal conditions or they just simply feign meeting legal conditions. It depends on the employee and his willingness with strength to undergo law enforcement of his claims at independent courts. This thesis examines in detail the institute of dismissal, pointing out the historical adjustments and adjustments in international documents, closely devoting the employment itself, object, subject, content, commencement and duration of the employment and also employer duties connected to the employment. Thereafter the thesis deals with legal actions and their validity or otherwise. Delivering, typical for directed legal...
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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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Segmentace mikroskopických snímků pomocí level-set metod
Bílková, Zuzana ; Kučera, Václav (advisor)
Název práce: Segmentace mikroskopických snímků pomocí level-set metod Autor: Zuzana Bílková Katedra: Katedra numerické matematiky Vedoucí diplomové práce: RNDr. Václav Kučera, Ph.D., KNM, MFF UK Konzultant: RNDr. Jindřich Soukup, ÚTIA, AV ČR Abstrakt: Tato diplomová práce představuje novou metodu pro segmentaci snímků pořízených mikroskopem s fázovým konrastem. Cílem je oddělit buňky od pozadí. Algoritmus je založen na variační formulaci level set metod, tedy na minimalizaci funkcionálu popisujícího level set funkci. Funkcionál je minimalizován gradientním tokem popsaným evoluční parciální diferenciální rovnicí. Nejdůležitější nové myšlenky jsou inicializace pomocí prahování a nové členy ve funkcionálu, které zrychlují konvergenci a zpřesňují výsledky. Také jsme použili nové funkce napsané v jazyce C k počítání gradientu a Laplaceova operátoru. Tato implementace je třikrát rychlejší než standardní funkce v MATLABu. Dosáhli jsme lepších výsledků než algoritmy, se kterými jsme metodu porovnávali. Klíčová slova: Segmentace, level set metody, aktivní kontury Title: Segmentation of microscopic images using level set methods Author: Zuzana Bílková Department: Department of Numerical Mathematics Supervisor: RNDr....
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Diet supplemantation among para troops in the Army of the Czech Republic
Kučera, Václav ; Michalička, Vladimír (advisor) ; Šťastný, Petr (referee)
Title Use of Dietary Supplements at the Czech Airborne Force. Objective The objective of my work was to examine the use of supplements in the Czech Airborne Battalions through the use of questionnaire method and analysis of professional literature. Professional soldiers, especially in airborne battalions, are expected to have a high level of psychological as well as physical readiness. Building on professional literature, articles, internet resources, and information from leading nutrition and supplementation experts I sought to compile a work that would provide basic information on the use of supplementation. Plan of processing At the very beginning, it was necessary to have a comprehensive idea of the implementation of the questionnaire method in the Czech Airborne Forces. A review of available materials was followed by a summary of the basic nutrients and supplements that are, in our opinion, the most used in paratroopers. As a next step a questionnaire was created. Then, we handed out and collected questionnaires at individual crews. This was followed by the questionnaire evaluation, charting, and finishing the bachelor's thesis. Results Supplementation is very popular with the Czech Army paratroopers. A total of 86.7 % of respondents use dietary supplements. The results show, that the main...
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The choice of the step in trust region methods
Rapavý, Martin ; Tichý, Petr (advisor) ; Kučera, Václav (referee)
The main goal of this thesis is the choice of steps in trust region methods for finding a minimum of a given function. The step corresponds to the problem of finding a minimum of a model function on a trust region. We characterize a solu- tion of this problem (Moré-Sorensen theorem) and consider various techniques for approximating a solution of this problem (the Cauchy point method, the dogleg method, the conjugate gradients method). In the case of the first two techniques we prove convergence of the optimization method. Finally, the above techniques are tested numerically in MATLAB on properly chosen functions and initial data. We comment on advantages and disadvantages of considered algorithms. 1
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