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Optimization Methods for Compensation of Shooting Inaccuracy
Horníček, Jan ; Novotný, Jan (referee) ; Popela, Pavel (advisor)
In this work there is performed analysis of inaccuracy shooting and its optimum corrections especially for playing darts. At rst the model describing inaccurate shooting is made. Using this model we received prerequisites for numerical solving of our problem. Computing algorithm was made and than was proven that our problem can be solved with arbitrary precision by this algorithm. The algorithm was implemented in MATLAB and modied using functional Analysis to minimize computing time. Our problem was solved by this algorithm and in conclusion was made simple application for visualization of received data.
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The Selected Stochastic Programs in Engineering Design
Čajánek, Michal ; Mrázková, Eva (referee) ; Popela, Pavel (advisor)
Two-stage stochastic programming problem with PDE constraint, specially elliptic equation is formulated. The computational scheme is proposed, whereas the emphasis is put on approximation techniques. We introduce method of approximation of random variables of stochastic problem and utilize suitable numerical methods, finite difference method first, then finite element method. There is also formulated a mathematical programming problem describing a membrane deflection with random load. It is followed by determination of the acceptableness of using stochastic optimization rather than deterministic problem and assess the quality of approximations based on Monte Carlo simulation method and the theory of interval estimates. The resulting mathematical models are implemented and solved in the general algebraic modeling system GAMS. Graphical and numerical results are presented.
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Mathematical description of vehicle motion trajectory
Lorenczyk, Jiří ; Popela, Pavel (referee) ; Porteš, Petr (advisor)
The goal of this thesis is to nd types of curves which would allow for the construction of a path that could be traversed by a vehicle. It seems that a minimal constraint for such a path is the continuity of curve's curvature. This leads to a closer look at the three types of curves: Clothoids, which are able to smoothly connect straights with arcs of a constant curvature, interpolation quintic splines, which are C2 smooth in the interpolation nodes and -splines, these belong to the family of quintic polynomial curves too, however, they are characterised by the vector of parameters which modies the shape of the curve. The thesis is accompanied by an application allowing for manual construction of the path composed of spline curves.
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Selected Optimization Models for Risk Reduction in Air Transport
Schwarz, Ondřej ; Bednář, Josef (referee) ; Popela, Pavel (advisor)
The diploma thesis deals with mathematical modeling of the situation when due to the increased risks occurring around Europe there is a need of immediate transport of Czech citizens back to the Czech Republic. The model takes into account the general level of random fluctuations in demand and transportation costs. The optimization model is then built on the ideas of stochastic programming and includes real and expert data in the field of air transportation, which are later implemented into GAMS. The results are discussed. The survey part of the diploma thesis is devoted to the possibility of applying linear and stochastic programming, the interpretation of the mathematical transportation theory. It describes the basic tools and development environment and the realization of mathematical model based on real data in GAMS. The second part of the thesis deals with a proposal of possible improvements in aviation by applying the real data into the model and by analyzing the obtained results.
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Mathematical models for transportation problems
Brzobohatý, Jan ; Hrabec, Dušan (referee) ; Popela, Pavel (advisor)
This bachelor's thesis deals with stochastic dominance. The goal is to lay the foundations for defining stochastic dominance, to describe its properties and to explain this term on simple examples. Another goal is to apply this term to network problems with random transport price. Examples in this thesis also contain solutions and code to find these solutions written in GAMS language.
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Analysis of Stocks and Shares Risk Level
Bohuněk, David ; Blažek, Petr (referee) ; Popela, Pavel (advisor)
This thesis deals with quantification of risk level of securities on Czech capital market. Part of quantification of risk is also assessing the impact of various scenarios on the price of securities. In theoretical part is briefly described scheme of securities market. Then is described history of Prague stock exchange, stock index included index of Prague stock exchange. The following section deals with describing fundamental investment instruments used in thesis. A significant part is devoted to mutual funds from a different angle than normal retail investor knows. The rest of the theoretical part describes the investment decisions based on the investment triangle including risk indicators applied in the practical part. At the beginning of practical are described selected investment instruments. Reader can also find out about their current and possible future market position. Further indicators of risk are applied, selected and described the scenario of events and their impact on the price of the instrument. In the last chapter is chosen one scenario in which risk indicators are calculated.
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Stochastic programming models with applications
Novotný, Jan ; Michálek, Jaroslav (referee) ; Popela, Pavel (advisor)
Diplomová práce se zabývá stochastickým programováním a jeho aplikací na problém mísení kameniva z oblasti stavebního inženýrství. Teoretická část práce je věnována odvození základních přístupů stochastického programování, tj. optimalizace se zohledněním náhodných vlivů v modelech. V aplikované části je prezentována tvorba vhodných optimalizačních modelů pro mísení kameniva, jejich implementace a výsledky. Práce zahrnuje původní aplikační výsledky docílené při řešení projektu GA ČR reg. čís. 103/08/1658 Pokročilá optimalizace návrhu složených betonových konstrukcí a teoretické výsledky projektu MŠMT České republiky čís. 1M06047 Centrum pro jakost a spolehlivost výroby.
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Optimization for Engineering Design
Fedorko, Tomáš ; Mauder, Tomáš (referee) ; Popela, Pavel (advisor)
This bachelor's thesis concerns the problem of finding an optimal cooling solution in respect to cooling costs. It describes the problem of heat transfer, its analytical solution, numerical solution of the boundary values problem and its implementation in MATLAB using the FMINCON function. Individual results are presented graphically, as well as, summarized in calculations and interpreted.
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Risk Modelling in Engineering Applications
Konečný, Antonín ; Karpíšek, Zdeněk (referee) ; Popela, Pavel (advisor)
The diploma thesis focuses on risk modelling in engineering applications. The knowledge obtained from statistical methods of operational research, specifically regression analysis, is used. Electrical quantities measured on the stator winding of an electrical machine (generator) are analyzed. Regression analysis and statistical hypothesis testing is performed in Statgraphics software. The success results of the regression analysis for risk reduction are documented in detail and the regression curves are visualized in graphs. Conclusions and recommendations for each generator, as well as general ones, are formulated.
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Spatial Decomposition for Differential Equation Constrained Stochastic Programs
Šabartová, Zuzana ; Mrázková, Eva (referee) ; Popela, Pavel (advisor)
Rozsáhlá třída inženýrských optimalizačních úloh vede na modely s omezeními ve tvaru obyčejných nebo parciálních diferenciálních rovnic (ODR nebo PDR). Protože diferenciálních rovnice je možné řešit analyticky jen v nejjednodušších případech, bylo k řešení použito numerických metod založených na diskretizaci oblasti. Zvolili jsme metodu konečných prvků, která umožňuje převod omezení ve tvaru diferenciálních rovnic na omezení ve tvaru soustavy lineárních rovnic. Reálné problémy jsou často velmi rozsáhlé a přesahují dostupnou výpočetní kapacitu. Výpočetní čas lze snížit pomocí progressive hedging algoritmu (PHA), který umožňuje paralelní implementaci. PHA je efektivní scénářová dekompoziční metoda pro řešení scénářových stochastických úloh. Modifikovaný PHA byl využit pro původní přístup prostorové dekompozice. Aproximace diferenciálních rovnic v modelu problému je dosaženo pomocí diskretizace oblasti. Diskretizace je dále využita pro prostorovou dekompozici modelu. Algoritmus prostorové dekompozice se skládá z několika hlavních kroků: vyřešení problému s hrubou diskretizací, rozdělení oblasti problému do překrývajících se částí a iterační řešení pomocí PHA s jemnější diskretizací s využitím hodnot z hrubé diskretizace jako okrajových podmínek. Prostorová dekompozice byla aplikována na základní testovací problém z oboru stavebního inženýrství, který se zabývá návrhem rozměrů průřezu nosníku. Algoritmus byl implementován v softwaru GAMS. Získané výsledky jsou zhodnoceny vzhledem k výpočetní náročnosti a délce překrytí.
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