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**Ullmann, Sebastian and Lang, Jens** (2018):

*Stochastic Galerkin reduced basis methods for parametrized linear elliptic PDEs.*

In: SIAM / ASA Journal on Uncertainty Quantification, [Online-Edition: https://arxiv.org/abs/1812.08519],

[Article]

## Abstract

We consider the estimation of parameter-dependent statistics of functional outputs of elliptic boundary value problems with parametrized random and deterministic inputs. For a given value of the deterministic parameter, a stochastic Galerkin finite element (SGFE) method can estimate the corresponding expectation and variance of a linear output at the cost of a single solution of a large block-structured linear algebraic system of equations. We propose a stochastic Galerkin reduced basis (SGRB) method as a means to lower the computational burden when statistical outputs are required for a large number of deterministic parameter queries. To derive an SGRB model, we project the spatial-stochastic weak solution of a parameter-dependent SGFE model onto a POD reduced basis generated from snapshots of the SGFE solution at representative values of the parameter. We propose residual-corrected estimates of the parameter-dependent expectation and variance of linear functional outputs and provide respective computable error bounds. We test the SGRB method numerically for a convection-diffusion-reaction problem, choosing the convective velocity as a deterministic parameter and the parametrized reactivity field as a random input. Compared to a standard reduced basis model embedded in a Monte Carlo sampling procedure, the SGRB model requires a similar number of reduced basis functions to meet a given tolerance requirement. However, only a single run of the SGRB model suffices to estimate a statistical output for a new deterministic parameter value, while the standard reduced basis model must be solved for each Monte Carlo sample.

Item Type: | Article |
---|---|

Erschienen: | 2018 |

Creators: | Ullmann, Sebastian and Lang, Jens |

Title: | Stochastic Galerkin reduced basis methods for parametrized linear elliptic PDEs |

Language: | German |

Abstract: | We consider the estimation of parameter-dependent statistics of functional outputs of elliptic boundary value problems with parametrized random and deterministic inputs. For a given value of the deterministic parameter, a stochastic Galerkin finite element (SGFE) method can estimate the corresponding expectation and variance of a linear output at the cost of a single solution of a large block-structured linear algebraic system of equations. We propose a stochastic Galerkin reduced basis (SGRB) method as a means to lower the computational burden when statistical outputs are required for a large number of deterministic parameter queries. To derive an SGRB model, we project the spatial-stochastic weak solution of a parameter-dependent SGFE model onto a POD reduced basis generated from snapshots of the SGFE solution at representative values of the parameter. We propose residual-corrected estimates of the parameter-dependent expectation and variance of linear functional outputs and provide respective computable error bounds. We test the SGRB method numerically for a convection-diffusion-reaction problem, choosing the convective velocity as a deterministic parameter and the parametrized reactivity field as a random input. Compared to a standard reduced basis model embedded in a Monte Carlo sampling procedure, the SGRB model requires a similar number of reduced basis functions to meet a given tolerance requirement. However, only a single run of the SGRB model suffices to estimate a statistical output for a new deterministic parameter value, while the standard reduced basis model must be solved for each Monte Carlo sample. |

Journal or Publication Title: | SIAM / ASA Journal on Uncertainty Quantification |

Divisions: | DFG-Collaborative Research Centres (incl. Transregio) DFG-Collaborative Research Centres (incl. Transregio) > Transregios DFG-Collaborative Research Centres (incl. Transregio) > Transregios > TRR 154 Mathematical Modelling, Simulation and Optimization using the Example of Gas Networks 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: | 20 Dec 2018 12:42 |

Official URL: | https://arxiv.org/abs/1812.08519 |

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