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Krylov Subspace Methods - Analysis and Application
Gergelits, Tomáš ; Strakoš, Zdeněk (advisor) ; Farrell, Patrick (referee) ; Herzog, Roland (referee)
Title: Krylov Subspace Methods - Analysis and Application Author: Tomáš Gergelits Department: Department of Numerical Mathematics Supervisor: prof. Ing. Zdeněk Strakoš, DrSc., Department of Numerical Mathematics Abstract: Convergence behavior of Krylov subspace methods is often studied for linear algebraic systems with symmetric positive definite matrices in terms of the condition number of the system matrix. As recalled in the first part of this thesis, their actual convergence behavior (that can be in practice also substantially affected by rounding errors) is however determined by the whole spectrum of the system matrix, and by the projections of the initial residual to the associated invariant subspaces. The core part of this thesis investigates the spectra of infinite dimensional operators −∇ · (k(x)∇) and −∇ · (K(x)∇), where k(x) is a scalar coefficient function and K(x) is a symmetric tensor function, preconditioned by the Laplace operator. Subsequently, the focus is on the eigenvalues of the matrices that arise from the discretization using conforming finite elements. Assuming continuity of K(x), it is proved that the spectrum of the preconditi- oned infinite dimensional operator is equal to the convex hull of the ranges of the diagonal function entries of Λ(x) from the spectral decomposition K(x) =...
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Mathematical Model of Membrane Distillation
Hvožďa, Jiří ; Komínek, Jan (referee) ; Kůdelová, Tereza (advisor)
Diplomová práce se zabývá membránovou destilací, především z matematické perspektivy. Jedná se o tepelně poháněný separační proces, ve kterém se pro rozdělení kapalné a plynné fáze používá porézní membrána. Kapalina se vypařuje a její plynná fáze prochází přes póry v membráně. Během tohoto procesu dochází k tepelné i látkové výměně, které jsou popsány systémem parciálních diferenciálnich rovnic. Další model je založen na analogii s elektrickými obvody, zákonu zachování energie, hmotnostní bilanci a empirických vztazích. Je ověřen s experimentálně naměřenými daty z nové alternativní destilační jednotky používající membránu a kondenzátor z polymerních dutých vláken. Výkon a účinnost jednotky jsou vyhodnoceny. Další možná vylepšení jsou navržena.
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Bifurcation in mathematical models in biology
Kozák, Michal ; Stará, Jana (referee)
Stationary, spatially inhomogenous solutions of reaction-diffusion systems are studied in this thesis. These systems appears in biological models based on a Tu- ring's idea of a diffusion driven instability. In the connection, a global behaviour of bifurcation branches of these stationary solutions is analyzed. The thesis in- sists on theory of differential equations and on (particularly topological) methods of nonlinear analysis. The existence, as well as non-compatness in one-dimensional space, of a bifurcation branch of general reaction-diffusion system leading to Tu- ring's efekt is proved. Further, a priori estimates of Thomas model are derived. The results tend to theorem, that forall diffusion coefficient from the preestab- lished set there exists at least one stacionary, spacially nontrivial solution of Tho- mas model.
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Adaptive methods for singularly perturbed partial differential equations
Lamač, Jan ; Knobloch, Petr (advisor)
This thesis deals with solving singularly perturbed convection- diffusion equations. Firstly, we construct a matched asymptotic expansion of the solution of the singularly perturbed convection-diffusion equation in 1D and derive a formula for the zeroth-order asymptotic expansion in several two- dimensional polygonal domains. Further, we present a set of stabilization meth- ods for solving singularly perturbed problems and prove the uniform convergence of the Il'in-Allen-Southwell scheme in 1D. Finally, we introduce a modification of the streamline upwind Petrov/Galerkin (SUPG) method on convection-oriented meshes. This new method enjoys several profitable properties such as the ful- filment of the discrete maximum principle. Besides the analysis of the method and derivation of a priori error estimates in respective energy norms we also carry out several numerical experiments verifying the theoretical results.
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Krylov Subspace Methods - Analysis and Application
Gergelits, Tomáš ; Strakoš, Zdeněk (advisor) ; Farrell, Patrick (referee) ; Herzog, Roland (referee)
Title: Krylov Subspace Methods - Analysis and Application Author: Tomáš Gergelits Department: Department of Numerical Mathematics Supervisor: prof. Ing. Zdeněk Strakoš, DrSc., Department of Numerical Mathematics Abstract: Convergence behavior of Krylov subspace methods is often studied for linear algebraic systems with symmetric positive definite matrices in terms of the condition number of the system matrix. As recalled in the first part of this thesis, their actual convergence behavior (that can be in practice also substantially affected by rounding errors) is however determined by the whole spectrum of the system matrix, and by the projections of the initial residual to the associated invariant subspaces. The core part of this thesis investigates the spectra of infinite dimensional operators −∇ · (k(x)∇) and −∇ · (K(x)∇), where k(x) is a scalar coefficient function and K(x) is a symmetric tensor function, preconditioned by the Laplace operator. Subsequently, the focus is on the eigenvalues of the matrices that arise from the discretization using conforming finite elements. Assuming continuity of K(x), it is proved that the spectrum of the preconditi- oned infinite dimensional operator is equal to the convex hull of the ranges of the diagonal function entries of Λ(x) from the spectral decomposition K(x) =...
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Numerical solution of the Ernst equation
Pospíšil, Marek ; Ledvinka, Tomáš (advisor) ; Svítek, Otakar (referee)
This work is concerned with solving the Ernst equation using numerical techniques, namely pseudospectral methods. In theoretical chapters, we summarize the properties of some black-hole space-times. The work then cites the derivation of the Ernst equation and the Kerr solution. Afterwards we present pseudospectral techniques on the example of a numerical solution of the Laplace equation with a boundary condition at infinity. Finally we solve a non-linear differential equation, thus proving, that pseudospectral methods might be used even on the Ernst equation. 1
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Beahvior of the solutions to the wave equation in compactified hyperboloidal slicing
Ivánek, Richard ; Ledvinka, Tomáš (advisor) ; Kofroň, David (referee)
In this bachelor thesis we discuss the effects of compactification and hyperboloidal slicing of spacetime in the numerical solution of wave equation primarily for their appli- cation in numerical relativity. The aim was to find the pros and cons of these concepts, to illustrate expected problems using diagrams and to rate the results obtained in spe- cific model problems. A brief explanation and demonstration of relevant numerical me- thods, hyperbolic Cauchy hypersurfaces, compactification and causal diagrams is a part of the thesis. As a conclusion, the effect of compactification and slicing on the accuracy of differential and integrational schemes was compared as well as the effect of discrete representation on the quality of initial data. 1
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Mathematical modelling and computational methods in applied sciences and engineering - Modelling 2019
Blaheta, Radim ; Starý, Jiří ; Sysala, Stanislav
Modelling 2019 is an international conference on Mathematical Modelling and Computational Methods in Applied Sciences and Engineering held in\nOlomouc, Czech Republic, in September 16 - 20, 2019. It aims to be a forum for an exchange of ideas, insights and experiences in different areas\nof mathematical modelling. It includes the fundamental formulation and analysis of mathematical models, the development of numerical methods and exploitation of capabilities of the contemporary high-performance computers or applications of mathematical modelling.\nThis conference belongs to a series of conferences previously held in Roznov in 2014 and 2009, in Pilsen in 2005 and 2001, and in Prague in 1998 and 1994. During the period of 25 years, the focus of the conference has been substantially enlarged. Besides the topics aiming at the development\nof numerical methods and analysis of mathematical models described by the partial differential equations, the conference relates to the inverse\nproblems, quantification of uncertainties in the input data, machine learning and exploitation of high-performance computing systems of petaflops and pre-exaflops performance. Increased attention is devoted to challenging industrial problems and collaboration with industry.
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