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Optimization using derivative-free and metaheuristic methods
Henclová, Kateřina ; Tichý, Petr (advisor)
Evolutionary algorithms have proved to be useful for tackling many practical black-box optimization problems. In this thesis, we describe one of the most powerful evolutionary algorithms of today, CMA- ES, and apply it in novel way to solve the problem of tuning multiple coupled PID controllers in combustion engine models. Powered by TCPDF (www.tcpdf.org)
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Možnosti energetického využití odpadů z pálenic
Tichý, Petr
This thesis is focused on waste made by smal distilleries and utilization of this waste for energetic use. In thesis, there are described physical and chemical properties of each type of waste snd the possibility of their recycling or utilization as a raw material. Current situation and situation in ways of waste destruction were analyzed in a case study made with several chosen small distilleries. Based on this study, distillery waste is used as a raw material for energetic in a limited way. Next chapter describes technology used for separation of waste fraction and technology for re-using its energetic potential. Separate chapter consist from case study which deals with utilization of waste in a one concrete small distillery for energetic use.
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Numerical range of an interval matrix
Ivičič, Michal ; Hladík, Milan (advisor) ; Tichý, Petr (referee)
The numerical range of a matrix is a set of complex numbers that contains all the eigen- values of the matrix. It is used for instance to estimate a matrix norm. This thesis is about the numerical range of an interval matrix. In the theoretical part, we examine its properties. We prove for example that it is NP-hard to find out whether a given point lies in the numerical range. On an example, we show that field of values of an interval matrix is not necessarily convex. The thesis contains descriptions of two algorithms for visualization of the convex hull of the numerical range. Both of them are only suitable for matrices of small sizes due to high time complexity. Therefore we also present a polyno- mial algorithm for computing the upper bound of the numerical range. In the practical part, we implement the algorithms as functions in the Matlab language. 1
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Least-squares problems with sparse-dense matrices
Riegerová, Ilona ; Tůma, Miroslav (advisor) ; Tichý, Petr (referee)
Problém nejmenších čtverc· (dále jen LS problém) je aproximační úloha řešení soustav lineárních algebraických rovnic, které jsou z nějakého d·vodu za- tíženy chybami. Existence a jednoznačnost řešení a metody řešení jsou známé pro r·zné typy matic, kterými tyto soustavy reprezentujeme. Typicky jsou ma- tice řídké a obrovských dimenzí, ale velmi často dostáváme z praxe i úlohy s maticemi o proměnlivé hustotě nenulových prvk·. Těmi se myslí řídké matice s jedním nebo více hustými řádky. Zde rozebíráme metody řešení tohoto LS pro- blému. Obvykle jsou založeny na rozdělení úlohy na hustou a řídkou část, které řeší odděleně. Tak pro řídkou část m·že přestat platit předpoklad plné sloupcové hodnosti, který je potřebný pro většinu metod. Proto se zde speciálně zabýváme postupy, které tento problém řeší. 1
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Global krylov methods for solving linear algebraic problems with matrix observations
Rapavý, Martin ; Hnětynková, Iveta (advisor) ; Tichý, Petr (referee)
In this thesis we study methods for solving systems of linear algebraic equati- ons with multiple right hand sides. Specifically we focus on block Krylov subspace methods and global Krylov subspace methods, which can be derived by various approaches to generalization of methods GMRES and LSQR for solving systems of linear equations with single right hand side. We describe the difference in construction of orthonormal basis in block methods and F-orthonormal basis in global methods, in detail. Finally, we provide numerical experiments for the deri- ved algorithms in MATLAB enviroment. On carefully selected test problems we compare convergence properties of the methods. 1
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The Lanczos method in finite precision arithmetic
Šimonová, Dorota ; Tichý, Petr (advisor) ; Hnětynková, Iveta (referee)
In this thesis we consider the Lanczos algoritm and its behaviour in finite precision. Having summarized theoretical properties of the algorithm and its connection to orthogonal polynomials, we recall the idea of the Lanczos method for approximating the matrix eigenvalues. As the behaviour of the algorithm is strongly influenced by finite precision arithmetic, the linear independence of the Lanczos vectors is usually lost after a few iterations. We use the most im- portant results from analysis of the finite precision Lanczos algorithm according to Paige, Greenbaum, Strakos and others. Based on that, we study formulation and properties of the mathematical model of finite presicion Lanczos computati- ons suggested by Greenbaum. We carry out numerical experiments in Matlab, which support the theoretical results.
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Line search in descent methods
Moravová, Adéla ; Tichý, Petr (advisor) ; Vlasák, Miloslav (referee)
In this thesis, we deal with descent methods for functional minimalization. We discuss three conditions for the choice of the step length (Armijo, Goldstein, and Wolfe condition) and four descent methods (The steepest descent method, Newton's method, Quasi-Newton's method BFGS and the conjugate gradient method). We discuss their convergence properties and their advantages and dis- advantages. Finally, we test these methods numerically in the GNU Octave pro- gramming system on three different functions with different number of variables. 1
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Incomplete Cholesky factorization
Hoang, Phuong Thao ; Tůma, Miroslav (advisor) ; Tichý, Petr (referee)
The thesis is about the incomplete Cholesky factorization and its va- riants, which are important for preconditioning a system with symmetric and positive definite matrix. Our main focus is on solving these systems, which arise in many technical applications and natural sciences, using preconditioned Con- jugate Gradients. Besides many other ways we can apply Cholesky factorization approximately, incompletely. In this thesis we study existence of the incomplete Cholesky factorization and we evaluate behaviour and potential of different vari- ants of the generic algorithm. 1
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The choice of the stopping criteria for Newton-like methods
Kurnas, Jakub ; Dolejší, Vít (advisor) ; Tichý, Petr (referee)
We formulate examples of partial differential equations which can be solved through their discretization and subsequent solution of derived algebraic system. A brief summary of Discontinuous Galerkin Discretization is given as well as definitions of algebraic and discretization errors. We derive the Newton method, which solves nonlinear algebraic systems by solving a sequence of linear problems, we modify the method and examine implementation options. We define stopping criteria for the Newton-like method using aforementioned errors and we explain how to keep accuracy of the solution of derived algebraic system and the original partial differential equation in balance. We present numerical experiments to illustrate theoretical background and mention several basic properties of the Newton- like method.
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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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