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  • © 2019

Tensor Spaces and Numerical Tensor Calculus

  • First monograph on this subject in the field of numerical mathematics

  • Covers algebraic and functional analysis aspects of tensor spaces

  • Focuses on the numerical treatment

Part of the book series: Springer Series in Computational Mathematics (SSCM, volume 56)

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eBook USD 119.00
Price excludes VAT (USA)
  • ISBN: 978-3-030-35554-8
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 159.99
Price excludes VAT (USA)
Hardcover Book USD 159.99
Price excludes VAT (USA)

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Table of contents (17 chapters)

  1. Front Matter

    Pages i-xxviii
  2. Algebraic Tensors

    1. Front Matter

      Pages 1-2
    2. Introduction

      • Wolfgang Hackbusch
      Pages 3-21
    3. Matrix Tools

      • Wolfgang Hackbusch
      Pages 23-47
    4. Algebraic Foundations of Tensor Spaces

      • Wolfgang Hackbusch
      Pages 49-94
  3. Functional Analysis of Tensor Spaces

    1. Front Matter

      Pages 95-96
    2. Banach Tensor Spaces

      • Wolfgang Hackbusch
      Pages 97-181
    3. General Techniques

      • Wolfgang Hackbusch
      Pages 183-200
    4. Minimal Subspaces

      • Wolfgang Hackbusch
      Pages 201-229
  4. Numerical Treatment

    1. Front Matter

      Pages 231-232
    2. r-Term Representation

      • Wolfgang Hackbusch
      Pages 233-256
    3. Tensor Subspace Representation

      • Wolfgang Hackbusch
      Pages 257-295
    4. r-Term Approximation

      • Wolfgang Hackbusch
      Pages 297-346
    5. Tensor Subspace Approximation

      • Wolfgang Hackbusch
      Pages 347-385
    6. Hierarchical Tensor Representation

      • Wolfgang Hackbusch
      Pages 387-451
    7. Matrix Product Systems

      • Wolfgang Hackbusch
      Pages 453-472
    8. Tensor Operations

      • Wolfgang Hackbusch
      Pages 473-505
    9. Tensorisation

      • Wolfgang Hackbusch
      Pages 507-540
    10. Multivariate Cross Approximation

      • Wolfgang Hackbusch
      Pages 541-558

About this book

Special numerical techniques are already needed to deal with n × n matrices for large n. Tensor data are of size n × n ×...× n=nd, where nd exceeds the computer memory by far. They appear for problems of high spatial dimensions. Since standard methods fail, a particular tensor calculus is needed to treat such problems. This monograph describes the methods by which tensors can be practically treated and shows how numerical operations can be performed. Applications include problems from quantum chemistry, approximation of multivariate functions, solution of partial differential equations, for example with stochastic coefficients, and more. In addition to containing corrections of the unavoidable misprints, this revised second edition includes new parts ranging from single additional statements to new subchapters. The book is mainly addressed to numerical mathematicians and researchers working with high-dimensional data. It also touches problems related to Geometric Algebra.

Keywords

  • high-dimensional problems
  • numerical tensor calculus
  • tensor formats
  • tensor spaces
  • tensorization

Authors and Affiliations

  • Max Planck Institute for Mathematics in the Sciences, Leipzig, Germany

    Wolfgang Hackbusch

About the author

Wolfgang Hackbusch is working in the field of numerical mathematics for partial differential equations and integral equations. He has published monographs, e.g., about the multi-grid method, about the numerical analysis of elliptic pdes, about iterative solution of large systems of equation, and about the technique of hierarchical matrices.

Bibliographic Information

Buying options

eBook USD 119.00
Price excludes VAT (USA)
  • ISBN: 978-3-030-35554-8
  • Instant PDF download
  • Readable on all devices
  • Own it forever
  • Exclusive offer for individuals only
  • Tax calculation will be finalised during checkout
Softcover Book USD 159.99
Price excludes VAT (USA)
Hardcover Book USD 159.99
Price excludes VAT (USA)