| Issue |
ESAIM: M2AN
Volume 52, Number 5, September–October 2018
|
|
|---|---|---|
| Page(s) | 1847 - 1873 | |
| DOI | https://doi.org/10.1051/m2an/2018009 | |
| Published online | 28 November 2018 | |
A posteriori snapshot location for POD in optimal control of linear parabolic equations★
1
PUC-Rio, Department of Mathematics,
Rua Marques de Sao Vicente, 225,
Gavea,
22451-900
Rio de Janeiro, Brazil.
2
University of Hamburg, Department of Mathematics,
20146
Hamburg, Germany.
* Corresponding author: This email address is being protected from spambots. You need JavaScript enabled to view it.
Received:
29
August
2016
Accepted:
15
January
2018
Abstract
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.
Mathematics Subject Classification: 49J20 / 65N12 / 78M34
Key words: Optimal control / model order reduction / proper orthogonal decomposition / optimal snapshot location
We would like to thank Z.J. Zhou from Shandong Normal University, China for providing the data and code of the space-time approximation in Gong et al. (2012). The first author also acknowledges the support of US Department of Energy (grant number DE-SC0009324).
© EDP Sciences, SMAI 2018
Current usage metrics show cumulative count of Article Views (full-text article views including HTML views, PDF and ePub downloads, according to the available data) and Abstracts Views on Vision4Press platform.
Data correspond to usage on the plateform after 2015. The current usage metrics is available 48-96 hours after online publication and is updated daily on week days.
Initial download of the metrics may take a while.