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• 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.
1

#### The Koiter shell equation in a coordinate free description

Meyer, Arnd 19 October 2012 (has links)
We give an alternate description of Koiter's shell equation that does not depend on the special mid surface coordinates, but uses differential operators defined on the mid surface.:1. Introduction 2. Basic differential geometry 3. The strain tensor and its simplifications 4. Linearization to small strain and coordinate free description 5. The resulting Koiter energy
2

#### Inverse Autoconvolution Problems with an Application in Laser Physics

Bürger, Steven 21 September 2016 (has links)
Convolution and, as a special case, autoconvolution of functions are important in many branches of mathematics and have found lots of applications, such as in physics, statistics, image processing and others. While it is a relatively easy task to determine the autoconvolution of a function (at least from the numerical point of view), the inverse problem, which consists in reconstructing a function from its autoconvolution is an ill-posed problem. Hence there is no possibility to solve such an inverse autoconvolution problem with a simple algebraic operation. Instead the problem has to be regularized, which means that it is replaced by a well-posed problem, which is close to the original problem in a certain sense. The outline of this thesis is as follows: In the first chapter we give an introduction to the type of inverse problems we consider, including some basic definitions and some important examples of regularization methods for these problems. At the end of the introduction we shortly present some general results about the convergence theory of Tikhonov-regularization. The second chapter is concerned with the autoconvolution of square integrable functions defined on the interval [0, 1]. This will lead us to the classical autoconvolution problems, where the term “classical” means that no kernel function is involved in the autoconvolution operator. For the data situation we distinguish two cases, namely data on [0, 1] and data on [0, 2]. We present some well-known properties of the classical autoconvolution operators. Moreover, we investigate nonlinearity conditions, which are required to show applicability of certain regularization approaches or which lead convergence rates for the Tikhonov regularization. For the inverse autoconvolution problem with data on the interval [0, 1] we show that a convergence rate cannot be shown using the standard convergence rate theory. If the data are given on the interval [0, 2], we can show a convergence rate for Tikhonov regularization if the exact solution satisfies a sparsity assumption. After these theoretical investigations we present various approaches to solve inverse autoconvolution problems. Here we focus on a discretized Lavrentiev regularization approach, for which even a convergence rate can be shown. Finally, we present numerical examples for the regularization methods we presented. In the third chapter we describe a physical measurement technique, the so-called SD-Spider, which leads to an inverse problem of autoconvolution type. The SD-Spider method is an approach to measure ultrashort laser pulses (laser pulses with time duration in the range of femtoseconds). Therefor we first present some very basic concepts of nonlinear optics and after that we describe the method in detail. Then we show how this approach, starting from the wave equation, leads to a kernel-based equation of autoconvolution type. The aim of chapter four is to investigate the equation and the corresponding problem, which we derived in chapter three. As a generalization of the classical autoconvolution we define the kernel-based autoconvolution operator and show that many properties of the classical autoconvolution operator can also be shown in this new situation. Moreover, we will consider inverse problems with kernel-based autoconvolution operator, which reflect the data situation of the physical problem. It turns out that these inverse problems may be locally well-posed, if all possible data are taken into account and they are locally ill-posed if one special part of the data is not available. Finally, we introduce reconstruction approaches for solving these inverse problems numerically and test them on real and artificial data.
3

#### The Koiter shell equation in a coordinate free description - extended

Meyer, Arnd January 2013 (has links)
We give an alternate description of Koiter's shell equation that does not depend on the special mid surface coordinates, but uses differential operators defined on the mid surface. This is the continuation of the preprint CSC12-02 (http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-96903) due to new additional simplifications of these operators.:1. Introduction 2. Basic differential geometry 3. The strain tensor and its simplifications 4. Linearization to small strain and coordinate free description 5. The resulting Koiter energy 6. Remarks on the differential operators
4

#### Inverse Autoconvolution Problems with an Application in Laser Physics

Bürger, Steven 21 October 2016 (has links) (PDF)
Convolution and, as a special case, autoconvolution of functions are important in many branches of mathematics and have found lots of applications, such as in physics, statistics, image processing and others. While it is a relatively easy task to determine the autoconvolution of a function (at least from the numerical point of view), the inverse problem, which consists in reconstructing a function from its autoconvolution is an ill-posed problem. Hence there is no possibility to solve such an inverse autoconvolution problem with a simple algebraic operation. Instead the problem has to be regularized, which means that it is replaced by a well-posed problem, which is close to the original problem in a certain sense. The outline of this thesis is as follows: In the first chapter we give an introduction to the type of inverse problems we consider, including some basic definitions and some important examples of regularization methods for these problems. At the end of the introduction we shortly present some general results about the convergence theory of Tikhonov-regularization. The second chapter is concerned with the autoconvolution of square integrable functions defined on the interval [0, 1]. This will lead us to the classical autoconvolution problems, where the term “classical” means that no kernel function is involved in the autoconvolution operator. For the data situation we distinguish two cases, namely data on [0, 1] and data on [0, 2]. We present some well-known properties of the classical autoconvolution operators. Moreover, we investigate nonlinearity conditions, which are required to show applicability of certain regularization approaches or which lead convergence rates for the Tikhonov regularization. For the inverse autoconvolution problem with data on the interval [0, 1] we show that a convergence rate cannot be shown using the standard convergence rate theory. If the data are given on the interval [0, 2], we can show a convergence rate for Tikhonov regularization if the exact solution satisfies a sparsity assumption. After these theoretical investigations we present various approaches to solve inverse autoconvolution problems. Here we focus on a discretized Lavrentiev regularization approach, for which even a convergence rate can be shown. Finally, we present numerical examples for the regularization methods we presented. In the third chapter we describe a physical measurement technique, the so-called SD-Spider, which leads to an inverse problem of autoconvolution type. The SD-Spider method is an approach to measure ultrashort laser pulses (laser pulses with time duration in the range of femtoseconds). Therefor we first present some very basic concepts of nonlinear optics and after that we describe the method in detail. Then we show how this approach, starting from the wave equation, leads to a kernel-based equation of autoconvolution type. The aim of chapter four is to investigate the equation and the corresponding problem, which we derived in chapter three. As a generalization of the classical autoconvolution we define the kernel-based autoconvolution operator and show that many properties of the classical autoconvolution operator can also be shown in this new situation. Moreover, we will consider inverse problems with kernel-based autoconvolution operator, which reflect the data situation of the physical problem. It turns out that these inverse problems may be locally well-posed, if all possible data are taken into account and they are locally ill-posed if one special part of the data is not available. Finally, we introduce reconstruction approaches for solving these inverse problems numerically and test them on real and artificial data.
5

#### Arbitrary Lagrangian-Eulerian Discontinous Galerkin methods for nonlinear time-dependent first order partial differential equations / Arbitrary Lagrangian-Eulerian Discontinous Galerkin-Methode für nichtlineare zeitabhängige partielle Differentialgleichungen erster Ordnung

Schnücke, Gero January 2016 (has links) (PDF)
The present thesis considers the development and analysis of arbitrary Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) methods with time-dependent approximation spaces for conservation laws and the Hamilton-Jacobi equations. Fundamentals about conservation laws, Hamilton-Jacobi equations and discontinuous Galerkin methods are presented. In particular, issues in the development of discontinuous Galerkin (DG) methods for the Hamilton-Jacobi equations are discussed. The development of the ALE-DG methods based on the assumption that the distribution of the grid points is explicitly given for an upcoming time level. This assumption allows to construct a time-dependent local affine linear mapping to a reference cell and a time-dependent finite element test function space. In addition, a version of Reynolds’ transport theorem can be proven. For the fully-discrete ALE-DG method for nonlinear scalar conservation laws the geometric conservation law and a local maximum principle are proven. Furthermore, conditions for slope limiters are stated. These conditions ensure the total variation stability of the method. In addition, entropy stability is discussed. For the corresponding semi-discrete ALE-DG method, error estimates are proven. If a piecewise $\mathcal{P}^{k}$ polynomial approximation space is used on the reference cell, the sub-optimal $\left(k+\frac{1}{2}\right)$ convergence for monotone fuxes and the optimal $(k+1)$ convergence for an upwind flux are proven in the $\mathrm{L}^{2}$-norm. The capability of the method is shown by numerical examples for nonlinear conservation laws. Likewise, for the semi-discrete ALE-DG method for nonlinear Hamilton-Jacobi equations, error estimates are proven. In the one dimensional case the optimal $\left(k+1\right)$ convergence and in the two dimensional case the sub-optimal $\left(k+\frac{1}{2}\right)$ convergence are proven in the $\mathrm{L}^{2}$-norm, if a piecewise $\mathcal{P}^{k}$ polynomial approximation space is used on the reference cell. For the fullydiscrete method, the geometric conservation is proven and for the piecewise constant forward Euler step the convergence of the method to the unique physical relevant solution is discussed. / Die vorliegende Arbeit beschäftigt sich mit der Entwicklung und Analyse von arbitrar Lagrangian-Eulerian discontinuous Galerkin (ALE-DG) Methoden mit zeitabhängigen Testfunktionen Räumen für Erhaltungs- und Hamilton-Jacobi Gleichungen. Grundlagen über Erhaltungsgleichungen, Hamilton-Jacobi Gleichungen und discontinuous Galerkin Methoden werden präsentiert. Insbesondere werden Probleme bei der Entwicklung von discontinuous Galerkin Methoden für die Hamilton-Jacobi Gleichungen untersucht. Die Entwicklung der ALE-DG Methode basiert auf der Annahme, dass die Verteilung der Gitterpunkte zu einem kommenden Zeitpunkt explizit gegeben ist. Diese Annahme ermöglicht die Konstruktion einer zeitabhängigen lokal affin-linearen Abbildung auf eine Referenzzelle und eines zeitabhängigen Testfunktionen Raums. Zusätzlich kann eine Version des Reynolds’schen Transportsatzes gezeigt werden. Für die vollständig diskretisierte ALE-DG Methode für nichtlineare Erhaltungsgleichungen werden der geometrischen Erhaltungssatz und ein lokales Maximumprinzip bewiesen. Des Weiteren werden Bedingungen für Limiter angegeben. Diese Bedingungen sichern die Stabilität der Methode im Sinne der totalen Variation. Zusätzlich wird die Entropie-Stabilität der Methode diskutiert. Für die zugehörige semi-diskretisierte ALE-DG Methode werden Fehlerabschätzungen gezeigt. Wenn auf der Referenzzelle ein Testfunktionen Raum, der stückweise Polynome vom Grad $k$ enthält verwendet wird, kann für einen monotonen Fluss die suboptimale Konvergenzordnung $\left(k+\frac{1}{2}\right)$ und für einen upwind Fluss die optimale Konvergenzordnung $\left(k+1\right)$ in der $\mathrm{L}^{2}$-Norm gezeigt werden. Die Leistungsfähigkeit der Methode wird anhand numerischer Beispiele für nichtlineare Erhaltungsgleichungen untersucht. Ebenso werden für die semi-diskretisierte ALE-DG Methode für nichtlineare Hamilton-Jacobi Gleichungen Fehlerabschätzungen gezeigt. Wenn auf der Referenzzelle ein Testfunktionen Raum, der stückweise Polynome vom Grad k enthält verwendet wird, kann im eindimensionalen Fall die optimale Konvergenzordnung $\left(k+1\right)$ und im zweidimensionalen Fall die suboptimale Konvergenzordnung $\left(k+\frac{1}{2}\right)$ in der $\mathrm{L}^{2}$-Norm gezeigt werden. Für die vollständig diskretisierte ALE-DG Methode werden der geometrischen Erhaltungssatz bewiesen und für die stückweise konstante explizite Euler Diskretisierung wird die Konvergenz gegen die eindeutige physikalisch relevante Lösung diskutiert.
6

#### The Koiter shell equation in a coordinate free description - extended

Meyer, Arnd 12 September 2013 (has links) (PDF)
We give an alternate description of Koiter's shell equation that does not depend on the special mid surface coordinates, but uses differential operators defined on the mid surface. This is the continuation of the preprint CSC12-02 (http://nbn-resolving.de/urn:nbn:de:bsz:ch1-qucosa-96903) due to new additional simplifications of these operators.
7

#### The Koiter shell equation in a coordinate free description

Meyer, Arnd 19 October 2012 (has links) (PDF)
We give an alternate description of Koiter's shell equation that does not depend on the special mid surface coordinates, but uses differential operators defined on the mid surface.
8

#### Hp-Finite Elements for PDE-Constrained Optimization

Wurst, Jan-Eric January 2015 (has links) (PDF)
9

#### Numerical schemes for multi-species BGK equations based on a variational procedure applied to multi-species BGK equations with velocity-dependent collision frequency and to quantum multi-species BGK equations / Numerische Verfahren für multispezies BGK Gleichungen mittels Variationsansatz angewandt auf multispezies BGK Gleichungen mit geschwindigkeitsabhängiger Stoßfrequenz sowie auf quantenmechanische multispezies BGK Gleichungen

Warnecke, Sandra January 2022 (has links) (PDF)
We consider a multi-species gas mixture described by a kinetic model. More precisely, we are interested in models with BGK interaction operators. Several extensions to the standard BGK model are studied. Firstly, we allow the collision frequency to vary not only in time and space but also with the microscopic velocity. In the standard BGK model, the dependence on the microscopic velocity is neglected for reasons of simplicity. We allow for a more physical description by reintroducing this dependence. But even though the structure of the equations remains the same, the so-called target functions in the relaxation term become more sophisticated being defined by a variational procedure. Secondly, we include quantum effects (for constant collision frequencies). This approach influences again the resulting target functions in the relaxation term depending on the respective type of quantum particles. In this thesis, we present a numerical method for simulating such models. We use implicit-explicit time discretizations in order to take care of the stiff relaxation part due to possibly large collision frequencies. The key new ingredient is an implicit solver which minimizes a certain potential function. This procedure mimics the theoretical derivation in the models. We prove that theoretical properties of the model are preserved at the discrete level such as conservation of mass, total momentum and total energy, positivity of distribution functions and a proper entropy behavior. We provide an array of numerical tests illustrating the numerical scheme as well as its usefulness and effectiveness. / Wir betrachten ein Gasgemisch, das aus mehreren Spezies zusammengesetzt ist und durch kinetische Modelle beschrieben werden kann. Dabei interessieren wir uns vor allem für Modelle mit BGK-Wechselwirkungsoperatoren. Verschiedene Erweiterungen des Standard-BGK-Modells werden untersucht. Im ersten Modell nehmen wir eine Abhängigkeit der Stoßfrequenzen von der mikroskopischen Geschwindigkeit hinzu. Im Standard-BGK-Modell wird diese Abhängigkeit aus Gründen der Komplexität vernachlässigt. Wir nähern uns der physikalischen Realität weiter an, indem wir die Abhängigkeit von der mikroskopischen Geschwindigkeit beachten. Die Struktur der Gleichungen bleibt erhalten, allerdings hat dies Auswirkungen auf die sogenannten Zielfunktionen im Relaxationsterm, welche sodann durch einen Variationsansatz definiert werden. Das zweite Modell berücksichtigt Quanteneffekte (für konstante Stoßfrequenzen), was wiederum die Zielfunktionen im Relaxationsterm beeinflusst. Diese unterscheiden sich abhängig von den jeweils betrachteten, quantenmechanischen Teilchentypen. In dieser Doktorarbeit stellen wir numerische Verfahren vor, die auf oben beschriebene Modelle angewandt werden können. Wir legen eine implizite-explizite Zeitdiskretisierung zu Grunde, da die Relaxationsterme für große Stoßfrequenzen steif werden können. Das Kernstück ist ein impliziter Löser, der eine gewisse Potenzialfunktion minimiert. Dieses Vorgehen imitiert die theoretische Herleitung in den Modellen. Wir zeigen, dass die Eigenschaften des Modells auch auf der diskreten Ebene vorliegen. Dies beinhaltet die Massen-, Gesamtimpuls- und Gesamtenergieerhaltung, die Positivität von Verteilungsfunktionen sowie das gewünschte Verhalten der Entropie. Wir führen mehrere numerische Tests durch, die die Eigenschaften, die Nützlichkeit und die Zweckmäßigkeit des numerischen Verfahrens aufzeigen. / Many applications require reliable numerical simulations of realistic set-ups e.g. plasma physics. This book gives a short introduction into kinetic models of gas mixtures describing the time evolution of rarefied gases and plasmas. Recently developed models are presented which extend existing literature by including more physical phenomena. We develop a numerical scheme for these more elaborated equations. The scheme is proven to maintain the physical properties of the models at the discrete level. We show several numerical test cases inspired by physical experiments.
10

#### Contributions to regularization theory and practice of certain nonlinear inverse problems

Hofmann, Christopher 23 December 2020 (has links)
The present thesis addresses both theoretical as well as numerical aspects of the treatment of nonlinear inverse problems. The first part considers Tikhonov regularization for nonlinear ill-posed operator equations in Hilbert scales with oversmoothing penalties. Sufficient as well as necessary conditions to establish convergence are introduced and convergence rate results are given for various parameter choice rules under a two sided nonlinearity constraint. Ultimately, both a posteriori as well as certain a priori parameter choice rules lead to identical converce rates. The theoretical results are supported and augmented by extensive numerical case studies. In particular it is shown, that the localization of the above mentioned nonlinearity constraint is not trivial. Incorrect localization will prevent convergence of the regularized to the exact solution. The second part of the thesis considers two open problems in inverse option pricing and electrical impedance tomography. While regularization through discretization is sufficient to overcome ill-posedness of the latter, the first requires a more sophisticated approach. It is shown, that the recovery of time dependent volatility and interest rate functions from observed option prices is everywhere locally ill-posed. This motivates Tikhonov-type or variational regularization with two parameters and penalty terms to simultaneously recover these functions. Two parameter choice rules using the L-hypersurface as well as a combination of L-curve and quasi-optimality are introduced. The results are again supported by extensive numerical case studies.

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