TU Darmstadt / ULB / TUbiblio

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

Ullmann, Sebastian ; Müller, Christopher ; Lang, Jens (2021)
Stochastic Galerkin Reduced Basis Methods for Parametrized Linear Convection−Diffusion−Reaction Equations.
In: Fluids, 6 (8)
doi: 10.3390/fluids6080263
Artikel, Bibliographie

Dies ist die neueste Version dieses Eintrags.

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: 2021
Autor(en): Ullmann, Sebastian ; Müller, Christopher ; Lang, Jens
Art des Eintrags: Bibliographie
Titel: Stochastic Galerkin Reduced Basis Methods for Parametrized Linear Convection−Diffusion−Reaction Equations
Sprache: Englisch
Publikationsjahr: 2021
Ort: Basel
Verlag: MDPI
Titel der Zeitschrift, Zeitung oder Schriftenreihe: Fluids
Jahrgang/Volume einer Zeitschrift: 6
(Heft-)Nummer: 8
Kollation: 24 Seiten
DOI: 10.3390/fluids6080263
Zugehörige Links:
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
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: 02 Aug 2024 13:15
Letzte Änderung: 02 Aug 2024 13:15
PPN:
Export:
Suche nach Titel in: TUfind oder in Google

Verfügbare Versionen dieses Eintrags

Frage zum Eintrag Frage zum Eintrag

Optionen (nur für Redakteure)
Redaktionelle Details anzeigen Redaktionelle Details anzeigen