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Stochastic Galerkin reduced basis methods for parametrized linear elliptic PDEs

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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