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A Review on Adaptive Low-Rank Approximation Techniques in the Hierarchical Tensor Format

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Extraction of Quantifiable Information from Complex Systems

Part of the book series: Lecture Notes in Computational Science and Engineering ((LNCSE,volume 102))

Abstract

The hierarchical tensor format allows for the low-parametric representation of tensors even in high dimensions d. On the one hand, this format provides a robust framework for approximate arithmetic operations with tensors based on rank truncations, which can be exploited in iterative algorithms. On the other hand, it can be used for the direct approximation of high-dimensional data stemming, e.g., from the discretisation of multivariate functions. In this review, we discuss several strategies for an adaptive approximation of tensors in the hierarchical format by black box type techniques, including problems of tensor reconstruction and tensor completion.

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Author information

Authors and Affiliations

  1. RWTH Aachen, Templergraben 55, 52056, Aachen, Germany

    Jonas Ballani, Lars Grasedyck & Melanie Kluge

Authors
  1. Jonas Ballani
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  2. Lars Grasedyck
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  3. Melanie Kluge
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Corresponding author

Correspondence to Jonas Ballani .

Editor information

Editors and Affiliations

  1. FB 12 Mathematik und Informatik, Philipps-Universität Marburg, Marburg, Germany

    Stephan Dahlke

  2. Inst. Geometrie & Praktische Mathemathik, RWTH Aachen University, Aachen, Germany

    Wolfgang Dahmen

  3. Universität Bonn Institut für Numerische Simulation, Bonn, Germany

    Michael Griebel

  4. Max-Planck-Inst. f. Mathem. in d. Naturwissenschaften, Leipzig, Sachsen, Germany

    Wolfgang Hackbusch

  5. Fachbereich Mathematik, Technische Universität Kaiserslautern, Kaiserslautern, Germany

    Klaus Ritter

  6. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Reinhold Schneider

  7. Seminar für Angewandte Mathematik, ETH Zürich, Zürich, Switzerland

    Christoph Schwab

  8. Institut für Mathematik, Technische Universität Berlin, Berlin, Germany

    Harry Yserentant

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Ballani, J., Grasedyck, L., Kluge, M. (2014). A Review on Adaptive Low-Rank Approximation Techniques in the Hierarchical Tensor Format. In: Dahlke, S., et al. Extraction of Quantifiable Information from Complex Systems. Lecture Notes in Computational Science and Engineering, vol 102. Springer, Cham. https://doi.org/10.1007/978-3-319-08159-5_10

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