On Properties of Adjoint Systems for Evolutionary PDEs
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866915233683996672 |
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| author | Tran, Brian K. Southworth, Ben S. Leok, Melvin |
| author_facet | Tran, Brian K. Southworth, Ben S. Leok, Melvin |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_02320 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On Properties of Adjoint Systems for Evolutionary PDEs Tran, Brian K. Southworth, Ben S. Leok, Melvin Optimization and Control Numerical Analysis 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. |
| title | On Properties of Adjoint Systems for Evolutionary PDEs |
| topic | Optimization and Control Numerical Analysis |
| url | https://arxiv.org/abs/2404.02320 |