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| Main Authors: | , , |
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| Format: | Preprint |
| Published: |
2023
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2303.00847 |
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| _version_ | 1866918037371748352 |
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| author | Castro, Paula Reyes, Juan Carlos De los Neitzel, Ira |
| author_facet | Castro, Paula Reyes, Juan Carlos De los Neitzel, Ira |
| contents | We carry out a rigorous analysis of four-dimensional variational data assimilation ($4D$-VAR) problems for linear and semilinear parabolic partial differential equations. Continuity of the state with respect to the spatial variable is required since pointwise observations of the state variable appear in the cost functional. Using maximal parabolic regularity tools, we prove this regularity for initial conditions with $L^β$-regularity guaranteed by control constraints, rather than Sobolev regularity of the controls ensured by artificial cost terms. We obtain existence of optimal controls and first order necessary optimality conditions for both the convex and nonconvex problem with spatial dimension $d=2,3$, as well as second order sufficient optimality conditions for the nonconvex problem for $d=2$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_00847 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Analysis of Four-Dimensional Variational Data Assimilation Problems in Low Regularity Spaces Castro, Paula Reyes, Juan Carlos De los Neitzel, Ira Optimization and Control 49J20, 49K20 We carry out a rigorous analysis of four-dimensional variational data assimilation ($4D$-VAR) problems for linear and semilinear parabolic partial differential equations. Continuity of the state with respect to the spatial variable is required since pointwise observations of the state variable appear in the cost functional. Using maximal parabolic regularity tools, we prove this regularity for initial conditions with $L^β$-regularity guaranteed by control constraints, rather than Sobolev regularity of the controls ensured by artificial cost terms. We obtain existence of optimal controls and first order necessary optimality conditions for both the convex and nonconvex problem with spatial dimension $d=2,3$, as well as second order sufficient optimality conditions for the nonconvex problem for $d=2$. |
| title | Analysis of Four-Dimensional Variational Data Assimilation Problems in Low Regularity Spaces |
| topic | Optimization and Control 49J20, 49K20 |
| url | https://arxiv.org/abs/2303.00847 |