%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 finite element schemes. Due to the characteristic offline-online decomposition, the reduced models often become suitable for a multi-query or real-time setting, where simulation results, such as field-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 offline-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 quantification of the resulting reduced simulation error. Numerical experiments on a parametrized convection problem, discretized with a finite 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 finite element schemes. Due to the characteristic offline-online decomposition, the reduced models often become suitable for a multi-query or real-time setting, where simulation results, such as field-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 offline-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 quantification of the resulting reduced simulation error. Numerical experiments on a parametrized convection problem, discretized with a finite 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 }