  We consider the reconstruction of a heterogeneous coefficient field in a
Robin boundary condition on an inaccessible part of the boundary in a Poisson
problem with an uncertain (or unknown) inhomogeneous conductivity field in the
interior of the domain. To account for model errors that stem from the
uncertainty in the conductivity coefficient, we treat the unknown conductivity
as a nuisance parameter and carry out approximative premarginalization over it,
and invert for the Robin coefficient field only. We approximate the related
modelling errors via the Bayesian approximation error (BAE) approach. The
uncertainty analysis presented here relies on a local linearization of the
parameter-to-observable map at the maximum a posteriori (MAP) estimates, which
leads to a normal (Gaussian) approximation of the parameter posterior density.
To compute the MAP point we apply an inexact Newton conjugate gradient approach
based on the adjoint methodology. The construction of the covariance is made
tractable by invoking a low-rank approximation of the data misfit component of
the Hessian. Two numerical experiments are considered: one where the prior
covariance on the conductivity is isotropic, and one where the prior covariance
on the conductivity is anisotropic. Results are compared to those based on
standard error models, with particular emphasis on the feasibility of the
posterior uncertainty estimates. We show that the BAE approach is a feasible
one in the sense that the predicted posterior uncertainty is consistent with
the actual estimation errors, while neglecting the related modelling error
yields infeasible estimates for the Robin coefficient. In addition, we
demonstrate that the BAE approach is approximately as computationally expensive
(measured in the number of PDE solves) as the conventional error approach.
