  In this work, the Bayesian approach to inverse problems is formulated in an
all-at-once setting. The advantages of the all-at-once formulation are known to
include the avoidance of a parameter-to-state map as well as numerical
improvements, especially when considering nonlinear problems. In the Bayesian
approach, prior knowledge is taken into account with the help of a prior
distribution. In addition, the error in the observation equation is formulated
by means of a distribution. This method naturally results in a whole posterior
distribution for the unknown target, not just point estimates. This allows for
further statistical analysis including the computation of credible intervals.
We combine the Bayesian setting with the all-at-once formulation, resulting in
a novel approach for investigating inverse problems. With this combination we
are able to chose a prior not only for the parameter, but also for the state
variable, which directly influences the parameter. Furthermore, errors not only
in the observation equation, but additionally, in the model can be taken into
account. %The aim of this approach is not only to accomplish reasonable
reconstructions of the unknown parameter but also to maximize the information
gained from measurements through combining it with prior knowledge, obtained
either from certain expertise or former investigation in the model. We analyze
this approach with the help of two linear standard examples, namely the inverse
source problem for the Poisson equation and the backwards heat equation, i.e. a
stationary and a time dependent problem. Appropriate function spaces and
derivation of adjoint operators are investigated. To assess the degree of
ill-posedness, we analyze the singular values of the corresponding all-at-once
forward operators. %as well as the convergence of the method. Finally, joint
priors are designed and numerically tested.
