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  • About
  • The Global ETD Search service is a free service for researchers to find electronic theses and dissertations. This service is provided by the Networked Digital Library of Theses and Dissertations.
    Our metadata is collected from universities around the world. If you manage a university/consortium/country archive and want to be added, details can be found on the NDLTD website.
31

A note on the second derivatives of rHCT basis functions

Weise, Michael January 2014 (has links)
We consider reduced Hsieh-Clough-Tocher basis functions with respect to a splitting into subtriangles at the barycenter of the original triangular element. This article gives a proof that the second derivatives of those functions, which in general may jump at the subtriangle boundaries, do not jump at the barycenter.:1 Introduction 2 Shape functions 3 Transformation of second derivatives 4 Second derivatives at the barycenter
32

Simplified calculation of rHCT basis functions for an arbitrary splitting

Weise, Michael January 2015 (has links)
Reduced Hsieh-Clough-Tocher elements are triangular C1-elements with only nine degrees of freedom. Simple formulas for the basis functions of reduced Hsieh-Clough-Tocher elements based on the edge vectors of the triangle have been given recently for a barycentric splitting. We generalise these formulas to the case of an arbitrary splitting point.:1 Introduction 2 Basic definitions 3 Mapping to the reference triangle 4 Construction of the rHCT shape functions
33

A note on the second derivatives of rHCT basis functions - extended

Weise, Michael January 2015 (has links)
We consider reduced Hsieh-Clough-Tocher basis functions with respect to a splitting into subtriangles at an arbitrary interior point of the original triangular element. This article gives a proof that the second derivatives of those functions, which in general may jump at the subtriangle boundaries, do not jump at the splitting point.:1 Introduction 2 Shape functions 3 Transformation of second derivatives 4 Second derivatives at the splitting point
34

Fast, exact and stable reconstruction of multivariate algebraic polynomials in Chebyshev form

Potts, Daniel, Volkmer, Toni 16 February 2015 (has links)
We describe a fast method for the evaluation of an arbitrary high-dimensional multivariate algebraic polynomial in Chebyshev form at the nodes of an arbitrary rank-1 Chebyshev lattice. Our main focus is on conditions on rank-1 Chebyshev lattices allowing for the exact reconstruction of such polynomials from samples along such lattices and we present an algorithm for constructing suitable rank-1 Chebyshev lattices based on a component-by-component approach. Moreover, we give a method for the fast, exact and stable reconstruction.
35

Balanced truncation model reduction for linear time-varying systems

Lang, Norman, Saak, Jens, Stykel, Tatjana January 2015 (has links)
A practical procedure based on implicit time integration methods applied to the differential Lyapunov equations arising in the square root balanced truncation method is presented. The application of high order time integrators results in indefinite right-hand sides of the algebraic Lyapunov equations that have to be solved within every time step. Therefore, classical methods exploiting the inherent low-rank structure often observed for practical applications end up in complex data and arithmetic. Avoiding the additional effort treating complex quantities, a symmetric indefinite factorization of both the right-hand side and the solution of the differential Lyapunov equations is applied.:1 Introduction 2 Balanced truncation for LTV systems 3 Solving differential Lyapunov equations 4 Solving the reduced-order system 5 Numerical experiments 6 Conclusion
36

Least Squares in Sampling Complexity and Statistical Learning

Bartel, Felix 19 January 2024 (has links)
Data gathering is a constant in human history with ever increasing amounts in quantity and dimensionality. To get a feel for the data, make it interpretable, or find underlying laws it is necessary to fit a function to the finite and possibly noisy data. In this thesis we focus on a method achieving this, namely least squares approximation. Its discovery dates back to around 1800 and it has since then proven to be an indispensable tool which is efficient and has the capability to achieve optimal error when used right. Crucial for the least squares method are the ansatz functions and the sampling points. To discuss them, we gather tools from probability theory, frame subsampling, and $L_2$-Marcinkiewicz-Zygmund inequalities. With that we give results in the worst-case or minmax setting, when a set of points is sought for approximating a class of functions, which we model as a generic reproducing kernel Hilbert space. Further, we give error bounds in the statistical learning setting for approximating individual functions from possibly noisy samples. Here, we include the covariate-shift setting as a subfield of transfer learning. In a natural way a parameter choice question arises for balancing over- and underfitting effect. We tackle this by using the cross-validation score, for which we show a fast way of computing as well as prove the goodness thereof.:1 Introduction 2 Least squares approximation 3 Reproducing kernel Hilbert spaces (RKHS) 4 Concentration inequalities 5 Subsampling of finite frames 6 L2 -Marcinkiewicz-Zygmund (MZ) inequalities 7 Least squares in the worst-case setting 8 Least squares in statistical learning 9 Cross-validation 10 Outlook
37

Chemnitz Symposium on Inverse Problems 2014

02 October 2014 (has links) (PDF)
Our symposium will bring together experts from the German and international 'Inverse Problems Community' and young scientists. The focus will be on ill-posedness phenomena, regularization theory and practice, and on the analytical, numerical, and stochastic treatment of applied inverse problems in natural sciences, engineering, and finance.
38

On Runge-Kutta discontinuous Galerkin methods for compressible Euler equations and the ideal magneto-hydrodynamical model / Runge-Kutta Discontinuous-Galerkin Verfahren für die kompressiblen Euler Gleichungen und das ideale magnetohydrodynamische Modell

Gallego Valencia, Juan Pablo January 2017 (has links) (PDF)
An explicit Runge-Kutta discontinuous Galerkin (RKDG) method is used to device numerical schemes for both the compressible Euler equations of gas dynamics and the ideal magneto- hydrodynamical (MHD) model. These systems of conservation laws are known to have discontinuous solutions. Discontinuities are the source of spurious oscillations in the solution profile of the numerical approximation, when a high order accurate numerical method is used. Different techniques are reviewed in order to control spurious oscillations. A shock detection technique is shown to be useful in order to determine the regions where the spurious oscillations appear such that a Limiter can be used to eliminate these numeric artifacts. To guarantee the positivity of specific variables like the density and the pressure, a positivity preserving limiter is used. Furthermore, a numerical flux, proven to preserve the entropy stability of the semi-discrete DG scheme for the MHD system is used. Finally, the numerical schemes are implemented using the deal.II C++ libraries in the dflo code. The solution of common test cases show the capability of the method. / Ein explizite Runge-Kutta discontinous Galerkin (RKDG) Verfahren wird angewendet, um numerische Diskretisierungen, sowohl für die kompressiblen Eulergleichungen der Gasdynamik, als auch für die idealen Magnetohydrodynamik (MHD) Gleichungen zu entwickeln. Es ist bekannt, dass diese System von Erhaltungsgleichungen unstetige Lösungen besitzen. Unstetigkeiten sind die Quelle von störenden Oszillationen im Lösungsprofil der numerischen Näherung, wenn ein numerisches Verfahren von hoher Ordnung verwendet wird. Verschiedene Techniken werden miteinander verglichen um störende Oszillationen zu kontrollieren, die bei der Approximation von Unstetigkeiten in der Lösung auftreten. Ein Verfahren zur Lokalisierung von Schockwellen wird vorgestellt und es wird gezeigt, dass dieses Verfahren nützlich ist um Regionen, in denen störende Oszillationen auftreten, zu bestimmen, so dass ein Limiter verwendet werden kann um diese numerischen Artefakte zu eliminieren. Um die Positivität spezieller Variablen, wie die Dichte und den Druck, zu bewahren, wird ein spezieller „positivitätserhaltender“ Limiter verwendet. Des Weiteren wird ein numerischer Fluss, für den bewiesenermaßen das semi-diskrete DG Verfahren für das MHD System Entropie-Stabil ist, verwendet. Abschließend werden die numerischen Verfahren unter Verwendung der deal.II C++ Bibliotheken im dflo code implementiert. Simulationen bekannter Testbeispiele zeigen das Potential dieses numerischen Verfahrens.
39

Fast Evaluation of Near-Field Boundary Integrals using Tensor Approximations / Schnelle Auswertung von Nahfeld-Randintegralen durch Tensorapproximationen

Ballani, Jonas 18 October 2012 (has links) (PDF)
In this dissertation, we introduce and analyse a scheme for the fast evaluation of integrals stemming from boundary element methods including discretisations of the classical single and double layer potential operators. Our method is based on the parametrisation of boundary elements in terms of a d-dimensional parameter tuple. We interpret the integral as a real-valued function f depending on d parameters and show that f is smooth in a d-dimensional box. A standard interpolation of f by polynomials leads to a d-dimensional tensor which is given by the values of f at the interpolation points. This tensor may be approximated in a low rank tensor format like the canonical format or the hierarchical format. The tensor approximation has to be done only once and allows us to evaluate interpolants in O(dr(m+1)) operations in the canonical format, or O(dk³ + dk(m + 1)) operations in the hierarchical format, where m denotes the interpolation order and the ranks r, k are small integers. In particular, we apply an efficient black box scheme in the hierarchical tensor format in order to adaptively approximate tensors even in high dimensions d with a prescribed (but heuristic) target accuracy. By means of detailed numerical experiments, we demonstrate that highly accurate integral values can be obtained at very moderate costs.
40

Numerische Simulation nahezu inkompressibler Materialien unter Verwendung von adaptiver, gemischter FEM / Numerical simulation of nearly incompressible material using adaptive, mixed FEM

Balg, Martina, Meyer, Arnd 02 November 2010 (has links) (PDF)
Ziel dieser Arbeit ist die Simulation der Deformation von Bauteilen, welche aus nahezu inkompressiblem Material bestehen. Dabei soll sich das Material sowohl linear als auch nichtlinear elastisch verhalten können. Zusätzlich soll die Belastung des Bauteils beliebig gewählt werden können, das heißt, es sollen kleine als auch große Deformationen möglich sein.

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