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

TL;DR

This paper introduces a novel time-adaptive optimal control method for 4D-var data assimilation in parabolic PDEs, reformulating optimality conditions into an elliptic PDE for direct solution and adaptive time grid creation.

Contribution

It presents a reformulation of 4D-var data assimilation as an elliptic PDE, enabling direct solution without iterative methods, and develops an adaptive time grid based on a posteriori error estimates.

Findings

Reformulation into an elliptic PDE simplifies computations.

Adaptive time grid improves measurement point selection.

Error estimation enhances data assimilation accuracy.

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.