In der vorliegenden Arbeit wird ein neues Tensorformat vorgestellt und eingehend analysiert. Das hierarchische Format verwendet einen binären Baum, um den Tensorraum der Ordnung d mit einer geschachtelten Unterraumstruktur zu versehen. Der Speicheraufwand für diese Darstellung ist von der Größenordnung O(dnr + dr^3), wobei n den Speicheraufwand in den Ansatzräumen kennzeichnet und r ein Rangparameter ist, der durch die Dimensionen der geschachtelten Unterräume bestimmt wird. Das hierarchische Format umfasst verschiedene Standardformate zur Tensordarstellung wie das kanonische oder r-Term-Format und die Unterraum-/Tucker-Darstellung. Die in dieser Arbeit entwickelte zugehörige Arithmetik inklusive mehrerer Approximationsmethoden basiert auf stabilen Methoden der Linearen Algebra, insbesondere die Singulärwertzerlegung und die QR-Zerlegung sind von zentraler Bedeutung. Die rechnerische Komplexität ist hierbei O(dnr^2+dr^4). Die lineare Abhängigkeit von der Ordnung d des Tensorraumes ist hervorzuheben. Für die verschiedenen Approximationsmethoden, deren Effizienz und Effektivität für die Anwendbarkeit des neuen Formates entscheidend sind, werden qualitative und quantitative Fehlerabschätzungen gezeigt. Umfassende numerische Experimente mit einem Fokus auf den Approximationsmethoden bestätigen zum einen die theoretischen Resultate und belegen die Stärken der neuen Tensordarstellung, zeigen aber zum anderen auch weitere, eher überraschende positive Eigenschaften der mit FastHOSVD bezeichneten schnellsten Kürzungsmethode. / In this dissertation we present and a new format for the representation of tensors and analyse its properties. The hierarchical format uses a binary tree in order to define a hierarchical structure of nested subspaces in the tensor space of order d. The strorage requirements are O(dnr+dr^3) where n is determined by the storage requirements in the ansatz spaces and r is a rank parameter determined by the dimensions of the nested subspaces. The hierarchichal representation contains the standard representation like canonical or r-term representation and subspace or Tucker representation. The arithmetical operations that have been developed in this work, including several approximation methods, are based on stable Linear Alebra methods, especially the singular value decomposition (SVD) and the QR decomposition are of importance. The computational complexity is O(dnr^2+dr^4). The linear dependence from the order d of the tensor space is important. The approximation methods are one of the key ingredients for the applicability of the new format and we present qualitative and quantitative error estimates. Numerical experiments approve the theoretical results and show some additional, but unexpected positive aspects of the fastest method called FastHOSVD.

In der vorliegenden Arbeit wird ein neues Tensorformat vorgestellt und eingehend analysiert. Das hierarchische Format verwendet einen binären Baum, um den Tensorraum der Ordnung d mit einer geschachtelten Unterraumstruktur zu versehen. Der Speicheraufwand für diese Darstellung ist von der Größenordnung O(dnr + dr^3), wobei n den Speicheraufwand in den Ansatzräumen kennzeichnet und r ein Rangparameter ist, der durch die Dimensionen der geschachtelten Unterräume bestimmt wird. Das hierarchische Format umfasst verschiedene Standardformate zur Tensordarstellung wie das kanonische oder r-Term-Format und die Unterraum-/Tucker-Darstellung. Die in dieser Arbeit entwickelte zugehörige Arithmetik inklusive mehrerer Approximationsmethoden basiert auf stabilen Methoden der Linearen Algebra, insbesondere die Singulärwertzerlegung und die QR-Zerlegung sind von zentraler Bedeutung. Die rechnerische Komplexität ist hierbei O(dnr^2+dr^4). Die lineare Abhängigkeit von der Ordnung d des Tensorraumes ist hervorzuheben. Für die verschiedenen Approximationsmethoden, deren Effizienz und Effektivität für die Anwendbarkeit des neuen Formates entscheidend sind, werden qualitative und quantitative Fehlerabschätzungen gezeigt. Umfassende numerische Experimente mit einem Fokus auf den Approximationsmethoden bestätigen zum einen die theoretischen Resultate und belegen die Stärken der neuen Tensordarstellung, zeigen aber zum anderen auch weitere, eher überraschende positive Eigenschaften der mit FastHOSVD bezeichneten schnellsten Kürzungsmethode. / In this dissertation we present and a new format for the representation of tensors and analyse its properties. The hierarchical format uses a binary tree in order to define a hierarchical structure of nested subspaces in the tensor space of order d. The strorage requirements are O(dnr+dr^3) where n is determined by the storage requirements in the ansatz spaces and r is a rank parameter determined by the dimensions of the nested subspaces. The hierarchichal representation contains the standard representation like canonical or r-term representation and subspace or Tucker representation. The arithmetical operations that have been developed in this work, including several approximation methods, are based on stable Linear Alebra methods, especially the singular value decomposition (SVD) and the QR decomposition are of importance. The computational complexity is O(dnr^2+dr^4). The linear dependence from the order d of the tensor space is important. The approximation methods are one of the key ingredients for the applicability of the new format and we present qualitative and quantitative error estimates. Numerical experiments approve the theoretical results and show some additional, but unexpected positive aspects of the fastest method called FastHOSVD.

In many applications that deal with high dimensional data, it is important to not store the high dimensional object itself, but its representation in a data sparse way. This aims to reduce the storage and computational complexity. There is a general scheme for representing tensors with the help of sums of elementary tensors, where the summation structure is defined by a graph/network. This scheme allows to generalize commonly used approaches in representing a large amount of numerical data (that can be interpreted as a high dimensional object) using sums of elementary tensors. The classification does not only distinguish between elementary tensors and non-elementary tensors, but also describes the number of terms that is needed to represent an object of the tensor space. This classification is referred to as tensor network (format). This work uses the tensor network based approach and describes non-linear block Gauss-Seidel methods (ALS and DMRG) in the context of the general tensor network framework. Another contribution of the thesis is the general conversion of different tensor formats. We are able to efficiently change the underlying graph topology of a given tensor representation while using the similarities (if present) of both the original and the desired structure. This is an important feature in case only minor structural changes are required. In all approximation cases involving iterative methods, it is crucial to find and use a proper initial guess. For linear iteration schemes, a good initial guess helps to decrease the number of iteration steps that are needed to reach a certain accuracy, but it does not change the approximation result. For non-linear iteration schemes, the approximation result may depend on the initial guess. This work introduces a method to successively create an initial guess that improves some approximation results. This algorithm is based on successive rank 1 increments for the r-term format. There are still open questions about how to find the optimal tensor format for a given general problem (e.g. storage, operations, etc.). For instance in the case where a physical background is given, it might be efficient to use this knowledge to create a good network structure. There is however, no guarantee that a better (with respect to the problem) representation structure does not exist.

In many applications that deal with high dimensional data, it is important to not store the high dimensional object itself, but its representation in a data sparse way. This aims to reduce the storage and computational complexity. There is a general scheme for representing tensors with the help of sums of elementary tensors, where the summation structure is defined by a graph/network. This scheme allows to generalize commonly used approaches in representing a large amount of numerical data (that can be interpreted as a high dimensional object) using sums of elementary tensors. The classification does not only distinguish between elementary tensors and non-elementary tensors, but also describes the number of terms that is needed to represent an object of the tensor space. This classification is referred to as tensor network (format). This work uses the tensor network based approach and describes non-linear block Gauss-Seidel methods (ALS and DMRG) in the context of the general tensor network framework. Another contribution of the thesis is the general conversion of different tensor formats. We are able to efficiently change the underlying graph topology of a given tensor representation while using the similarities (if present) of both the original and the desired structure. This is an important feature in case only minor structural changes are required. In all approximation cases involving iterative methods, it is crucial to find and use a proper initial guess. For linear iteration schemes, a good initial guess helps to decrease the number of iteration steps that are needed to reach a certain accuracy, but it does not change the approximation result. For non-linear iteration schemes, the approximation result may depend on the initial guess. This work introduces a method to successively create an initial guess that improves some approximation results. This algorithm is based on successive rank 1 increments for the r-term format. There are still open questions about how to find the optimal tensor format for a given general problem (e.g. storage, operations, etc.). For instance in the case where a physical background is given, it might be efficient to use this knowledge to create a good network structure. There is however, no guarantee that a better (with respect to the problem) representation structure does not exist.