During the last decades, reduced basis (RB) methods have been developed
to a wide methodology for model reduction of problems that are governed
by parametrized partial differential equations ($P^2DE$s ). In particular
equations of elliptic and parabolic type for linear, low polynomial
or monotonic nonlinearities have been treated successfully by RB
methods using ï¬nite element schemes. Due to the characteristic
ofï¬ine-online decomposition, the reduced models often become suitable
for a multi-query or real-time setting, where simulation results,
such as ï¬eld-variables or output estimates, can be approximated
reliably and rapidly for varying parameters. In the current study,
we address a certain class of time-dependent evolution schemes with
explicit discretization operators that are arbitrarily parameter
dependent. We extend the RB-methodology to these cases by applying
the empirical interpolation method to localized discretization operators.
The main technical ingredients are: (i) generation of a collateral
reduced basis modelling the effects of the discretization operator
under parameter variations in the ofï¬ine-phase and (ii) an online
simulation scheme based on a numerical subgrid and localized evaluations
of the evolution operator. We formulate an a-posteriori error estimator
for quantiï¬cation of the resulting reduced simulation error. Numerical
experiments on a parametrized convection problem, discretized with
a ï¬nite volume scheme, demonstrate the applicability of the model
reduction technique. We obtain a parametrized reduced model, which
enables parameter variation with fast simulation response. We quantify
the computational gain with respect to the non-reduced model and
investigate the error convergence.
%0 Report
%1 HOR07
%A Haasdonk, Bernard
%A Ohlberger, M.
%A Rozza, G.
%D 2007
%K anm from:britsteiner ians imported
%N 09/07 - N, FB 10
%T A Reduced Basis Method for Evolution Schemes with Parameter-Dependent Explicit Operators
%X During the last decades, reduced basis (RB) methods have been developed
to a wide methodology for model reduction of problems that are governed
by parametrized partial differential equations ($P^2DE$s ). In particular
equations of elliptic and parabolic type for linear, low polynomial
or monotonic nonlinearities have been treated successfully by RB
methods using ï¬nite element schemes. Due to the characteristic
ofï¬ine-online decomposition, the reduced models often become suitable
for a multi-query or real-time setting, where simulation results,
such as ï¬eld-variables or output estimates, can be approximated
reliably and rapidly for varying parameters. In the current study,
we address a certain class of time-dependent evolution schemes with
explicit discretization operators that are arbitrarily parameter
dependent. We extend the RB-methodology to these cases by applying
the empirical interpolation method to localized discretization operators.
The main technical ingredients are: (i) generation of a collateral
reduced basis modelling the effects of the discretization operator
under parameter variations in the ofï¬ine-phase and (ii) an online
simulation scheme based on a numerical subgrid and localized evaluations
of the evolution operator. We formulate an a-posteriori error estimator
for quantiï¬cation of the resulting reduced simulation error. Numerical
experiments on a parametrized convection problem, discretized with
a ï¬nite volume scheme, demonstrate the applicability of the model
reduction technique. We obtain a parametrized reduced model, which
enables parameter variation with fast simulation response. We quantify
the computational gain with respect to the non-reduced model and
investigate the error convergence.
@techreport{HOR07,
abstract = {During the last decades, reduced basis (RB) methods have been developed
to a wide methodology for model reduction of problems that are governed
by parametrized partial differential equations ($P^2DE$s ). In particular
equations of elliptic and parabolic type for linear, low polynomial
or monotonic nonlinearities have been treated successfully by RB
methods using ï¬nite element schemes. Due to the characteristic
ofï¬ine-online decomposition, the reduced models often become suitable
for a multi-query or real-time setting, where simulation results,
such as ï¬eld-variables or output estimates, can be approximated
reliably and rapidly for varying parameters. In the current study,
we address a certain class of time-dependent evolution schemes with
explicit discretization operators that are arbitrarily parameter
dependent. We extend the RB-methodology to these cases by applying
the empirical interpolation method to localized discretization operators.
The main technical ingredients are: (i) generation of a collateral
reduced basis modelling the effects of the discretization operator
under parameter variations in the ofï¬ine-phase and (ii) an online
simulation scheme based on a numerical subgrid and localized evaluations
of the evolution operator. We formulate an a-posteriori error estimator
for quantiï¬cation of the resulting reduced simulation error. Numerical
experiments on a parametrized convection problem, discretized with
a ï¬nite volume scheme, demonstrate the applicability of the model
reduction technique. We obtain a parametrized reduced model, which
enables parameter variation with fast simulation response. We quantify
the computational gain with respect to the non-reduced model and
investigate the error convergence.},
added-at = {2021-09-29T14:35:10.000+0200},
author = {Haasdonk, Bernard and Ohlberger, M. and Rozza, G.},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/27c62d7b2961a5864555e2b8dcb452128/mathematik},
file = {:PDF/HOR07_www_preprint_EOI_explicit_operators_before_ETNA.pdf:PDF},
groups = {haasdonk_misc, haasdonk_all_papers},
institution = {University of M{\"u}nster},
interhash = {0b876cfac7f9ad4d9eacc5aac369b8eb},
intrahash = {7c62d7b2961a5864555e2b8dcb452128},
keywords = {anm from:britsteiner ians imported},
note = {Accepted by ETNA.},
number = {09/07 - N, FB 10},
owner = {haasdonk},
timestamp = {2021-10-15T08:38:35.000+0200},
title = {A Reduced Basis Method for Evolution Schemes with Parameter-Dependent Explicit Operators},
year = 2007
}