TU Darmstadt / ULB / TUbiblio

A weighted reduced basis method for parabolic PDEs with random data

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: 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.

Item Type: Book Section
Erschienen: 2018
Editors: Schäfer, Michael and Behr, Marek and Mehl, Miriam and Wohlmuth, Barbara
Creators: Spannring, Christopher and Ullmann, Sebastian and 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.

Title of Book: Recent Advances in Computational Engineering
Series Name: Lecture Notes in Computational Science and Engineering
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
Official URL: https://link.springer.com/chapter/10.1007/978-3-319-93891-2_...
Related URLs:
Export:

Optionen (nur für Redakteure)

View Item View Item