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Inverse Problems in a Bayesian Setting

  • Hermann G. MatthiesEmail author
  • Elmar Zander
  • Bojana V. Rosić
  • Alexander Litvinenko
  • Oliver Pajonk
Chapter
Part of the Computational Methods in Applied Sciences book series (COMPUTMETHODS, volume 41)

Abstract

In a Bayesian setting, inverse problems and uncertainty quantification (UQ)—the propagation of uncertainty through a computational (forward) model—are strongly connected. In the form of conditional expectation the Bayesian update becomes computationally attractive. We give a detailed account of this approach via conditional approximation, various approximations, and the construction of filters. Together with a functional or spectral approach for the forward UQ there is no need for time-consuming and slowly convergent Monte Carlo sampling. The developed sampling-free non-linear Bayesian update in form of a filter is derived from the variational problem associated with conditional expectation. This formulation in general calls for further discretisation to make the computation possible, and we choose a polynomial approximation. After giving details on the actual computation in the framework of functional or spectral approximations, we demonstrate the workings of the algorithm on a number of examples of increasing complexity. At last, we compare the linear and nonlinear Bayesian update in form of a filter on some examples.

Keywords

Posterior Distribution Monte Carlo Kalman Filter Conditional Expectation Uncertainty Quantification 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

Notes

Acknowledgments

This work was partially supported by the German research foundation “Deutsche Forschungsgemeinschaft” (DFG).

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Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Hermann G. Matthies
    • 1
    Email author
  • Elmar Zander
    • 1
  • Bojana V. Rosić
    • 1
  • Alexander Litvinenko
    • 2
  • Oliver Pajonk
    • 3
    • 4
  1. 1.TU BraunschweigBrunswickGermany
  2. 2.KAUSTThuwalSaudi Arabia
  3. 3.ElektrobitBraunschweigGermany
  4. 4.Schlumberger Information Solutions ASKjellerNorway

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