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arXiv · 2404.02320

On Properties of Adjoint Systems for Evolutionary PDEs

Brian K. Tran·Ben S. Southworth·Melvin Leok
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Abstract

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.

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Brian K. Tran, Ben S. Southworth, Melvin Leok. 2024-04-02. On Properties of Adjoint Systems for Evolutionary PDEs. https://doi.org/10.1007/s00332-024-10071-1

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