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Solving bordered linear systems
Štrausová, Jitka ; Janovský, Vladimír (advisor) ; Zítko, Jan (referee)
The comparison of two algorithms for solving bordered linear systems is considered. The matrix of this system consists of four blocks (matrices A,B,C,D), the upper left one is a sparse matrix A, which is ill-conditioned and structured. The other blocks (B,C,D) are dense. We say that the matrix A is bordered with the matrices B,C,D. It is desirable to preserve the block structure of the matrix and take advantage of sparsity and structure of the matrix A. The literature suggests to use two different algorithms: The first one is the method BEM for matrices with the borders of width equal to one. The recursive alternative for matrices with wider borders is called BEMW. The second algorithm is an iterative method. Both techniques are based on different variants of the block LU-decomposition.
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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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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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Numerical solution of convection-diffusion problems by discontinuous Galerkin method
Vlasák, Miloslav ; Dolejší, Vít (advisor) ; Janovský, Vladimír (referee) ; Vejchodský, Tomáš (referee)
This work is concerned with the theoretical analysis of the discontinuous Galerkin finite element method. We use a discontinuous Galerkin formulation for a scalar convection-diffusion equation with nonlinear convective term. The resulting semidiscretized equations with symmetric (SIPG) or nonsymmetric (NIPG) diffusive term are then discretized in time by Backward Differential formulae (BDF), implicit Runge-Kutta methods and Time discontinuous Galerkin. All of these schemes are linearized by a suitable explicit extrapolations to avoid nonlinearity in the convective term. These final schemes are theoretically analyzed and error estimates are derived. We also present some superconvergence result for Time discontinuous Galerkin for nonsymmetric operator. Numerical experiments verify the theoretical results.
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Periodic solutions of ordinary differential equations
Mitro, Erik ; Janovský, Vladimír (advisor) ; Felcman, Jiří (referee)
The thesis deals with periodic solutions of ordinary differential equations and examining of their stability. We are mainly limited to scalar differential equations. The first chapter is devoted to the stability of periodic solutions that is related to the Poincaré map. The aim is to decide on the asymptotic stability/instability of the fixed point of this map. To this end we need to compute derivatives of the Poincaré map of the first order or, possibly, of the higher orders. In the second chapter we introduce the concept of bifurcation and we examine the population model. In the third chapter we briefly mention the Van der Pol oscillator i.e the system of two equations. We illustrate the theory by examples.
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Filippov dynamical systems with applications
Šimonová, Dorota ; Janovský, Vladimír (advisor) ; Ratschan, Stefan (referee)
The thesis is motivated by problems of contact mechanics with friction. At the beginning we describe a class of piecewise smooth systems with discontinuous vector field called Filippov systems. We also show how to solve them. The rest of this thesis is focused on applications, especially dry friction model and finite element model of Coulomb friction with one contact point. We propose a technique for simulation of the second mentioned model which combines sovling methods for Filippov systems and impact oscillators. Powered by TCPDF (www.tcpdf.org)
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Numerická optimalizace
Márová, Kateřina ; Janovský, Vladimír (advisor) ; Lukšan, Ladislav (referee)
This thesis addresses the topic of unconstrained optimization. It describes seven derivative-free optimization methods for objective functions of multiple variables. Three groups of methods are distinguished. The Alternating Variable method and the method of Hooke and Jeeves represent the pattern search methods. Then there are two simplex algorithms: one by Spendley, Hext and Himsworth and the amoeba algorithm of Nelder and Mead. The family of methods with adaptive sets of search directions consists of Rosenbrock's method, the method of Davies, Swann and Campey, and Powell's method. All algorithms are implemented in MATLAB and tested on three functions of two variables. Their progression is illustrated by multiple figures and their comparative analysis is given. Powered by TCPDF (www.tcpdf.org)
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On a model of corruption in a democratic society
Splítek, Martin ; Janovský, Vladimír (advisor) ; Mlčoch, Lubomír (referee)
The aim of this work is to study the behavior of serious social pheno- menon - corruption, and we do this through a mathematical model of corruption in a democratic society, published in [1]. The model is a dynamical system of three differential equations, specified by three variables and ten parameters. The model is studied by means of numerical analysis, namely, the method of nume- rical integration of ordinary differential equations and the method of numerical continuation. We used toolbox Matcont [2], which works in the environment of program MATLAB [3]. The result is commented parametric study of the pheno- menon of corruption. Keywords: ordinary diferential equations, dynamic systems, bifurcation ana- lysis 1
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Steady states of dynamical systems
Šerý, David ; Janovský, Vladimír (advisor) ; Vlasák, Miloslav (referee)
In the thesis we analyse qualitative properties of dynamical systems near equilibria. We mainly deal with planar equations. The key notion is the stability of steady state. The stability analysis is closely connected to linearisation, which in many cases doesn't suffice. In that case Lyapunov function may help. We define stable and unstable manifold, basin of attraction, topological equivalence of equations and demonstrate their significance in qualitative analysis. The theory will be illustrated on examples. In the third chapter we briefly mention numerical continuation of steady states with respect to a parameter. 1
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