POD for optimal control of the Cahn-Hilliard system using spatially adapted snapshots

TL;DR

This paper develops a reduced-order modeling approach using POD with spatially adapted snapshots for efficient optimal control of a convective Cahn-Hilliard system, demonstrating significant computational speedups.

Contribution

It introduces a POD-based reduced-order model with adaptive finite element snapshots for the optimal control of the Cahn-Hilliard system, improving computational efficiency.

Findings

POD-ROM achieves large speedup factors over high-fidelity finite element methods.

Adaptive snapshot selection enhances the accuracy of the reduced-order model.

The method effectively controls the convective Cahn-Hilliard system with reduced computational cost.

Abstract

The present work considers the optimal control of a convective Cahn-Hilliard system, where the control enters through the velocity in the transport term. We prove the existence of a solution to the considered optimal control problem. For an efficient numerical solution, the expensive high-dimensional PDE systems are replaced by reduced-order models utilizing proper orthogonal decomposition (POD-ROM). The POD modes are computed from snapshots which are solutions of the governing equations which are discretized utilizing adaptive finite elements. The numerical tests show that the use of POD-ROM combined with spatially adapted snapshots leads to large speedup factors compared with a high-fidelity finite element optimization.