A~posteriori error analysis for optimization with PDE constraints
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arXiv
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| Main Authors: | , , , |
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| Format: | Preprint |
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2025
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| _version_ | 1866929762231910400 |
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| author | Gaspoz, Fernando Kreuzer, Christian Veeser, Andreas Wollner, Winnifried |
| author_facet | Gaspoz, Fernando Kreuzer, Christian Veeser, Andreas Wollner, Winnifried |
| contents | We consider finite element solutions to optimization problems, where the state depends on the possibly constrained control through a linear partial differential equation. Basing upon a reduced and rescaled optimality system, we derive a posteriori bounds capturing the approximation of the state, the adjoint state, the control and the observation. The upper and lower bounds show a gap, which grows with decreasing cost or Tikhonov regularization parameter. This growth is mitigated compared to previous results and can be countered by refinement if control and observation involve compact operators. Numerical results illustrate these properties for model problems with distributed and boundary control. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2503_13170 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A~posteriori error analysis for optimization with PDE constraints Gaspoz, Fernando Kreuzer, Christian Veeser, Andreas Wollner, Winnifried Numerical Analysis 49M25, 49M41, 65N15, 65N20, 65N30 We consider finite element solutions to optimization problems, where the state depends on the possibly constrained control through a linear partial differential equation. Basing upon a reduced and rescaled optimality system, we derive a posteriori bounds capturing the approximation of the state, the adjoint state, the control and the observation. The upper and lower bounds show a gap, which grows with decreasing cost or Tikhonov regularization parameter. This growth is mitigated compared to previous results and can be countered by refinement if control and observation involve compact operators. Numerical results illustrate these properties for model problems with distributed and boundary control. |
| title | A~posteriori error analysis for optimization with PDE constraints |
| topic | Numerical Analysis 49M25, 49M41, 65N15, 65N20, 65N30 |
| url | https://arxiv.org/abs/2503.13170 |