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In this paper we study the approximation of an optimal control problem for linear parabolic PDEs with model order reduction based on Proper Orthogonal Decomposition (POD-MOR). POD-MOR is a Galerkin approach where the basis functions are obtained upon information contained in time snapshots of the parabolic PDE related to given input data. In the present work we show that for POD-MOR in optimal control of parabolic equations it is important to have knowledge about the controlled system at the right time instances. We propose to determine the time instances (snapshot locations) by an a posteriori error control concept. The proposed method is based on a reformulation of the optimality system of the underlying optimal control problem as a second order in time and fourth order in space elliptic system which is approximated by a space-time finite element method. Finally, we present numerical tests to illustrate our approach and show the effectiveness of the method in comparison to existing approaches.
Alla, Alessandro 1 ; Grässle, Carmen 1 ; Hinze, Michael 1
@article{M2AN_2018__52_5_1847_0,
author = {Alla, Alessandro and Gr\"assle, Carmen and Hinze, Michael},
title = {A posteriori snapshot location for {POD} in optimal control of linear parabolic equations},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis },
pages = {1847--1873},
publisher = {EDP-Sciences},
volume = {52},
number = {5},
year = {2018},
doi = {10.1051/m2an/2018009},
zbl = {1417.49040},
mrnumber = {3880599},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018009/}
}
TY - JOUR AU - Alla, Alessandro AU - Grässle, Carmen AU - Hinze, Michael TI - A posteriori snapshot location for POD in optimal control of linear parabolic equations JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2018 SP - 1847 EP - 1873 VL - 52 IS - 5 PB - EDP-Sciences UR - http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018009/ DO - 10.1051/m2an/2018009 LA - en ID - M2AN_2018__52_5_1847_0 ER -
%0 Journal Article %A Alla, Alessandro %A Grässle, Carmen %A Hinze, Michael %T A posteriori snapshot location for POD in optimal control of linear parabolic equations %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2018 %P 1847-1873 %V 52 %N 5 %I EDP-Sciences %U http://geodesic.mathdoc.fr/articles/10.1051/m2an/2018009/ %R 10.1051/m2an/2018009 %G en %F M2AN_2018__52_5_1847_0
Alla, Alessandro; Grässle, Carmen; Hinze, Michael. A posteriori snapshot location for POD in optimal control of linear parabolic equations. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 52 (2018) no. 5, pp. 1847-1873. doi: 10.1051/m2an/2018009
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