An adaptive POD approximation method for the control of advection-diffusion equations

TL;DR

This paper introduces an adaptive Proper Orthogonal Decomposition (POD) method combined with Dynamic Programming to efficiently solve finite horizon optimal control problems for advection-diffusion equations, enhancing computational adaptivity and accuracy.

Contribution

It proposes a novel adaptive POD approach integrated with Hamilton-Jacobi equation approximation for improved control of advection-diffusion systems.

Findings

Effective basis function computation with error indicators

Adaptive time subdivision improves solution accuracy

Test problems demonstrate method's efficiency and accuracy

Abstract

We present an algorithm for the approximation of a finite horizon optimal control problem for advection-diffusion equations. The method is based on the coupling between an adaptive POD representation of the solution and a Dynamic Programming approximation scheme for the corresponding evolutive Hamilton-Jacobi equation. We discuss several features regarding the adaptivity of the method, the role of error estimate indicators to choose a time subdivision of the problem and the computation of the basis functions. Some test problems are presented to illustrate the method.