Towards time adaptive observations for model order reduction in data assimilation

In this work, we focus on two aspects of 4D-var data assimilation (DA) governed by parabolic partial differential equations (PDEs). First, we are interested on how to set up adaptive time grids for DA problems and in what extend DA benefits from it. Second, we study the application of model order reduction (MOR) for DA problems. Since solving DA problems requires to solve the involved PDEs repeatedly, the use of MOR techniques is an obvious approach. We apply the methods Proper Orthogonal Decomposition (POD) and Dynamic Mode Decomposition (DMD) and investigate whether the previously introduced adaptive time grid facilitates the MOR with respect to accuracy and efficiency. In order to construct an adaptive time grid, we interpret the DA problem in the context of optimal control and use a reformulation of the optimality conditions. Following Gong et al. (2012), we transferred their idea of deriving a-posteriori error estimates to the 4D-var problem in Graßle and Marquardt (2024). In this work, we extend our previous results where we derived an error estimate for the adjoint state by additionally considering an estimate for the state. The resulting time grid is used for MOR, which has already been done for distributed control problems in order to identify suitable snapshot locations, see Alla et al. (2016, 2018). We conclude with a numerical example.