  This paper develops a probabilistic numerical method for solution of partial
differential equations (PDEs) and studies application of that method to
PDE-constrained inverse problems. This approach enables the solution of
challenging inverse problems whilst accounting, in a statistically principled
way, for the impact of discretisation error due to numerical solution of the
PDE. In particular, the approach confers robustness to failure of the numerical
PDE solver, with statistical inferences driven to be more conservative in the
presence of substantial discretisation error. Going further, the problem of
choosing a PDE solver is cast as a problem in the Bayesian design of
experiments, where the aim is to minimise the impact of solver error on
statistical inferences; here the challenge of non-linear PDEs is also
considered. The method is applied to parameter inference problems in which
discretisation error in non-negligible and must be accounted for in order to
reach conclusions that are statistically valid.
