POD for Optimal Control of the Cahn-Hilliard System Using Spatially Adapted Snapshots
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.
We like to thank Christian Kahle for providing many libraries which we could use for the coding. The first author gratefully acknowledges the financial support by the DFG through the priority program SPP 1962. The third author gratefully acknowledges the financial support by the DFG through the Collaborative Research Center SFB/TRR 181.
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