  We consider the Riemann manifold Hamiltonian Monte Carlo (RMHMC) method for
solving statistical inverse problems governed by partial differential equations
(PDEs). The power of the RMHMC method is that it exploits the geometric
structure induced by the PDE constraints of the underlying inverse problem.
Consequently, each RMHMC posterior sample is almost independent from the others
providing statistically efficient Markov chain simulation. We reduce the cost
of forming the Fisher information matrix by using a low rank approximation via
a randomized singular value decomposition technique. This is efficient since a
small number of Hessian-vector products are required. The Hessian-vector
product in turn requires only two extra PDE solves using the adjoint technique.
The results suggest RMHMC as a highly efficient simulation scheme for sampling
from PDE induced posterior measures.
