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Parametric and Uncertainty Computations with Tensor Product Representations

  • Hermann G. Matthies
  • Alexander Litvinenko
  • Oliver Pajonk
  • Bojana V. Rosić
  • Elmar Zander
Part of the IFIP Advances in Information and Communication Technology book series (IFIPAICT, volume 377)

Abstract

Computational uncertainty quantification in a probabilistic setting is a special case of a parametric problem. Parameter dependent state vectors lead via association to a linear operator to analogues of covariance, its spectral decomposition, and the associated Karhunen-Loève expansion. From this one obtains a generalised tensor representation The parameter in question may be a tuple of numbers, a function, a stochastic process, or a random tensor field. The tensor factorisation may be cascaded, leading to tensors of higher degree. When carried on a discretised level, such factorisations in the form of low-rank approximations lead to very sparse representations of the high dimensional quantities involved. Updating of uncertainty for new information is an important part of uncertainty quantification. Formulated in terms or random variables instead of measures, the Bayesian update is a projection and allows the use of the tensor factorisations also in this case.

Keywords

uncertainty quantification parametric problems low-rank tensor approximation Bayesian updating 

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Copyright information

© IFIP International Federation for Information Processing 2012

Authors and Affiliations

  • Hermann G. Matthies
    • 1
  • Alexander Litvinenko
    • 1
  • Oliver Pajonk
    • 1
  • Bojana V. Rosić
    • 1
  • Elmar Zander
    • 1
  1. 1.Institute of Scientific ComputingTechnische UniversitätBraunschweigGermany

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