In this thesis, bifurcation phenomena in dynamical systems with cosymmetry and Hamiltonian structure were investigated using numerical methods. Several numerical continuation methods and test functions for detecting bifurcations were presented. The numerical results for various examples are given using a numerical bifurcation analysis toolbox.

We study the structure of the set of extreme points of the compact convex set of matrix-valued holomorphic functions with positive real part on a finitely-connected planar domain 𝐑 normalized to have value equal to the identity matrix at some prescribed point t₀ ∈ 𝐑. This leads to an integral representation for such functions more general than what would be expected from the result for the scalar-valued case. After Cayley transformation, this leads to a integral Agler decomposition for the matrix Schur class over 𝐑 (holomorphic contractive matrix-valued functions over 𝐑). Application of a general theory of abstract Schur-class generated by a collection of test functions leads to a transfer-function realization for the matrix Schur-class over 𝐑, extending results known up to now only for the scalar case. We also explain how these results provide a new perspective for the dilation theory for Hilbert space operators having 𝐑 as a spectral set. / Ph. D.

This thesis is divided into two independent parts. The first part concerns the 3D Navier-Stokes equations. The second part deals with regularity issues for a family of integro-differential equations. In the first part of this thesis, we consider weak solutions of the 3D Navier-Stokes equations with L² initial data. We prove that ([Nabla superscript alpha])u is locally integrable in space-time for any real [alpha] such that 1 < [alpha] < 3. Up to now, only the second derivative ([Nabla]²)u was known to be locally integrable by standard parabolic regularization. We also present sharp estimates of those quantities in local weak-L[superscript (4/([alpha]+1))]. These estimates depend only on the L² norm of the initial data and on the domain of integration. Moreover, they are valid even for [alpha] ≥ 3 as long as u is smooth. The proof uses a standard approximation of Navier-Stokes from Leray and blow-up techniques. The local study is based on De Giorgi techniques with a new pressure decomposition. To handle the non-locality of fractional Laplacians, Hardy space and Maximal functions are introduced. In the second part of this thesis, we consider non-local integro-differential equations under certain natural assumptions on the kernel, and obtain persistence of Hölder continuity for their solutions. In other words, we prove that a solution stays in C[superscript beta] for all time if its initial data lies in C[superscript beta]. Also, we prove a C[superscript beta]-regularization effect from [mathematical equation] initial data. It provides an alternative proof to the result of Caffarelli, Chan and Vasseur [10], which was based on De Giorgi techniques. This result has an application for a fully non-linear problem, which is used in the field of image processing. In addition, we show Hölder regularity for solutions of drift diffusion equations with supercritical fractional diffusion under the assumption [mathematical equation]on the divergent-free drift velocity. The proof is in the spirit of Kiselev and Nazarov [48] where they established Hölder continuity of the critical surface quasi-geostrophic (SQG) equation by observing the evolution of a dual class of test functions. / text

Over the last three decades, CFD simulations have become commonplace as a tool in the engineering and design of high-speed aircraft. Experiments are often complemented by computational simulations, and CFD technologies have proved very useful in both the reduction of aircraft development cycles, and in the simulation of conditions difficult to reproduce experimentally. Great advances have been made in the field since its introduction, especially in areas of meshing, computer architecture, and solution strategies. Despite this, there still exist many computational limitations in existing CFD methods; in particular, reliable higher order and hp-adaptive methods for the Navier-Stokes equations that govern viscous compressible flow. Solutions to the equations of viscous flow can display shocks and boundary layers, which are characterized by localized regions of rapid change and high gradients. The use of adaptive meshes is crucial in such settings -- good resolution for such problems under uniform meshes is computationally prohibitive and impractical for most physical regimes of interest. However, the construction of "good" meshes is a difficult task, usually requiring a-priori knowledge of the form of the solution. An alternative to such is the construction of automatically adaptive schemes; such methods begin with a coarse mesh and refine based on the minimization of error. However, this task is difficult, as the convergence of numerical methods for problems in CFD is notoriously sensitive to mesh quality. Additionally, the use of adaptivity becomes more difficult in the context of higher order and hp methods. Many of the above issues are tied to the notion of robustness, which we define loosely for CFD applications as the degradation of the quality of numerical solutions on a coarse mesh with respect to the Reynolds number, or nondimensional viscosity. For typical physical conditions of interest for the compressible Navier-Stokes equations, the Reynolds number dictates the scale of shock and boundary layer phenomena, and can be extremely high -- on the order of 10⁷ in a unit domain. For an under-resolved mesh, the Galerkin finite element method develops large oscillations which prevent convergence and pollute the solution. The issue of robustness for finite element methods was addressed early on by Brooks and Hughes in the SUPG method, which introduced the idea of residual-based stabilization to combat such oscillations. Residual-based stabilizations can alternatively be viewed as modifying the standard finite element test space, and consequently the norm in which the finite element method converges. Demkowicz and Gopalakrishnan generalized this idea in 2009 by introducing the Discontinous Petrov-Galerkin (DPG) method with optimal test functions, where test functions are determined such that they minimize the discrete linear residual in a dual space. Under the ultra-weak variational formulation, these test functions can be computed locally to yield a symmetric, positive-definite system. The main theoretical thrust of this research is to develop a DPG method that is provably robust for singular perturbation problems in CFD, but does not suffer from discretization error in the approximation of test functions. Such a method is developed for the prototypical singular perturbation problem of convection-diffusion, where it is demonstrated that the method does not suffer from error in the approximation of test functions, and that the L² error is robustly bounded by the energy error in which DPG is optimal -- in other words, as the energy error decreases, the L² error of the solution is guaranteed to decrease as well. The method is then extended to the linearized Navier-Stokes equations, and applied to the solution of the nonlinear compressible Navier-Stokes equations. The numerical work in this dissertation has focused on the development of a 2D compressible flow code under the Camellia library, developed and maintained by Nathan Roberts at ICES. In particular, we have developed a framework allowing for rapid implementation of problems and the easy application of higher order and hp-adaptive schemes based on a natural error representation function that stems from the DPG residual. Finally, the DPG method is applied to several convection diffusion problems which mimic difficult problems in compressible flow simulations, including problems exhibiting both boundary layers and singularities in stresses. A viscous Burgers' equation is solved as an extension of DPG to nonlinear problems, and the effectiveness of DPG as a numerical method for compressible flow is assessed with the application of DPG to two benchmark problems in supersonic flow. In particular, DPG is used to solve the Carter flat plate problem and the Holden compression corner problem over a range of Mach numbers and laminar Reynolds numbers using automatically adaptive schemes, beginning with very under-resolved/coarse initial meshes. / text

Tato práce se zabývá možností paralelizace Genetického Algoritmu a jeho ná-sledné evaluace pomocí testovacích účelových funkcí. První část je teoretická a shrnuje základní poznatky z oblasti Genetických Algoritmů, paralelních archi-tektur, paralelních výpočtů a optimalizace. A dále je tato část doplněna o mož-nosti paralelizace Genetického Algoritmu. V následující praktické části je rozebrán algoritmus paralelního Genetického Algoritmu, jenž je použitý při experimentu a také je diskutována struktura a účel zvoleného experimentu. Následně jsou diskutovány výsledky získané z běhu experimentu na Eridani Clusteru z pohledu zrychlení výpočtu, kvality nalezeného řešení a závislosti kvality řešení na migračním schématu.

Diploma thesis is focused on global optimization methods and their utilization for medical image registration. The main aim is creation of the genetic algorithm and test its functionality on synthetic data. Besides test functions and test figures algorithm was subjected to real medical images. For this purpose was created graphical user interface with choise of parameters according to actual requirement. After adding an iterative gradient method it became of hybrid genetic algorithm.

Master thesis is focused on optimization of cogeneration system for purpose of rating optimization methods and evaluating properties of these methods. For each method there is description together with block schemes. First part of thesis is devoted to description of methods and their comparison. Second part consists of development of hybrid algorithm, which is used to optimize cogeneration systém model. Each algorithm compared is together with hybrid algorithms included in annexes.