Spannring, Christopher and Ullmann, Sebastian and Lang, Jens Schäfer, Michael and Behr, Marek and Mehl, Miriam and Wohlmuth, Barbara (eds.) (2018):
A weighted reduced basis method for parabolic PDEs with random data.
In: Lecture Notes in Computational Science and Engineering, In: Recent Advances in Computational Engineering, Cham, Springer International Publishing, pp. 145-161, DOI: 10.1007/978-3-319-93891-2_9,
[Online-Edition: https://link.springer.com/chapter/10.1007/978-3-319-93891-2_...],
[Book Section]

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.

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