# Reduced basis method for finite volume approximations of parametrized linear evolution equations

Bernard Haasdonk; Mario Ohlberger

ESAIM: Mathematical Modelling and Numerical Analysis (2008)

- Volume: 42, Issue: 2, page 277-302
- ISSN: 0764-583X

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topHaasdonk, Bernard, and Ohlberger, Mario. "Reduced basis method for finite volume approximations of parametrized linear evolution equations." ESAIM: Mathematical Modelling and Numerical Analysis 42.2 (2008): 277-302. <http://eudml.org/doc/250341>.

@article{Haasdonk2008,

abstract = {
The model order reduction methodology of reduced basis (RB)
techniques offers efficient treatment of parametrized partial differential
equations (P2DEs) by providing both approximate solution procedures and
efficient error estimates.
RB-methods have so far mainly been applied to finite element schemes
for elliptic and parabolic problems. In the current study
we extend the methodology to general linear evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations. The new theoretic contributions are the formulation of a reduced basis approximation scheme for these general evolution problems and
the derivation of rigorous a-posteriori error estimates in various norms. Algorithmically, an offline/online decomposition of the scheme and the error estimators is
realized in case of affine parameter-dependence of the problem.
This is the basis for a rapid online computation in case of multiple simulation requests.
We introduce a new offline basis-generation algorithm based on our
a-posteriori error estimator which combines ideas from existing approaches.
Numerical experiments for an instationary convection-diffusion problem
demonstrate the efficient applicability of the approach.
},

author = {Haasdonk, Bernard, Ohlberger, Mario},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Model reduction; reduced basis methods; finite volume methods; a-posteriori error estimates.; model reduction; a-posteriori error estimates},

language = {eng},

month = {3},

number = {2},

pages = {277-302},

publisher = {EDP Sciences},

title = {Reduced basis method for finite volume approximations of parametrized linear evolution equations},

url = {http://eudml.org/doc/250341},

volume = {42},

year = {2008},

}

TY - JOUR

AU - Haasdonk, Bernard

AU - Ohlberger, Mario

TI - Reduced basis method for finite volume approximations of parametrized linear evolution equations

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2008/3//

PB - EDP Sciences

VL - 42

IS - 2

SP - 277

EP - 302

AB -
The model order reduction methodology of reduced basis (RB)
techniques offers efficient treatment of parametrized partial differential
equations (P2DEs) by providing both approximate solution procedures and
efficient error estimates.
RB-methods have so far mainly been applied to finite element schemes
for elliptic and parabolic problems. In the current study
we extend the methodology to general linear evolution schemes such as finite volume schemes for parabolic and hyperbolic evolution equations. The new theoretic contributions are the formulation of a reduced basis approximation scheme for these general evolution problems and
the derivation of rigorous a-posteriori error estimates in various norms. Algorithmically, an offline/online decomposition of the scheme and the error estimators is
realized in case of affine parameter-dependence of the problem.
This is the basis for a rapid online computation in case of multiple simulation requests.
We introduce a new offline basis-generation algorithm based on our
a-posteriori error estimator which combines ideas from existing approaches.
Numerical experiments for an instationary convection-diffusion problem
demonstrate the efficient applicability of the approach.

LA - eng

KW - Model reduction; reduced basis methods; finite volume methods; a-posteriori error estimates.; model reduction; a-posteriori error estimates

UR - http://eudml.org/doc/250341

ER -

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## Citations in EuDML Documents

top- Bernard Haasdonk, Convergence Rates of the POD–Greedy Method
- Alexandre Janon, Maëlle Nodet, Clémentine Prieur, Certified reduced-basis solutions of viscous Burgers equation parametrized by initial and boundary values
- Mark Kärcher, Martin A. Grepl, A Posteriori Error Estimation for Reduced Order Solutions of Parametrized Parabolic Optimal Control Problems

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