@phdthesis{Baehr2021, author = {B{\"a}hr, Martin}, title = {Efficient time integration methods for linear parabolic partial differential equations with applications}, doi = {10.26127/BTUOpen-5753}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:co1-opus4-57532}, school = {BTU Cottbus - Senftenberg}, year = {2021}, abstract = {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.}, subject = {Parabolic equations; Time integration methods; Fast explicit methods; Model order reduction; Visual computing; Parabolische Gleichungen; Zeitintegrationsverfahren; Schnelle explizite Verfahren; Modellordnungsreduktion; Lineare parabolische Differentialgleichung; Modellordnungsreduktion; Zeitintegrationsverfahren}, language = {en} } @phdthesis{Clausnitzer2024, author = {Clausnitzer, Julian}, title = {Numerical solution of SPDEs on a class of two-dimensional domains and of random differential equations using PCE with exponential time differencing}, doi = {10.26127/BTUOpen-7020}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:co1-opus4-70206}, school = {BTU Cottbus - Senftenberg}, year = {2024}, abstract = {The first part of this thesis deals with the numerical solution of semilinear parabolic stochastic partial differential equations on general two-dimensional C²-bounded domains. The existing exponential Euler time stepping scheme is used on two-dimensional domains, using a spectral approximation in space. Since the base functions are not given analytically, they are numerically approximated using a boundary element method combined with a contour integral method to solve nonlinear eigenvalue problems. An error analysis is given, and numerical experiments conclude the investigation of the method. The second part of the thesis compares the performance of non-intrusive and intrusive polynomial chaos expansion methods based on exponential time differencing schemes for a range of random ordinary and partial differential equations. It is shown in comprehensive numerical experiments that the two approaches are competitive for a range of different equations, but intrusive polynomial chaos becomes less efficient for polynomial nonlinearities of degree greater than two and breaks down for a reaction-diffusion equation exhibiting complex pattern formation behavior.}, subject = {SPDE; Boundary integral method; Random differential equation; Polynomial chaos; Exponential time differencing; Dissertation; Stochastische partielle Differentialgleichungen; Randintegralmethode; Spektralmethode; Polynomielles Chaos; Numerisches Verfahren; Stochastische partielle Differentialgleichung; Zuf{\"a}llige Differentialgleichung; Wiener-Chaos-Zerlegung}, language = {en} } @phdthesis{Khanian2018, author = {Khanian, Maryam}, title = {3D reconstruction using generalized perspective photometric stereo}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:co1-opus4-47191}, school = {BTU Cottbus - Senftenberg}, year = {2018}, abstract = {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.}, subject = {Artificial Intelligence; Computer vision; 3D Reconstruction; Neural networks; Photometric stereo; K{\"u}nstliche Intelligenz; Computer-Vision; 3D-Rekonstruktion; Neuronale Netze; Photometrisches Stereo; Dreidimensionale Rekonstruktion; Maschinelles Sehen; Neuronales Netz}, language = {en} } @phdthesis{SharifiBoroujerdi2018, author = {Sharifi Boroujerdi, Ali}, title = {Linguistic interpretation of visual contents via Deep Learning}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:co1-opus4-47184}, school = {BTU Cottbus - Senftenberg}, year = {2018}, abstract = {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.}, subject = {Deep Learning; Computer vision; Linguistic Interpretation; Machine Learning; Texture description; Deep Learning; Computer Vision; Linguistische Interpretation; Maschinelles Lernen; Textur Beschreibung; Maschinelles Sehen; Computerlinguistik; Deep Learning}, language = {en} } @phdthesis{CastilloAlejandre2019, author = {Castillo Alejandre, Susana}, title = {Digital humanity: the temporal and semantic structure of dynamic conversational facial expressions}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:co1-opus4-51562}, school = {BTU Cottbus - Senftenberg}, year = {2019}, abstract = {This thesis focuses on establishing a - theoretically founded and empirically derived - novel methodological pipeline to provide Embodied Conversational Agents (ECAs) with natural facial expressions and desired personality traits. After giving an overview on the content of this thesis, we dedicate its second part to derive the Semantic Space (SSp) for facial expressions, finding that the same space is used for expression words, expression videos, and motion-capture-based point-cloud animations. The process involved the creation of a new facial expression database using Motion Capture (MoCap) technology. Our technique can be used to empirically map specific motion trajectories (including their frequency-decomposition) onto specific perceptual attributes, allowing the targeted creation of novel animations with the desired perceptual traits, as exemplified in the third part of this thesis. Before addressing our final conclusions and, on the grounds that the systematic differences between individuals while performing the same facial expressions are related to their personality, we devote the fourth part of this thesis to the study of the mapping between personality and expressive facial motions.}, subject = {Facial Animation; Cognitive-based behavioural modelling; Motion capture; Personality; Gesichtsanimation; Kognitiv-basierte Verhaltensmodellierung; Pers{\"o}nlichkeit; Merkmalsextraktion; Motion Capturing; Mimik; Computeranimation}, language = {en} } @phdthesis{Schneidereit2022, author = {Schneidereit, Toni}, title = {The solution of time-dependent ordinary differential equations using neural networks : collocation polynomial neural forms and adaptive neural domain refinement}, doi = {10.26127/BTUOpen-6389}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:co1-opus4-63896}, school = {BTU Cottbus - Senftenberg}, year = {2022}, abstract = {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.}, subject = {Neural forms; Differential equations; Collocation polynomial neural forms; Domain segmentation; Adaptive neural domain refinement; Neuronale Formen; Differentialgleichungen; Neuronale Kollokationspolynome; Adaptive Neuronale Gebietsverfeinerung; Differentialgleichung; Kollokation; Numerisches Verfahren; Polynom; Differential equations; Collocation methods}, language = {en} } @misc{Kahra2020, type = {Master Thesis}, author = {Kahra, Marvin}, title = {Maximumprinzipien f{\"u}r harmonische Abbildungen in Riemann'schen Mannigfaltigkeiten}, url = {http://nbn-resolving.de/urn:nbn:de:kobv:co1-opus4-55889}, school = {BTU Cottbus - Senftenberg}, year = {2020}, abstract = {Der Begriff des Maximumprinzips stellt eine spezielle Eigenschaft von L{\"o}sungen gewisser Differentialgleichungen dar. Dabei handelt es sich grunds{\"a}tzlich um die Eigenschaft, dass das (globale) Maximum am Rand des Definitionsbereiches der Abbildung, welche das Maximumprinzip erf{\"u}llt, angenommen wird. Die gr{\"o}ßte Anwendung findet das Maximumprinzip dabei bei der Eindeutigkeit und Stabilit{\"a}t der L{\"o}sungen von Dirichlet-Problemen, welcher sich diese Arbeit widmet. Hierzu werden einige grundlegende Annahmen bez{\"u}glich zweidimensionaler Riemann'scher Mannigfaltigkeiten und deren Metrik genutzt, um daraus Absch{\"a}tzungen f{\"u}r Jacobi-Felder zu gewinnen. Dabei werden die Jacobi-Felder in einen Tangential- und einen Normalanteil aufgespalten und insbesondere der Normalanteil sowohl von unten als auch von oben beschr{\"a}nkt, wodurch sich eine Beschr{\"a}nkung der Riemann'schen Metrik f{\"u}r beliebige zweidimensionale Vektoren ergibt. Diese Absch{\"a}tzungen bilden den Grundstein f{\"u}r die Anwendung der direkten Variationsmethode von Hildebrandt, Kaul und Widman auf den Fall einer Kreissscheibe in einer Riemann'schen Mannigfaltigkeit. Dabei wird eine schwache Existenzaussage bzgl. des Minimierungsproblems des Energiefunktionals gezeigt, welche ohne weitere Regularit{\"a}tsannahmen auskommt, und anschließend auf schwachharmonische Abbildungen mit vorgeschriebenen Randwerten ausgeweitet. Ein weiteres Resultat stellt die Eindeutigkeit und Stabilit{\"a}t von L{\"o}sungen des Dirichlet-Problems f{\"u}r harmonische Abbildungen in Form des Maximumprinzips von J{\"a}ger und Kaul dar, welches durch Absch{\"a}tzungen von Jacobi-Feldern gewonnen wird. Als eine alternative Herangehensweise dazu wird in dieser Arbeit das Maximumprinzip der konvexen H{\"u}lle vorgestellt. Die wichtigsten Unterschiede bestehen darin, dass hierbei von einem stabilen Riemann'schen Gebiet ausgegangen wird und mittels dessen eine Familie von Geod{\"a}tischen betrachtet wird, die in Form eines Zentralfeldes eine einfache {\"U}berdeckung der Kreisscheibe bilden. Des Weiteren wird mit dieser Methode die Eineindeutigkeit der Dirichlet-L{\"o}sung gew{\"a}hrleistet. Diese Aussage l{\"a}sst sich in dem Sinne erweitern, dass die L{\"o}sung unter entsprechenden Bedingungen an die Metrik einen C^(2+α)-Diffeomorphismus {\"u}ber das gesamte Gebiet darstellt. Schließlich f{\"u}hren wir diese Methode am Beispiel der Poincar{\´e}'schen Halbebene vor.}, subject = {Maximumprinzip; Mannigfaltigkeit; Harmonische Abbildung; Maximum principle; Manifold; Harmonic mapping; Mannigfaltigkeit; Maximumprinzip; Harmonische Abbildung; Partielle Differentialgleichung; Dirichlet-Problem}, language = {de} }