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

The all-at-once-formulation for deterministic inverse problems has recently been considered. Our goal is to put this approach in a Bayesian framework. The advantages of our approach are the additional choice of a prior also for the state variable and the possibility to take into account perturbations in the model. By means of the inverse source problem and the backward heat equation we test the all-at-once formulation in appropriate function spaces, derive adjoint operators, investigate in different priors for the state and do numerical tests.