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This is followed by a reformulation of the optimality conditions into an elliptic PDE, which is only dependent on the adjoint state and can therefore be solved directly without the need for e.g. gradient methods or related iterative procedures. Furthermore, we derive an a-posteriori error estimation for this system as well as its initial condition. We utilize this estimate to formulate a procedure for the creation of an adaptive grid in time for the adjoint state. This is used for 4D-var data assimilation in order to identify suitable time points to take measurements.<\/jats:p>","DOI":"10.1007\/s10915-025-02995-x","type":"journal-article","created":{"date-parts":[[2025,7,28]],"date-time":"2025-07-28T04:38:49Z","timestamp":1753677529000},"update-policy":"https:\/\/doi.org\/10.1007\/springer_crossmark_policy","source":"Crossref","is-referenced-by-count":0,"title":["A time adaptive optimal control approach for 4D-var data assimilation problems governed by parabolic PDEs"],"prefix":"10.1007","volume":"104","author":[{"ORCID":"https:\/\/orcid.org\/0000-0003-0318-0740","authenticated-orcid":false,"given":"Carmen","family":"Gr\u00e4\u00dfle","sequence":"first","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]},{"ORCID":"https:\/\/orcid.org\/0009-0008-0248-7533","authenticated-orcid":false,"given":"Jannis","family":"Marquardt","sequence":"additional","affiliation":[],"role":[{"role":"author","vocabulary":"crossref"}]}],"member":"297","published-online":{"date-parts":[[2025,7,28]]},"reference":[{"key":"2995_CR1","doi-asserted-by":"publisher","unstructured":"Ainsworth, M., Oden, J.T.: A Posteriori Error Estimation in Finite Element Analysis. 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