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A Weighted Reduced Basis Method for Parabolic PDEs with Random Data

  • Christopher SpannringEmail author
  • Sebastian Ullmann
  • Jens Lang
Conference paper
Part of the Lecture Notes in Computational Science and Engineering book series (LNCSE, volume 124)

Abstract

This work considers a weighted POD-greedy method to estimate statistical outputs parabolic PDE problems with parametrized random data. The key idea of weighted reduced basis methods is to weight the parameter-dependent error estimate according to a probability measure in the set-up of the reduced space. The error of stochastic finite element solutions is usually measured in a root mean square sense regarding their dependence on the stochastic input parameters. An orthogonal projection of a snapshot set onto a corresponding POD basis defines an optimum reduced approximation in terms of a Monte Carlo discretization of the root mean square error. The errors of a weighted POD-greedy Galerkin solution are compared against an orthogonal projection of the underlying snapshots onto a POD basis for a numerical example involving thermal conduction. In particular, it is assessed whether a weighted POD-greedy solutions is able to come significantly closer to the optimum than a non-weighted equivalent. Additionally, the performance of a weighted POD-greedy Galerkin solution is considered with respect to the mean absolute error of an adjoint-corrected functional of the reduced solution.

Keywords

Weighted reduced basis method Uncertainty quantification Model order reduction Proper orthogonal decomposition Weighted POD-greedy 

Notes

Acknowledgements

This work is supported by the ‘Excellence Initiative’ of the German federal and state governments and the Graduate School of Computational Engineering at the Technische Universität Darmstadt.

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Copyright information

© Springer International Publishing AG, part of Springer Nature 2018

Authors and Affiliations

  • Christopher Spannring
    • 1
    • 2
    Email author
  • Sebastian Ullmann
    • 1
    • 2
  • Jens Lang
    • 1
    • 2
  1. 1.Graduate School of Computational EngineeringTechnische Universität DarmstadtDarmstadtGermany
  2. 2.Department of MathematicsTechnische Universität DarmstadtDarmstadtGermany

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