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: alla@mat.puc-rio.br

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.