The nonlocal Kelvin principle and the dual approach to nonlocal control in the conduction coefficients
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arXiv
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| Hauptverfasser: | , |
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| Format: | Preprint |
| Veröffentlicht: |
2021
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| _version_ | 1866909090371862528 |
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| author | Evgrafov, Anton Bellido, Jose C. |
| author_facet | Evgrafov, Anton Bellido, Jose C. |
| contents | We explore the dual approach to nonlocal optimal design, specifically for a classical min-max problem which in this study is associated with a nonlocal scalar diffusion equation. We reformulate the optimal design problem utilizing a dual variational principle, which is expressed in terms of nonlocal two-point fluxes. We introduce the proper functional space framework to deal with this formulation, and establish its well-posedness. The key ingredient is the inf-sup (Ladyzhenskaya--Babuska--Brezzi) condition, which holds uniformly with respect to small nonlocal horizons. As a byproduct of this, we are able to prove convergence of nonlocal to local optimal design problems in a straightforward fashion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_06031 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | The nonlocal Kelvin principle and the dual approach to nonlocal control in the conduction coefficients Evgrafov, Anton Bellido, Jose C. Optimization and Control Analysis of PDEs 49J21, 49J45, 49J35, 80M50 We explore the dual approach to nonlocal optimal design, specifically for a classical min-max problem which in this study is associated with a nonlocal scalar diffusion equation. We reformulate the optimal design problem utilizing a dual variational principle, which is expressed in terms of nonlocal two-point fluxes. We introduce the proper functional space framework to deal with this formulation, and establish its well-posedness. The key ingredient is the inf-sup (Ladyzhenskaya--Babuska--Brezzi) condition, which holds uniformly with respect to small nonlocal horizons. As a byproduct of this, we are able to prove convergence of nonlocal to local optimal design problems in a straightforward fashion. |
| title | The nonlocal Kelvin principle and the dual approach to nonlocal control in the conduction coefficients |
| topic | Optimization and Control Analysis of PDEs 49J21, 49J45, 49J35, 80M50 |
| url | https://arxiv.org/abs/2106.06031 |