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
Artikel, Zweitveröffentlichung, Verlagsversion
 | Es ist eine neuere Version dieses Eintrags verfügbar. |
Kurzbeschreibung (Abstract)
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.
| Typ des Eintrags: | Artikel |
|---|---|
| Erschienen: | 2023 |
| Autor(en): | Ullmann, Sebastian ; Müller, Christopher ; Lang, Jens |
| Art des Eintrags: | Zweitveröffentlichung |
| Titel: | Stochastic Galerkin Reduced Basis Methods for Parametrized Linear Convection−Diffusion−Reaction Equations |
| Sprache: | Englisch |
| Publikationsjahr: | 14 November 2023 |
| Ort: | Darmstadt |
| Publikationsdatum der Erstveröffentlichung: | 2021 |
| Ort der Erstveröffentlichung: | Basel |
| Verlag: | MDPI |
| Titel der Zeitschrift, Zeitung oder Schriftenreihe: | Fluids |
| Jahrgang/Volume einer Zeitschrift: | 6 |
| (Heft-)Nummer: | 8 |
| Kollation: | 24 Seiten |
| DOI: | 10.26083/tuprints-00019561 |
| URL / URN: | https://tuprints.ulb.tu-darmstadt.de/19561 |
| Zugehörige Links: | |
| Herkunft: | Zweitveröffentlichung DeepGreen |
| Kurzbeschreibung (Abstract): | 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. |
| Freie Schlagworte: | 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: | Verlagsversion |
| URN: | urn:nbn:de:tuda-tuprints-195616 |
| Zusätzliche Informationen: | This article belongs to the Special Issue Reduced Order Models for Computational Fluid Dynamics |
| Sachgruppe der Dewey Dezimalklassifikatin (DDC): | 500 Naturwissenschaften und Mathematik > 510 Mathematik |
| Fachbereich(e)/-gebiet(e): | 04 Fachbereich Mathematik 04 Fachbereich Mathematik > Numerik und wissenschaftliches Rechnen |
| Hinterlegungsdatum: | 14 Nov 2023 13:43 |
| Letzte Änderung: | 15 Nov 2023 08:27 |
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| Suche nach Titel in: | TUfind oder in Google |
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- Stochastic Galerkin Reduced Basis Methods for Parametrized Linear Convection−Diffusion−Reaction Equations. (deposited 14 Nov 2023 13:43) [Gegenwärtig angezeigt]
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