This work presents four new methods for the analysis and visualization of tensor fields. The focus is on tensor fields which arise in the context of structural mechanics simulations. The first method deals with the design of components made of short fiber reinforced polymers using injection molding. The stability of such components depends on the fiber orientations, which are affected by the production process. For this reason, the stresses under load as well as the fiber orientations are analyzed. The stresses and fiber orientations are each given as tensor fields. For the analysis four features are defined. The features indicate if the component will resist the load or not, and if the respective behavior depends on the fiber orientation or not. For an in depth analysis a glyph was developed, which shows the admissible fiber orientations as well as the given fiber orientation. With these visualizations the engineer can rate a given fiber orientation and gets hints for improving the fiber orientation. The second method depicts gradients of stress tensors using glyphs. A thorough understanding of the stress gradient is desirable, since there is some evidence that not only the stress but also its gradient influences the stability of a material. Gradients of stress tensors are third order tensors, the visualization is therefore a great challenge and there is very little research on this subject so far. The objective of the third method is to analyse the complete invariant part of the tensor field. Scalar invariants play an important role in many applications, but proper selection of such invariants is often difficult. For the analysis of the complete invariant part the notion of 'extremal point' is introduced. An extremal point is characterized by the fact that there is a scalar invariant which has a critical point at this position. Moreover it will be shown that the extrema of several common invariants are contained in the set of critical points. The fourth method presented in this work uses the Heat Kernel Signature (HKS) for the visualization of tensor fields. The HKS is computed from the heat kernel and was originally developed for surfaces. It characterizes the metric of the surface under weak assumptions. i.e. the shape of the surfaces is determined up to isometric deformations. The fact that every positive definite tensor field can be considered as the metric of a Riemannian manifold allows to apply the HKS on tensor fields.

The recent state of art in the magnetic resonance area putsforward the electron paramagnetic resonance, EPR, and nuclearmagnetic resonance, NMR, experiments on prominent positions forinvestigations of molecular and electronic structure. A mostdifficult aspect of such experiments is usually the properinterpretation of data obtained from high-resolution spectra,that, however, at the same time opens a great challenge forpure theoretical methods to interpret the spectral features.This thesis constitutes an effort in this respect, as itpresents and discusses calculations of EPR and NMR parametersof paramagnetic molecules. The calculations are based on newmethodology for determination of properties of paramagneticmolecules in the framework of the density functional theory,which has been developed in our laboratory. Paramagnetic molecules are, in some sense, very special. Thepresence of unpaired electrons essentially modifies theirspectra. The experimental determination of the magneticresonance parameters of such molecules is, especially in theNMR case, quite complicated and requires special techniques ofspectral detection. The significant efforts put into suchexperiments are completely justi fied though by the importantroles of paramagnetic species playing in many areas, such as,for example, molecular magnets, active centers in biologicalsystems, and defects in inorganic conductive materials. The first two papers of this thesis deal with thetheoretical determination of NMR parameters, such as thenuclear shielding tensors and the chemical shifts, inparamagnetic nitroxides that form core units in molecularmagnets. The developed methodology aimed to realize highaccuracy in the calculations in order to achieve successfulapplications for the mentioned systems. Theeffects of hydrogenbonding are also described in that context. Our theory forevaluation of nuclear shielding tensors in paramagneticmolecules is consistent up to the second order in the finestructure constant and considers orbital, fully isotropicdipolar, and isotropic contact contributions to the shieldingtensor. The next three projects concern electron paramagneticresonance. The wellknown EPR parameters, such as the g-tensorsand the hyperfine coupling constants are explored. Calculationsof electronic g-tensors were carried out in the framework of aspin-restricted open-shell Kohn-Sham method combined with thelinear response theory recently developed in our laboratory.The spincontamination problem is then automatically avoided.The solvent effects, described by the polarizable continuummodel, are also considered. For calculations of the hyperfinecoupling constants a so-called restricted-unrestricted approachhas been developed in the context of density functional theory.Comparison of experimentally and theoretically determinedparameters shows that qualitative mutual agreement of the twosets of data can be easily achieved by employing the proposedformalisms.

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Materialien wie Kristalle, biologisches Gewebe oder elektroaktive Polymere kommen häufig in verschiedenen Anwendung, wie dem Prothesenbau oder der Simulation von künstlicher Muskulatur vor. Diese und viele weitere Materialien haben gemeinsam, dass sie unter gewissen Umständen ihre Form und andere Materialeigenschaften ändern. Um diese Veränderung beschreiben zu können, werden, abhängig von der Anwendung, verschiedene Tensoren unterschiedlicher Ordnung benutzt. Durch die Komplexität und die starke Abhängigkeit der Tensorbedeutung von der Anwendung, gibt es bisher kein Verfahren Tensoren höherer Ordnung darzustellen, welches standardmäßig benutzt wird. Auch bezogen auf einzelne Anwendungen gibt es nur sehr wenig Arbeiten, die sich mit der visuellen Darstellung dieser Tensoren auseinandersetzt. Diese Arbeit beschäftigt sich mit diesem Problem. Es werden drei verschiedene Methoden präsentiert, Tensoren höherer Ordnung zu analysieren und zu visualisieren. Alle drei Methoden basieren auf der sogenannte deviatorischen Zerlegung und der Multipoldarstellung. Mit Hilfe der Multipole können die Symmetrien des Tensors und damit des beschriebenen Materials bestimmt werden. Diese Eigenschaft wird in für die Visualisierung des Steifigkeitstensors benutzt. Die zweite Methode basiert direkt auf den Multipolen und kann damit beliebige Tensoren in drei Dimensionen darstellen. Dieses Verfahren wird anhand des Kopplungs Tensors, ein Tensor dritter Ordnung, vorgestellt. Die ersten zwei Verfahren sind lokale Glyph-basierte Verfahren. Das dritte Verfahren ist ein erstes globales Tensorvisualisierungsverfahren, welches Tensoren beliebiger Ordnung und Symmetry in drei Dimensionen mit Hilfe eines linienbasierten Verfahrens darstellt. / Materials like crystals, biological tissue or electroactive polymers are frequently used in applications like prosthesis construction or the simulation of artificial musculature. These and many other materials have in common that they change their shape and other material properties under certain circumstances. To describe these changes, different tensors of different order, dependent of the application, are used. Due to the complexity and the strong dependency of the tensor meaning of the application, there is, by now, no visualization method that is used by default. Also for specific applications there are only a few methods that address the visual analysis of higher-order tensors. This work adresses this problem. Three different methods to analyse and visualize tensors of higher order will be provided. All three methods are based on the so called deviatoric decomposition and the multipole representation. Using the multipoles the symmetries of a tensor and, therefore, of the described material, can be calculated. This property is used to visualize the stiffness tensor. The second method uses the multipoles directly and can be used for each tensor of any order in three dimensions. This method is presented by analysing the third-order coupling tensor. These two techniques are glyph-based visualization methods. The third one, a line-based method, is, according to our knowledge, a first global visualization method that can be used for an arbitrary tensor in three dimensions.

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The ever-increasing applications of Big Data in improving networking application performancehave motivated the networking community to deploy it in SDN (Software defined network) toconstruct flexible, scalable, self-aware, and self-managing networks. The primary purpose ofthis research is to investigate the validity of tensor-decomposition, a well-knownmathematical approach for data reduction, to catch patterns in network traffic as an initialstep toward the network's intelligence.Using only three-dimensional (cubic) tensors (Source, Destination, Bandwidth). Theconducted research used both offline (not simulated) and online (Mininet and RYU controllersimulation) network traffic of the GEANT (TOTEM) dataset. From the tensor decompositionanalysis on the adjacency matricies, we caught traffic intensity patterns between nodes(switches), which provided suggestions that helps rebuild the topology (which nodes shouldbe physically connected to the others). However, capturing the patterns in the time revolutionwas invalid due to limitations in the three-dimensional tensor.

Modern robotics and automation systems require high-level reasoning capability in representing, identifying, and interpreting the three-dimensional data of the real world. Understanding the world's geometric structure by visual data is known as 3D perception. The necessity of analyzing irregular and complex 3D data has led to the development of high-dimensional frameworks for data learning. Here, we design several sparse learning-based approaches for high-dimensional data that effectively tackle multiple perception problems, including data filtering, data recovery, and data retrieval. The frameworks offer generative solutions for analyzing complex and irregular data structures without prior knowledge of data. The first part of the dissertation proposes a novel method that simultaneously filters point cloud noise and outliers as well as completing missing data by utilizing a unified framework consisting of a novel tensor data representation, an adaptive feature encoder, and a generative Bayesian network. In the next section, a novel multi-level generative chaotic Recurrent Neural Network (RNN) has been proposed using a sparse tensor structure for image restoration. In the last part of the dissertation, we discuss the detection followed by localization, where we discuss extracting features from sparse tensors for data retrieval. / Doctor of Philosophy / The development of automation systems and robotics brought the modern world unrivaled affluence and convenience. However, the current automated tasks are mainly simple repetitive motions. Tasks that require more artificial capability with advanced visual cognition are still an unsolved problem for automation. Many of the high-level cognition-based tasks require the accurate visual perception of the environment and dynamic objects from the data received from the optical sensor. The capability to represent, identify and interpret complex visual data for understanding the geometric structure of the world is 3D perception. To better tackle the existing 3D perception challenges, this dissertation proposed a set of generative learning-based frameworks on sparse tensor data for various high-dimensional robotics perception applications: underwater point cloud filtering, image restoration, deformation detection, and localization. Underwater point cloud data is relevant for many applications such as environmental monitoring or geological exploration. The data collected with sonar sensors are however subjected to different types of noise, including holes, noise measurements, and outliers. In the first chapter, we propose a generative model for point cloud data recovery using Variational Bayesian (VB) based sparse tensor factorization methods to tackle these three defects simultaneously. In the second part of the dissertation, we propose an image restoration technique to tackle missing data, which is essential for many perception applications. An efficient generative chaotic RNN framework has been introduced for recovering the sparse tensor from a single corrupted image for various types of missing data. In the last chapter, a multi-level CNN for high-dimension tensor feature extraction for underwater vehicle localization has been proposed.

Die Berechnung von Tensorintegralen ist eines der komplizierteren Probleme bei der Berechnung von Einschleifen-Feynmandiagrammen. In dieser Arbeit wird die Computerprogrammbibliothek PJFry entwickelt, mit der Tensorintegrale mit bis zu fünf äusseren Beinen und unter Zugrundelegung beliebiger Kinematik numerisch ausgewertet werden können. Im Programm PJFry sind Algorithmen implementiert, mit denen bei der Reduktion von Pentagon-Tensoren inverse Potenzen der Gramdeterminanten vermieden werden können. Gramdeterminanten der Boxdiagramme werden unter Verwendung von Rekursionsrelationen mit variabler Raum-Zeit-Dimension in einem Satz neuer Basisintegrale isoliert. Die neuen Basisintegrale werden ebenfalls durch Rekursionsrelationen mit variabler Raum-Zeit-Dimension oder durch Entwicklung in kleinen Gramdeterminanten ausgewertet. Die Konvergenz letzterer wird durch Padé-Extrapolation erheblich beschleunigt. Ein Cache-System erlaubt die mehrfache Verwendung von numerischen Bausteinen und erhöht zusätzlich die Effizienz des Programmpakets. Ausser ausführlichen Tests von Struktur und Genauigkeit der Algorithmen wird eine nichtriviale Beispielanwendung ausgearbeitet und mit dem Programm NGluon verglichen: die Berechnung von fünf-Gluon-Helizitätsamplituden. Schließlich werden die virtuellen Einschleifenkorrekturen zur Myonpaarproduktion mit Emission energiereicher ("harter") Photonen berechnet. Die Methode wird erläutert, wie auch Renormierung und Behandlung der Polstruktur in dimensionaler Regularisierung. Numerische Vorhersagen für differentielle Wirkungsquerschnitte werden berechnet, unter Zugrundelegung der kinematischen Situationen, wie sie bei den Detektoren KLOE (DAFNE, Frascati) und BaBar (SLAC) typisch sind. / In this thesis we discuss the problem of evaluation of tensor integrals appearing in a typical one-loop Feynman diagram calculation. We present a computer library for the numerical evaluation of tensor integrals with up to 5 legs and arbitrary kinematics. The code implements algorithms based on the formalism which avoids the appearance of inverse Gram determinants in the reduction of pentagon diagrams. The Gram determinants of box integrals are isolated in the set of new basis integrals by using dimensional recurrence relations. These integrals are then evaluated by dimensional recurrence or expansion in small Gram determinant, which is improved by Padé extrapolation. A cache system allows reuse of identical building blocks and increases the efficiency. After describing the cross checks and accuracy tests, we show a sample application to the evaluation of five gluon helicity amplitudes, which is compared with the output of the program NGluon. In the last part the program is applied to the calculation of the one-loop virtual corrections to the muon pair production with hard photon emission. The computation method is explained, followed by a discussion of renormalization and pole structure. Finally, we present numerical results for differential cross sections with kinematics of the KLOE and BaBar detectors.

This master's thesis addresses numerical methods of computing the typical ranks of tensors over the real numbers and explores some properties of tensors over finite fields. We present three numerical methods to compute typical tensor rank. Two of these have already been published and can be used to calculate the lowest typical ranks of tensors and an approximate percentage of how many tensors have the lowest typical ranks (for some tensor formats), respectively. The third method was developed by the authors with the intent to be able to discern if there is more than one typical rank. Some results from the method are presented but are inconclusive. In the area of tensors over nite filds some new results are shown, namely that there are eight GLq(2) GLq(2) GLq(2)-orbits of 2 2 2 tensors over any finite field and that some tensors over Fq have lower rank when considered as tensors over Fq2 . Furthermore, it is shown that some symmetric tensors over F2 do not have a symmetric rank and that there are tensors over some other finite fields which have a larger symmetric rank than rank.