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Title: Application of hp-adaptive discontinuous Galerkin method to com- pressible flow simulation Author: Karol Tarčák Department: Department of Numerical Mathematics Supervisor: prof. RNDr. Vít Dolejší, Ph.D., DSc. Abstract: In the present work we study an residuum estimate of disconti- nuous Galerkin method for the solution of Navier-Stokes equations. Firstly we summarize the construction of the viscous compressible flow model via Navier-Stokes partial differential equation and discontinuous Galerkin met- hod. Then we propose an extension of an already known residuum estimate for stationary problems to non-stationary problems. We observe the beha- vior of the proposed estimate and modify an existing hp-adaptive algorithm to use our estimate. Finally we apply the modified algorithm on test cases and present adapted meshes from the numerical experiments. Keywords: discontinuous Galerkin method, adaptivity, error estimate 4

The thesis is concerned with theoretical and practical aspects of the hp- adaptive finite element method for solving elliptic and electromagnetic prob- lems described by partial differential equations in three spatial dimensions. Besides the standard element refinements, the hp-adaptivity allows indepen- dent adaptation of degrees of the polynomial approximation as well. This leads to exponentially fast convergence even for problems with singularities. The efficiency of the hp-adaptivity is enhanced even more by the ability of the algorithm to work with meshes with arbitrary-level hanging nodes. This generality, however, leads to great complexity of the implementation. There- fore, the thesis concentrates on the mathematical analysis of algorithms that have led to successful implementation of the method. In addition, the the- sis discusses the numerical integration in 3D and the implementation of the method itself. Finally, numerical results obtained by this new implemen- tation are presented. They confirm advantages of hp-adaptivity on meshes with arbitrary-level hanging nodes. 1

Title: Novel Methods for Natural Language Generation in Spoken Dialogue Systems Author: Ondřej Dušek Department: Institute of Formal and Applied Linguistics Supervisor: Ing. Mgr. Filip Jurčíček, Ph.D., Institute of Formal and Applied Linguistics Abstract: This thesis explores novel approaches to natural language generation (NLG) in spoken dialogue systems (i.e., generating system responses to be presented the user), aiming at simplifying adaptivity of NLG in three respects: domain portability, language portability, and user-adaptive outputs. Our generators improve over state-of-the-art in all of them: First, our gen- erators, which are based on statistical methods (A* search with perceptron ranking and sequence-to-sequence recurrent neural network architectures), can be trained on data without fine-grained semantic alignments, thus simplifying the process of retraining the generator for a new domain in comparison to previous approaches. Second, we enhance the neural-network-based gener- ator so that it takes preceding dialogue context into account (i.e., user's way of speaking), thus producing user-adaptive outputs. Third, we evaluate sev- eral extensions to the neural-network-based generator designed for producing output in morphologically rich languages, showing improvements in Czech generation. In...

Title: Estimation of the algebraic error and stopping criteria in numerical solution of partial differential equations Author: Jan Papež Department: Department of Numerical Mathematics Supervisor of the master thesis: Zdeněk Strakoš Abstract: After introduction of the model problem and its properties we describe the Conjugate Gradient Method (CG). We present the estimates of the energy norm of the error and a heuristic for the adaptive refinement of the estimate. The difference in the local behaviour of the discretization and the algebraic error is illustrated by numerical experiments using the given model problem. A posteriori estimates for the discretization and the total error that take into account the inexact solution of the algebraic system are then discussed. In order to get a useful perspective, we briefly recall the multigrid method. Then the Cascadic Conjugate Gradient Method of Deuflhard (CCG) is presented. Using the estimates for the error presented in the preceding parts of the thesis, the new stopping criteria for CCG are proposed. The CCG method with the new stopping criteria is then tested. Keywords: numerical PDE, discretization error, algebraic error, error es- timates, locality of the error, adaptivity

The compressible Euler equations describe the motion of compressible inviscid fluids. They are used in many areas ranging from aerospace, automotive, and nuclear engineering to chemistry, ecology, climatology, and others. Mathematically, the compressible Euler equations represent a hyperbolic system consisting of several nonlinear partial differential equations (conservation laws). These equations are solved most frequently by means of Finite Volume Methods (FVM) and low-order Finite Element Methods (FEM). However, both these approaches are lacking higher order accuracy and moreover, it is well known that conforming FEM is not the optimal tool for the discretization of first-order equations. The most promissing approach to the approximate solution of the compressible Euler equations is the discontinuous Galerkin method that combines the stability of FVM, with excellent approximation properties of higher-order FEM. The objective of this Master Thesis was to develop, implement and test new adaptive algorithms for the nonstationary compressible Euler equations based on higher-order discontinuous Galerkin (hp-DG) methods. The basis for the new methods were the discontinuous Galerkin methods and space-time adaptive hp-FEM algorithms on dynamical meshes for nonstationary second-order problems. The new algorithms...

The diploma thesis deals with an orientation mechanism of industrial robots for mounting of pivots and screws into the mounting holes. In the first phase there is a research of inventions mainly realized in the United States Patent and Trademark Office database. Subsequently the conditions of the assembly are solved. On the base of the conditions the functional requirements are described. The diploma thesis includes seven conceptual variants, which are proposed as a 3D design. Using the basical multi-criteria method the best conceptual variant is specified. That is specified due to the accepted criterias. In the next part of the diploma thesis there is a calculation, that comprises this technically and economically most promising conceptual variant. The calculation consists of a statics, a kinematics and a dynamics of the propel, a stenght control of the mechanism and kinematics analysis of the whole mechanism. Results of the calculation are used by the detailed design of orientation mechanism, that also comprises technological production theory and a correct assembly. Economical analysis, which comprises cost of produce of this newly developed automation equipment, is worked up in the conclusion of the diploma thesis. The diploma thesis also comprises drawings. The variant solves a new conception, which can be patented as an utility model.

Title: Algebraic Error in Matrix Computations in the Context of Numerical Solution of Partial Differential Equations Author: Jan Papež Department: Department of Numerical Mathematics Supervisor: prof. Ing. Zdeněk Strakoš, DrSc., Department of Numerical Mathe- matics Abstract: Solution of algebraic problems is an inseparable and usually the most time-consuming part of numerical solution of PDEs. Algebraic computations are, in general, not exact, and in many cases it is even principally desirable not to perform them to a high accuracy. This has consequences that have to be taken into account in numerical analysis. This thesis investigates in this line some closely related issues. It focuses, in particular, on spatial distribution of the errors of different origin across the solution domain, backward error interpretation of the algebraic error in the context of function approximations, incorporation of algebraic errors to a posteriori error analysis, influence of algebraic errors to adaptivity, and construction of stopping criteria for (preconditioned) iterative algebraic solvers. Progress in these issues requires, in our opinion, understanding the interconnections between the phases of the overall solution process, such as discretization and algebraic computations. Keywords: Numerical solution of partial...

This thesis is concerned with several issues of a posteriori error estimates for linear problems. In its first part error estimates for the heat conduction equation discretized by the backward Euler method in time and discontinuous Galerkin method in space are derived. In the second part guaranteed and locally efficient error estimates involving algebraic error for Poisson equation discretized by the discontinuous Galerkin method are derived. The technique is based on the flux reconstruction where meshes with hanging nodes and variable polynomial degree are allowed. An adaptive strategy combining both adaptive mesh refinement and stopping criteria for iterative algebraic solvers is proposed. In the last part a numerical method for computing guaranteed lower and upper bounds of principal eigenvalues of symmetric linear elliptic differential operators is presented. 1