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 ?nite element schemes. Due to the characteristic of?ine-online
decomposition, the reduced models often become suitable for a multi-query
or real-time setting, where simulation results, such as ?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 of?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 quanti?cation of
the resulting reduced simulation error. Numerical experiments on
a parametrized convection problem, discretized with a ?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 haasdonk2007reduced
%A Haasdonk, B.
%A Ohlberger, M.
%A Rozza, G.
%D 2007
%K imported vorlaeufig
%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 ?nite element schemes. Due to the characteristic of?ine-online
decomposition, the reduced models often become suitable for a multi-query
or real-time setting, where simulation results, such as ?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 of?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 quanti?cation of
the resulting reduced simulation error. Numerical experiments on
a parametrized convection problem, discretized with a ?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{haasdonk2007reduced,
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 ?nite element schemes. Due to the characteristic of?ine-online
decomposition, the reduced models often become suitable for a multi-query
or real-time setting, where simulation results, such as ?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 of?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 quanti?cation of
the resulting reduced simulation error. Numerical experiments on
a parametrized convection problem, discretized with a ?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 = {2018-07-20T10:54:15.000+0200},
author = {Haasdonk, B. and Ohlberger, M. and Rozza, G.},
biburl = {https://puma.ub.uni-stuttgart.de/bibtex/2486bbeed38237fa93b0598f7ddcfe227/mhartmann},
file = {:http\://www.mathematik.uni-stuttgart.de/fak8/ians/publications/files/HOR07_www_preprint_EOI_explicit_operators_before_ETNA.pdf:PDF},
institution = {University of M{\"u}nster},
interhash = {0b876cfac7f9ad4d9eacc5aac369b8eb},
intrahash = {486bbeed38237fa93b0598f7ddcfe227},
keywords = {imported vorlaeufig},
note = {Accepted by ETNA.},
number = {09/07 - N, FB 10},
owner = {haasdonk},
timestamp = {2018-07-20T08:54:15.000+0200},
title = {A Reduced Basis Method for Evolution Schemes with Parameter-Dependent
Explicit Operators},
year = 2007
}