A-posteriori snapshot location for POD in optimal control of linear parabolic equations

TL;DR

This paper introduces an a-posteriori error control method to optimally select snapshot locations in POD-based model order reduction for linear parabolic PDE control problems, improving approximation accuracy.

Contribution

It proposes a novel approach to determine snapshot times using an a-posteriori error estimate, enhancing POD-MOR effectiveness in optimal control of parabolic equations.

Findings

Improved accuracy in POD-MOR with optimized snapshot locations.

Numerical tests demonstrate the method's effectiveness.

Outperforms existing snapshot selection approaches.

Abstract

In this paper we study the approximation of an optimal control problem for linear para\-bolic 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. This 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 to show the effectiveness of the method in comparison to existing approaches.