On Properties of Adjoint Systems for Evolutionary PDEs
<|reference_start|>On Properties of Adjoint Systems for Evolutionary PDEs: We investigate the geometric structure of adjoint systems associated with evolutionary partial differential equations at the fully continuous, semi-discrete, and fully discrete levels and the relations between these levels. We show that the adjoint system associated with an evolutionary partial differential equation has an infinite-dimensional Hamiltonian structure, which is useful for connecting the fully continuous, semi-discrete, and fully discrete levels. We subsequently address the question of discretize-then-optimize versus optimize-then-discrete for both semi-discretization and time integration, by characterizing the commutativity of discretize-then-optimize methods versus optimize-then-discretize methods uniquely in terms of an adjoint-variational quadratic conservation law. For Galerkin semi-discretizations and one-step time integration methods in particular, we explicitly construct these commuting methods by using structure-preserving discretization techniques.<|reference_end|>
@article{tran2024on,
    title={On Properties of Adjoint Systems for Evolutionary PDEs},
    author={Brian K. Tran, Ben S. Southworth, Melvin Leok},
    journal={arXiv preprint arXiv:2404.02320},
    year={2024},
    doi={10.1007/s00332-024-10071-1},
    archivePrefix={arXiv},
    eprint={2404.02320},
    primaryClass={math.OC cs.NA math.NA}
}