A time adaptive optimal control approach for 4D-var data assimilation problems governed by parabolic PDEs

2024/07/30 by Carmen Gräßle, Gräßle, Carmen, Jannis Marquardt +1
Earth and Planetary Sciences · #49J20 #49K20 #49M41 #65M15 #65M50 #FOS: Mathematics #Meteorological Phenomena and Simulations #Optimization and Control (math.OC)

paper · pdf · doi:10.48550/arxiv.2407.20866

openalex publication_date 2024/07/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We interpret the 4D-var data assimilation problem for a parabolic partial differential equation (PDE) in the context of optimal control and revisit the process of deriving optimality conditions for an initial control problem. 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.

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