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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2402.14417 |
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| _version_ | 1866908372484227072 |
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| author | Lentz, Anna Wachsmuth, Daniel |
| author_facet | Lentz, Anna Wachsmuth, Daniel |
| contents | We investigate time-dependent optimization problems in fractional Sobolev spaces with the sparsity promoting $L^p$-pseudo norm for $0<p<1$ in the objective functional. In order to avoid computing the fractional Laplacian on the time-space cylinder $I\times Ω$, we introduce an auxiliary function $w$ on $Ω$ that is an upper bound for the function $u\in L^2(I\timesΩ)$. We prove existence and regularity results and derive a necessary optimality condition. This is done by smoothing the $L^p$-pseudo norm and by penalizing the inequality constraint regarding $u$ and $w$. The problem is solved numerically with an iterative scheme whose weak limit points satisfy a weaker form of the necessary optimality condition. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2402_14417 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Spatially sparse optimization problems in fractional order Sobolev spaces Lentz, Anna Wachsmuth, Daniel Optimization and Control 49K30, 49M20 We investigate time-dependent optimization problems in fractional Sobolev spaces with the sparsity promoting $L^p$-pseudo norm for $0<p<1$ in the objective functional. In order to avoid computing the fractional Laplacian on the time-space cylinder $I\times Ω$, we introduce an auxiliary function $w$ on $Ω$ that is an upper bound for the function $u\in L^2(I\timesΩ)$. We prove existence and regularity results and derive a necessary optimality condition. This is done by smoothing the $L^p$-pseudo norm and by penalizing the inequality constraint regarding $u$ and $w$. The problem is solved numerically with an iterative scheme whose weak limit points satisfy a weaker form of the necessary optimality condition. |
| title | Spatially sparse optimization problems in fractional order Sobolev spaces |
| topic | Optimization and Control 49K30, 49M20 |
| url | https://arxiv.org/abs/2402.14417 |