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1 
The Koiter shell equation in a coordinate free descriptionMeyer, 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 PhysicsBürger, 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 illposed 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 wellposed 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 Tikhonovregularization.
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 wellknown 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 socalled SDSpider, which leads to an inverse problem of autoconvolution type. The SDSpider 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 kernelbased 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 kernelbased 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 kernelbased autoconvolution operator, which reflect the data situation of the physical problem. It turns out that these inverse problems may be locally wellposed, if all possible data are taken into account and they are locally illposed 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  extendedMeyer, 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 CSC1202 (http://nbnresolving.de/urn:nbn:de:bsz:ch1qucosa96903) 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 
Contributions to regularization theory and practice of certain nonlinear inverse problemsHofmann, 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 illposed 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 illposedness 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 illposed. This motivates Tikhonovtype or variational regularization with two parameters and penalty terms to simultaneously recover these functions. Two parameter choice rules using the Lhypersurface as well as a combination of Lcurve and quasioptimality are introduced. The results are again supported by extensive numerical case studies.

5 
Lowrank Tensor Methods for PDEconstrained OptimizationBünger, Alexandra 14 December 2021 (has links)
Optimierungsaufgaben unter Partiellen Differentialgleichungen (PDGLs) tauchen in verschiedensten Anwendungen der Wissenschaft und Technik auf. Wenn wir ein PDGL Problem formulieren, kann es aufgrund seiner Größe unmöglich werden, das Problem mit konventionellen Methoden zu lösen. Zusätzlich noch eine Optimierung auszuführen birgt zusätzliche Schwierigkeiten. In vielen Fällen können wir das PDGL Problem in einem kompakteren Format formulieren indem wir der zugrundeliegenden KroneckerProdukt Struktur zwischen Raum und Zeitdimension Aufmerksamkeit schenken. Wenn die PDGL zusätzlich mit Isogeometrischer Analysis diskretisiert wurde, können wir zusätlich eine NiedrigRang Approximation zwischen den einzelnen Raumdimensionen erzeugen. Diese NiedrigRang Approximation lässt uns die Systemmatrizen schnell und speicherschonend aufstellen. Das folgende PDGLProblem lässt sich als Summe aus KroneckerProdukten beschreiben, welche als eine NiedrigRang Tensortrain Formulierung interpretiert werden kann. Diese kann effizient im NiedrigRang Format gelöst werden. Wir illustrieren dies mit unterschiedlichen, anspruchsvollen Beispielproblemen.:Introduction
Tensor Train Format
Isogeometric Analysis
PDEconstrained Optimization
Bayesian Inverse Problems
A lowrank tensor method for PDEconstrained optimization with Isogeometric Analysis
A lowrank matrix equation method for solving PDEconstrained optimization problems
A lowrank tensor method to reconstruct sparse initial states for PDEs with Isogeometric Analysis
Theses and Summary
Bibilography / Optimization problems governed by Partial Differential Equations (PDEs) arise in various applications of science and engineering. If we formulate a discretization of a PDE problem, it may become infeasible to treat the problem with conventional methods due to its size. Solving an optimization problem on top of the forward problem poses additional difficulties. Often, we can formulate the PDE problem in a more compact format by paying attention to the underlying Kronecker product structure between the space and time dimension of the discretization. When the PDE is discretized with Isogeometric Analysis we can additionally formulate a lowrank representation with Kronecker products between its individual spatial dimensions. This lowrank formulation gives rise to a fast and memory efficient assembly for the system matrices. The PDE problem represented as a sum of Kronecker products can then be interpreted as a lowrank tensor train formulation, which can be efficiently solved in a lowrank format. We illustrate this for several challenging PDEconstrained problems.:Introduction
Tensor Train Format
Isogeometric Analysis
PDEconstrained Optimization
Bayesian Inverse Problems
A lowrank tensor method for PDEconstrained optimization with Isogeometric Analysis
A lowrank matrix equation method for solving PDEconstrained optimization problems
A lowrank tensor method to reconstruct sparse initial states for PDEs with Isogeometric Analysis
Theses and Summary
Bibilography

6 
Lowrank iterative methods of periodic projected Lyapunov equations and their application in model reduction of periodic descriptor systemsBenner, Peter, Hossain, MohammadSahadet, Stykel, Tatjana January 2011 (has links)
We discuss the numerical solution of largescale sparse projected discretetime periodic Lyapunov equations in lifted form which arise in model reduction of periodic descriptor systems. We extend the alternating direction implicit method and the Smith method to such equations. Lowrank versions of these methods are also presented, which can be used to compute lowrank approximations to the solutions of projected periodic Lyapunov equations in lifted form with lowrank righthand side. Moreover, we consider an application of the Lyapunov solvers to balanced truncation model reduction of periodic discretetime descriptor systems. Numerical results are given to illustrate the efficiency and accuracy of the proposed methods.:1 Introduction
2 Periodic descriptor systems
3 ADI method for causal lifted Lyapunov equations
4 Smith method for noncausal lifted Lyapunov equations
5 Application to model order reduction
6 Numerical results
7 Conclusions

7 
Fast Evaluation of NearField Boundary Integrals using Tensor Approximations: Fast Evaluation of NearField Boundary Integralsusing Tensor ApproximationsBallani, Jonas 10 October 2012 (has links)
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 ddimensional parameter tuple. We interpret the integral as a realvalued function f depending on d parameters and show that f is smooth in a ddimensional box. A standard interpolation of f by polynomials leads to a ddimensional 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.

8 
Programmbeschreibung SPCPM3AdHXX  Teil 1Meyer, Arnd 11 March 2014 (has links)
Beschreibung der Finite Elemente SoftwareFamilie SPCPM3AdHXX
für: (S)cientific (P)arallel (C)omputing  (P)rogramm(M)odul (3)D (ad)aptiv (H)exaederelemente.
Für XX stehen die einzelnen Spezialvarianten, die in Teil 2 detailliert geschildert werden.
Stand: Ende 2013:1 Allgemeine Vorbemerkungen
2 Grundstruktur
3 Datenstrukturen
4 Gesamtablauf
5 Parallelisierung
6 Die Grundvariante A3D_Original und ihre Bibliotheken

9 
Chemnitz Symposium on Inverse Problems 2014Hofmann, Bernd January 2014 (has links)
Our symposium will bring together experts from the German and international 'Inverse Problems Community' and young scientists. The focus will be on illposedness phenomena, regularization theory and practice, and on the analytical, numerical, and stochastic treatment of applied inverse problems in natural sciences, engineering, and finance.

10 
About an autoconvolution problem arising in ultrashort laser pulse characterizationBürger, Steven January 2014 (has links)
We are investigating a kernelbased autoconvolution problem, which has its origin in the physics of ultra short laser pulses. The task in this problem is to reconstruct a complexvalued function $x$ on a finite interval from measurements of its absolute value and a kernelbased autoconvolution of the form [[F(x)](s)=int k(s,t)x(st)x(t)de t.]
This problem has not been studied in the literature. One reason might be that one has more information than in the classical autoconvolution case, where only the right hand side is available. Nevertheless we show that ill posedness phenomena may occur. We also propose an algorithm to solve the problem numerically and demonstrate its performance with artificial data. Since the algorithm fails to produce good results with real data and we suspect that the data for $F(x)$ are not dependable we also consider the whole problem with only $arg(F(x))$ given instead of $F(x)$.

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