All-at-once formulations of inverse problems within the Bayesian appr...

The all-at-once-formulation for deterministic inverseproblems has recently been considered.

Our goal is to put this approach in a Bayesianframework. The

advantages of our approach are the additional choiceof a prior also for the state variable and

the possibility to take into account perturbations inthe model. By means of the backward heat

equation we test the all-at-once formulation inappropriate function spaces, derive adjoint

operators, investigate in different priors for thestate and do numerical tests.