Sampling-Strategien zur Approximation hochdimensionaler Tensoren im hierarchischen Niedrigrangformat

• Sampling strategies for high-dimensional tensor approximation in hierarchical low rank format

Kluge, Melanie; Grasedyck, Lars (Thesis advisor); Schneider, Reinhold (Thesis advisor)

Aachen (2016)
Dissertation / PhD Thesis

Dissertation, RWTH Aachen, 2016

Abstract

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.