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Shadowing property of numerical methods for partial differential equations
Brichta, Ondřej ; Kučera, Václav (advisor) ; Knobloch, Petr (referee)
This thesis is focused on the shadowing property of numerical methods for partial differential equations. The goals of this thesis are the application of shadowing theory to the case of linear maps, modification of standard techniques for the purpose of this application and the application of the adapted theory to multistep schemes. In the in- troductory overview, we first focus on a study of the standard shadowing theory, then we formulate basic statements and prove a relationship between a conctractivity and shadowing. Afterwards we find the characterization of the shadowing property for linear maps. In the next sections, we adapt definitions of the shadowing theory to requirements of multistep methods. As an example, we apply the adapted theory to the Dufort-Frankel scheme in the third chapter. At the end of this thesis, remarks on shadowing in general multistep methods and remarks on a relationship between the shadowing property and stability are presented. 1
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Interpolace hladkých funkcí pomocí kvadratických a kubických splinů
Eckstein, Jiří ; Kučera, Václav (advisor) ; Dolejší, Vít (referee)
In this thesis, we study properties of cubic and quadratic spline interpolation. First, we define the notions of spline and interpolation. We then merge them to study cubic and quadratic spline interpolations. We go through the individual spline interpolation types, show an algorithm for constructing selected types and sum up their basic properties. We then present a computer program based on the provided algorithms. We use it to construct spline interpolations of some sample functions and we calculate errors of these interpolation and compare them with theoretical estimates.
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Removal of JPEG compression artefacts in image data
Lopata, Jan ; Kučera, Václav (advisor) ; Šroubek, Filip (referee)
This thesis is concerned with the removal of artefacts typical for JPEG im- age compression. First, we describe the mathematical formulation of the JPEG format and the problem of artefact removal. We then formulate the problem as an optimization problem, where the minimized functional is obtained via Bayes' theorem and complex wavelets. We describe proximal operators and algorithms and apply them to the minimization of the given functional. The final algorithm is implemented in MATLAB and tested on several test problems. 1
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Numerical solution of equations describing the dynamics of flocking
Živčáková, Andrea ; Kučera, Václav (advisor) ; Janovský, Vladimír (referee)
This work is devoted to the numerical solution of equations describing the dynamics of flocks of birds. Specifically, we pay attention to the Euler equations 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 numerical scheme was implemented and numerical experiments showing the robustness of the scheme were carried out. Powered by TCPDF (www.tcpdf.org)
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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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Exact and approximate Riemann solvers for the Euler equations
Živčáková, Andrea ; Kučera, Václav (advisor) ; Felcman, Jiří (referee)
In this work we deal with the solution and implementation of the problem of solving a partial differential equation with a piecewise constant initial condition, the so-called Riemann's problem. Specifically, we study the equations of conservation laws describing inviscid adiabatic flow of an ideal gas - the Euler equations. After some investigation, we show that these equations can be transformed to a quasilinear hyperbolic partial differential equation of first order. We are especially interested in the one-dimensional Euler equations for which we want to get an analytically exact Riemann's solver. The solution is found by investigation of properties of waves, namely rarefaction waves, shock waves and contact discontinuities were treated. The output of this work is a program in C for finding the exact Riemann's solver for one-dimensional Euler equations. The program is based on a theoretical analysis summarized in the first two chapters, and is tested on standard test data. The theory is based on the books [1] and [2].
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Metody vyššího řádu založené na rekonstrukci
Dominik, Oldřich ; Kučera, Václav (advisor) ; Dolejší, Vít (referee)
This work is concerned with the introduction of a new higher order numerical scheme based on the discontinuous Galerkin method (DGM). We follow the methodology of higher order finite volume (HOFV) and spectral volume (SV) schemes and introduce a reconstruction operator into the DGM. This operator constructs higher order piecewise polynomial reconstructions from the lower order DGM scheme. We present two variants: the generalization of standard HOFV schemes, already proposed by Dumbser et al. (2008) and the generalization of the SV method introduced by Wang (2002). Theoretical aspects are discussed and numerical experiments with the focus on a 2D advection problem are carried out. Powered by TCPDF (www.tcpdf.org)
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Parameter optimization in COVID-19 epidemiological models
Martínek, Josef ; Kučera, Václav (advisor) ; Kopfová, Jana (referee)
This work is concerned with modelling of the spread of infectious diseases with em- phasis on the current COVID-19 pandemic. Our goal is to estimate unknown parameters in epidemiological models from real data on the spread of the disease in the Czech Repub- lic. To model the evolution of the epidemic, we consider compartmental models, which lead to a system of ordinary differential equations. We then formulate a non-linear least squares problem for the optimization of the model parameters to fit the model outcome to the observed data. We numerically optimize by the Levenberg-Marquardt method, which requires the Jacobian of the vector of residuals. This is obtained by deriving and solving the sensitivity equations corresponding to the considered model. We test the method on noisy artificial data and on a well documented English boarding school in- fluenza epidemic. Finally, we apply the method to Czech COVID-19 data and discuss the results. One of the conclusions of this work is the introduction of the concept of effective population size, to overcome the unrealistic assumption of complete homogeneity of the population. Thus the population size is not apriori given, but is an unknown parameter to be optimized. This leads to much better agreement of the models and real data. This appears to be a new concept. 1
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