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A weighted reduced basis method for parabolic PDEs with random data

Spannring, Christopher ; Ullmann, Sebastian ; Lang, Jens
Schäfer, Michael ; Behr, Marek ; Mehl, Miriam ; Wohlmuth, Barbara (eds.) (2018):
A weighted reduced basis method for parabolic PDEs with random data.
In: Lecture Notes in Computational Science and Engineering, 124, In: Recent Advances in Computational Engineering, pp. 145-161, Cham, Springer International Publishing, ISBN 978-3-319-93891-2,
DOI: 10.1007/978-3-319-93891-2_9,
[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.

Item Type: Book Section
Erschienen: 2018
Editors: Schäfer, Michael ; Behr, Marek ; Mehl, Miriam ; Wohlmuth, Barbara
Creators: Spannring, Christopher ; Ullmann, Sebastian ; Lang, Jens
Title: A weighted reduced basis method for parabolic PDEs with random data
Language: English
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.

Book Title: Recent Advances in Computational Engineering
Series: Lecture Notes in Computational Science and Engineering
Series Volume: 124
Place of Publication: Cham
Publisher: Springer International Publishing
ISBN: 978-3-319-93891-2
Divisions: Exzellenzinitiative
Exzellenzinitiative > Graduate Schools
Exzellenzinitiative > Graduate Schools > Graduate School of Computational Engineering (CE)
Exzellenzinitiative > Graduate Schools > Graduate School of Energy Science and Engineering (ESE)
04 Department of Mathematics
04 Department of Mathematics > Numerical Analysis and Scientific Computing
Date Deposited: 21 Dec 2017 08:58
DOI: 10.1007/978-3-319-93891-2_9
URL / URN: https://link.springer.com/chapter/10.1007/978-3-319-93891-2_...
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