A residual based snapshot location strategy for POD in distributed optimal control of linear parabolic equations

TL;DR

This paper introduces a residual-based snapshot location strategy for POD in distributed optimal control of linear parabolic PDEs, improving the selection of time instances for basis functions and enhancing model reduction accuracy.

Contribution

It proposes an a-posteriori error control method for optimal snapshot selection based on a reformulated elliptic system, advancing POD-MOR effectiveness in control problems.

Findings

The proposed snapshot strategy improves model accuracy.

Numerical tests demonstrate the method's effectiveness.

The approach outperforms existing snapshot selection methods.

Abstract

In this paper we study the approximation of a distributed 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. For the determination of the time instances (snapshot locations) we propose an a-posteriori error control concept which 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.