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**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_... |

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