FG Angewandte Mathematik
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Die vorliegende Arbeit behandelt für eine breite Klasse zweidimensionaler Variationsprobleme eine Existenz- und Regularitätstheorie, die der Lösung von Randwertproblemen partieller Differentialgleichungssysteme dient. Dabei werden bekannte Ergebnisse gründlich untersucht und umfassend aufgearbeitet. Teilweise wird eine geeignete Anpassung der Voraussetzungen einiger Resultate vorgenommen. Speziell wird die Theorie auf das Plateausche Problem für Flächen vorgeschriebener mittlerer Krümmung im R3 angewendet. Eingangs wird das Konzept der direkten Methoden der Variationsrechnung erläutert. Über einen fundamentalen Satz zur schwachen Unterhalbstetigkeit von Funktionalen wird die Existenz eines Minimierers für eine breite Klasse von Variationsproblemen nachgewiesen. Da die Existenztheorie in Sobolev-Räumen agiert, sind Untersuchungen zur Regularität eines Minimierers notwendig. Im Rahmen der Regularitätstheorie wird das Dirichletsche Wachstumstheorem von Morrey gezeigt, welches ein hinreichendes Kriterium für die Hölder-Stetigkeit einer Funktion X ∈ W1,2(G) liefert. Zum Nachweis der Anwendbarkeit des Dirichletschen Wachstumstheorems auf einen Minimierer wird ein Wachstumslemma verwendet. Dabei werden eine Verknüpfung des Dirichlet-Integrals mit Fourierreihen sowie das Dirichlet- Integral der harmonischen Ersetzung einer Funktion mit L2-Randwerten genutzt. Infolgedessen ergibt sich die Hölder-Stetigkeit im Inneren für einen Minimierer. Zudem wird die Stetigkeit des Minimierers bis zum Rand gezeigt, sodass die Existenz eines stetigen Minimierers für eine breite Klasse von Variationsproblemen gesichert ist. Anschließend erfolgt die Berechnung der ersten Variation sowie die damit verbundene Herleitung der schwachen Euler-Lagrange-Gleichung, welche eine Beziehung zur Lösungstheorie von Differentialgleichungen im Sinne des Dirichletschen Prinzips herstellt.
Ausgehend von der schwachen Euler-Lagrange-Gleichung wird die Hölder-Stetigkeit
der ersten Ableitungen eines Minimierers bewiesen. Für die Lösung von Randwertproblemen mit partiellen Differentialgleichungen zweiter Ordnung mithilfe der Variationsrechnung ist der Nachweis stetiger zweiter Ableitungen des Minimierers notwendig. Dabei wird der Fokus auf sogenannte Minimierer vom Poissonschen Typ gelegt. Eine C2,σ-Rekonstruktion, die auf der Schaudertheorie basiert, liefert in diesem Fall die gewünschte Regularität. In Anwendung dessen werden das Randwertproblem harmonischer Abbildungen in Riemannschen Räumen sowie das Dirichletproblem des H-Flächen-Systems behandelt, indem jeweils ein geeignetes Variationsproblem aufgestellt wird. Als Erweiterung des Dirichletproblems des H-Flächen-Systems wird das allgemeine
Plateausche Problem für Flächen vorgeschriebener mittlerer Krümmung mit den erarbeiteten Methoden der Variationsrechnung untersucht. Zudem wird das Plateausche Problem für Flächen vorgeschriebener mittlerer Krümmung in Kugeln, in Zylindern sowie insbesondere im Einheitskegel gelöst. Insgesamt wird eine ausführliche, in sich geschlossene und gut verständliche Existenzund Regularitätstheorie der zweidimensionalen Variationsrechnung zur Behandlung von Randwertproblemen partieller Differentialgleichungssysteme dargestellt, welche sich in besonderer Weise bei der Lösung des Plateauschen Problems für Flächen vorgeschriebener mittlerer Krümmung entfaltet.
A classic approach for solving differential equations with neural networks builds upon neural forms, which employ the differential equation with a discretisation of the solution domain. Making use of neural forms for time-dependent differential equations, one can apply the recently developed method of domain segmentation. That is, the domain may be split into several subdomains, on which the optimisation problem is solved. In classic adaptive numerical methods, the mesh as well as the domain may be refined or decomposed, in order to improve the accuracy. Also, the degree of approximation accuracy may be adapted. Therefore, it is desirable to transfer such important and successful strategies to the field of neural-network-based solutions. In the presented work, we propose a novel adaptive neural approach to meet this aim for solving time-dependent problems. To this end, each subdomain is reduced in size until the optimisation is resolved up to a predefined training accuracy. In addition, while the neural networks employed are by default small, we propose a means to adjust also the number of neurons in an adaptive way. We introduce conditions to automatically confirm the solution reliability and optimise computational parameters whenever it is necessary. Results are provided for several initial-value problems that illustrate important computational properties of the method.
Solving differential equations is still a topic of major interest, due to their appearance in many fields of science and engineering and a classic approach with neural networks builds upon trial solutions, the so-called neural forms. The latter are incorporated in a cost function that is subject to minimisation, to train the involved neural networks. Neural forms represent general and flexible tools for solving ordinary differential equations, partial differential equations as well as systems of each. However, the computational approach is in general highly dependent on a variety of computational parameters and the choice of the optimisation methods. Studying the solution of a simple but fundamental stiff ordinary differential equations with small feedforward neural networks and first order optimisation shows, that it is possible to identify preferable choices for parameters and methods. The neural network weight initialisation appears to be a sensitive topic, while having a major impact on the solution accuracy. Especially the use of non-random (deterministic) weights partially shows poor performance, but removes a stochastic component. Further research reveals, that a new polynomial representation of the neural forms can significantly increase the reliability of a deterministic initialisation (all weights have initially the same values assigned). In order to maintain smaller neural network architectures and solve the differential equation, even on fairly large domains, a new technique called domain segmentation (for initial value problems) is introduced. The solution domain splits into equidistant subdomains and the above-mentioned collocation polynomial neural forms are solved separately in each domain fragment. At the boundary of any subdomain, a new initial value is provided by the neural forms solution and directly incorporated in the adjacent one. In classic adaptive numerical methods for solving differential equations, the mesh as well as the domain may be refined or decomposed, respectively, in order to improve numerical accuracy. The subdomain distribution can also be connected with an adaptive refinement. That is, the neural network training status is combined with an adaptive subdomain size reduction in the new adaptive neural domain refinement algorithm. That is, each subdomain is reduced in size until the optimisation is resolved up to a predefined training accuracy. In addition, while the neural networks are by default small, the number of neurons may also be adjusted in an adaptive way. Conditions are introduced to automatically confirm the solution reliability and optimise computational parameters whenever it is necessary.
The basic filters in mathematical morphology are dilation and erosion. They are defined by a structuring element that is usually shifted pixel-wise over an image, together with a comparison process that takes place within the corresponding mask. This comparison is made in the grey value case by means of maximum or minimum formation. Hence, there is easy access to max-plus algebra and, by means of an algebra change, also to the theory of linear algebra.
We show that an approximation of the maximum function forms a commutative semifield (with respect to multiplication) and corresponds to the maximum again in the limit case. In this way, we demonstrate a novel access to the logarithmic connection between the Fourier transform and the slope transformation. In addition, we prove that the dilation by means of a Fast Fourier Transform depends only on the size of the structuring element used. Moreover, we derive a bound above which the Fourier approximation yields results that are exact in terms of grey value quantization.
In this thesis we study efficient time integration methods for linear parabolic PDEs to solve practical problems that arise in a variety of real-world applications. The classical construction of numerical methods for solving PDEs is based on the method of lines, which leads to a large sparse semi-discretised system of ODEs to which any numerical method for initial value ODE problems can be applied. When dealing with parabolic-type problems, the underlying ODE systems are known to be stiff. Therefore, in the context of linear model problems, the use of implicit schemes is usually considered to be the best choice in practice.
However, this statement is not correct for some relevant real-world applications. In particular, implicit schemes can cause high computational costs that are equipped with certain model conditions. The model problems considered here are coupled with various settings, ranging from many different initial conditions over long-term simulation with relatively frequent model updates, to dealing with very large-scale problems for which the matrix size can exceed several millions. For this reason, we are interested in sophisticated and computationally efficient numerical methods that bring the aspects of approximation accuracy as well as computational and storage complexity into balance.
Even nowadays it is still a challenging task to devise a numerical method that combines high accuracy, robustness and computational efficiency for the model problem to be solved. Therefore, the main objective is to find an easy and efficient ODE integration scheme for each individual model problem. On this basis, we first give a comprehensive introduction to the state-of-the-art methods that are often used for practical purposes. In this framework, we will investigate very detailed the theoretical and numerical foundations of two popular techniques, namely the fast explicit methods and the model order reduction techniques. This is primarily important in order to fully understand the numerical methods, and also absolutely essential in finding the best numerical method that is specifically suitable for the intended purpose.
Our second goal is then to efficiently solve the practical problems that arise in connection with shape correspondence, geothermal energy storage and image osmosis filtering. For each application we specify a complete setup, and in order to provide an efficient and accurate numerical approximation, we give a thorough discussion of the various numerical solvers along with many technical details and own adaptations. We validate our numerical findings through many experiments using synthetic and real-world data. In addition, the thesis provides a complete and detailed description of the powerful methods that can be very useful for tackling similar problems that are the subject of interest in many applications.
This thesis deals with a novel approach for analyzing and computing interior transmission eigenvalues of (piecewise) homogeneous media in two dimensions. It is based on approximating boundary data of respective eigenfunctions by the method of fundamental solutions. However, since a straightforward implementation would solely exploit ill-conditioned matrices and thus evoke spurious results, a stabilization scheme is incorporated. The combined method is then studied with a distinction between isotropic and anisotropic materials, and complemented by novel approximation theory each. Numerical validations complete the investigations for different wave type scenarios
The main part of the research outlined in this thesis is to develop Deep Learning models for the linguistic interpretation of the visual contents. This part is split into two research problems: interactive region segmentation and captioning, and selective texture labeling. In the first attempt, we proposed a novel hybrid Deep Learning architecture whereby the user is able to specify an arbitrary region of the image that should be highlighted and described. The proposed model alternates the bounding box indications of the standard object localization process with the output of a deep interactive segmentation module to achieve a better understanding of the dense image captioning and improve the object localization accuracy. The idea of the next part is to establish a bidirectional correlation between deep texture representation and its linguistic description via a hybrid CNN-RNN model that enables end-to-end learning of the selective texture labeling. This novel architecture provides new opportunities to describe, search, and also retrieve texture images from their linguistic descriptions. To be able to train such a model, we generated a multi-label texture dataset that covers color, material, and pattern labeling simultaneously. Our contribution to the automatic generation of texture descriptions provides an excellent opportunity to enrich the existing vocabulary of the image captioning. Such a conceptual extension can be used for fine-grained captioning applicable in geology, meteorology and other natural sciences where fine-grained image structures are of importance to understand complicated patterns. Apart from Deep Learning technologies, in the final section of the thesis, we proposed a novel approach to define mathematical morphology on color images. To this end, we converted common RGB-values of the color images into a new biconal color space and then combined two approaches of mathematical morphology to give meaning to the maximum and the minimum of the matrix field data and formulate our novel strategy.
Reconstruction of the 3D shape information is a fundamental problem in computer vision. Among different shape recovering technologies, photometric stereo is highlighted for its capability to produce high quality 3D reconstruction. This dissertation generalizes photometric stereo in different aspects towards creating a practical 3D reconstruction. The proposed techniques can be considered as a fundamental support to develop future cameras offering 3D shapes for various applications such as movie and video game industry, medical sciences, virtual reality, automotive driving and etc. The first generalization is developed for addressing specularities in 3D reconstructions and also involving the perspective projection. These attempts lead to remove the limitation of working with diffuse materials and confined projected scenes. We will prove the applicability of our approach using complex scenes like endoscopy images. In the second proposed approach, we will offer a real-time 3D reconstruction of micro-details with a more generalized reflectance model. Moreover, a recurrent optimization network will be provided. These innovations lead to presenting the 3D reconstruction of details which are even invisible to human eyes like micro-prints on the banknote. This information recovery can be used in various areas such as detecting security items on financial documents for fraud detection and also the quality control of any industrial productions including delicate details such as printed circuits. In the third proposed model, we develop a PS reconstruction technique using neural networks for the uncalibrated PS where the light direction is not available. Finally, for the first time, benefiting from deep neural networks and meta heuristic algorithms, we will devise an approach which can deliver high qualified 3D shape from the internet and out-door images, without any pre-necessary knowledge.