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(Keine Zusammenfassung in deutscher Sprache hinterlegt.) The modeling of physical properties often leads to the task of solving partial differential equations or integral equations. The results of some discretisation and linearisation process are matrix equations or linear systems of equations with special features. In the case of partial differential equations one exploits the local character of the differentiation by using some finite element method or finite difference scheme and gains a sparse system matrix. In the case of (nonlocal) integral operators low rank approximations seem to be the method of choice. These are either given explicitly by some multipole method or panel clustering technique or implicitly by rank revealing decompositions. Both types of matrices can be represented as so-called H-matrices. In this thesis we investigate algorithms that perform the addition, multiplication and inversion of H-matrices approximately. Under moderate assumptions the complexity of these new arithmetics is almost linear (linear up to logarithmic terms of order 1 to 3). The arithmetic operations can be performed adaptively, that is up to some given accuracy epsilon the relative error of the operations is zero. The question arises under which circumstances the inverse of an H-matrix can be approximated by an H-matrix. For the techniques used in this thesis we need very restrictive assumptions, but the numerical examples in the last part indicate that the approximability does not depend on these assumptions.

Tensors are obtained, e.g., by the consideration of a physical phenomena in a serie of measurements or by the discretisation of a multivariate function. Therein, the phenomena can be time-dependent, e.g., in the case of meteorological data or the evaluation of a functional value can be very expensive, e.g., if the solution of a partiell differential equation ist given by a huge dense linear system. These different situations lead to different restrictions for the generation of the tensor entries, which are grouped into three categories.In this work, we present three different strategies to approximate a high-dimensional tensor, which selects a set of approriate tensor entries for one of the three situations. Combined with our developed calculation routine, the tensor-cross-approximation, an approximation in the hierarchical format is available. The approximation is of significantly less storage complexity than the original tensor and can be used for further calculation instead of the orinigal one.We consider in detail how much the approximation accuracy ist influenced by changing the occuring variables of the tensor and the strategy. On this basis, we point their strengths and weaknesses out. We also looking beyond our calculation routine by comparing it with different optimization methods.All considerations and observations are joined together by given an advice in which case which method with which strategy is the most suitable one and how to use the strengths of the selected strategy in the best way.