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Low Rank Approximation of Multidimensional Data

  • Mejdi AzaïezEmail author
  • Lucas Lestandi
  • Tomás Chacón Rebollo
Chapter
Part of the CISM International Centre for Mechanical Sciences book series (CISM, volume 592)

Abstract

In the last decades, numerical simulation has experienced tremendous improvements driven by massive growth of computing power. Exascale computing has been achieved this year and will allow solving ever more complex problems. But such large systems produce colossal amounts of data which leads to its own difficulties. Moreover, many engineering problems such as multiphysics or optimisation and control, require far more power that any computer architecture could achieve within the current scientific computing paradigm. In this chapter, we propose to shift the paradigm in order to break the curse of dimensionality by introducing decomposition to reduced data. We present an extended review of data reduction techniques and intends to bridge between applied mathematics community and the computational mechanics one. The chapter is organized into two parts. In the first one bivariate separation is studied, including discussions on the equivalence of proper orthogonal decomposition (POD, continuous framework) and singular value decomposition (SVD, discrete matrices). Then, in the second part, a wide review of tensor formats and their approximation is proposed. Such work has already been provided in the literature but either on separate papers or into a pure applied mathematics framework. Here, we offer to the data enthusiast scientist a description of Canonical, Tucker, Hierarchical and Tensor train formats including their approximation algorithms. When it is possible, a careful analysis of the link between continuous and discrete methods will be performed.

Keywords

Data reduction Model Reduction Singular Values Decomposition Data MOR POD HOSVD Low rank approximation tensors Tensor train 

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Copyright information

© CISM International Centre for Mechanical Sciences 2019

Authors and Affiliations

  • Mejdi Azaïez
    • 1
    Email author
  • Lucas Lestandi
    • 2
    • 3
  • Tomás Chacón Rebollo
    • 4
  1. 1.Bordeaux Institut National Polytechnique de BordeauxBordeauxFrance
  2. 2.Université de BordeauxBordeauxFrance
  3. 3.Institut de Mécanique et d’IngénierieBordeauxFrance
  4. 4.Instituto de Matemáticas de la Universidad de Sevilla - IMUSSevillaSpain

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