  The conventional way of formulating inverse problems such as identification
of a (possibly infinite dimensional) parameter, is via some forward operator,
which is the concatenation of the observation operator with the
parameter-to-state-map for the underlying model. Recently, all-at-once
formulations have been considered as an alternative to this reduced
formulation, avoiding the use of a parameter-to-state map, which would
sometimes lead to too restrictive conditions. Here the model and the
observation are considered simultaneously as one large system with the state
and the parameter as unknowns. A still more general formulation of inverse
problems, containing both the reduced and the all-at-once formulation, but also
the well-known and highly versatile so-called variational approach (also called
Kohn-Vogelius functional approach) as special cases, is to formulate the
inverse problem as a minimization problem -- instead of an equation -- for the
state and parameter. Regularization can be incorporated via imposing
constraints and/or adding regularization terms to the objective. In this paper,
after giving a motivation by formulating the electrical impedance tomography
(EIT) problem by means of the classical Kohn-Vogelius functional, we will dwell
on the regularization aspects for such variational formulations in an abstract
setting. Indeed, combination of regularization by constraints and by
penalization leads to new methods that are applicable without solving forward
problems. In particular, for the EIT problem we will consider a method
employing box constraints in a very natural manner to incorporate the
discrepancy principle for regularization parameter choice as well as a priori
information on the searched for conductivity.
