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Stochastic Galerkin Reduced Basis Methods for Parametrized Linear Convection−Diffusion−Reaction Equations

Ullmann, Sebastian ; Müller, Christopher ; Lang, Jens (2023)
Stochastic Galerkin Reduced Basis Methods for Parametrized Linear Convection−Diffusion−Reaction Equations.
In: Fluids, 2021, 6 (8)
doi: 10.26083/tuprints-00019561
Article, Secondary publication, Publisher's Version

Copyright Information: CC BY 4.0 International - Creative Commons, Attribution.

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Item Type: Article
Type of entry: Secondary publication
Title: Stochastic Galerkin Reduced Basis Methods for Parametrized Linear Convection−Diffusion−Reaction Equations
Language: English
Date: 14 November 2023
Place of Publication: Darmstadt
Year of primary publication: 2021
Place of primary publication: Basel
Publisher: MDPI
Journal or Publication Title: Fluids
Volume of the journal: 6
Issue Number: 8
Collation: 24 Seiten
DOI: 10.26083/tuprints-00019561
Corresponding Links:
Origin: Secondary publication DeepGreen

We consider the estimation of parameter-dependent statistics of functional outputs of steady-state convection–diffusion–reaction equations with parametrized random and deterministic inputs in the framework of linear elliptic partial differential equations. For a given value of the deterministic parameter, a stochastic Galerkin finite element (SGFE) method can estimate the statistical moments of interest of a linear output at the cost of solving a single, large, block-structured linear 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. Our working assumption is that we have access to the computational resources necessary to set up such a reduced-order model for a spatial-stochastic weak formulation of the parameter-dependent model equations. In this scenario, the complexity of evaluating the SGRB model for a new value of the deterministic parameter only depends on the reduced dimension. To derive an SGRB model, we project the spatial-stochastic weak solution of a parameter-dependent SGFE model onto a reduced basis generated by a proper orthogonal decomposition (POD) of snapshots of SGFE solutions 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 or diffusivity 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.

Uncontrolled Keywords: model order reduction, proper orthogonal decomposition, stochastic galerkin, finite elements, parametrized partial differential equation, Monte Carlo, reduced basis method, MSC: 65C30, 65N30, 65N35, 60H35, 35R60
Status: Publisher's Version
URN: urn:nbn:de:tuda-tuprints-195616
Additional Information:

This article belongs to the Special Issue Reduced Order Models for Computational Fluid Dynamics

Classification DDC: 500 Science and mathematics > 510 Mathematics
Divisions: 04 Department of Mathematics > Numerical Analysis and Scientific Computing
Date Deposited: 14 Nov 2023 13:43
Last Modified: 17 Nov 2023 10:39
SWORD Depositor: Deep Green
URI: https://tuprints.ulb.tu-darmstadt.de/id/eprint/19561
PPN: 513280715
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