  Parameter identification problems typically consist of a model equation, e.g.
a (system of) ordinary or partial differential equation(s), and the observation
equation. In the conventional reduced setting, the model equation is eliminated
via the parameter-to-state map. Alternatively, one might consider both sets of
equations (model and observations) as one large system, to which some
regularization method is applied. The choice of the formulation (reduced or
all-at-once) can make a large difference computationally, depending on which
regularization method is used: Whereas almost the same optimality system arises
for the reduced and the all-at-once Tikhonov method, the situation is different
for iterative methods, especially in the context of nonlinear models. In this
paper we will exemplarily provide some convergence results for all-at-once
versions of variational, Newton type and gradient based regularization methods.
Moreover we will compare the implementation requirements for the respective
all-at-one and reduced versions and provide some numerical comparison.
