  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.
