  We give accurate and ergodic numerical methods for semilinear, second-order
Langevin stochastic partial differential equations (SPDE). As a byproduct, we
also give good geometric numerical methods for their infinite-dimensional
Hamiltonian counterpart. These methods are suitable for Hamiltonian Monte Carlo
on Hilbert spaces without preconditioning the underlying Hamiltonian dynamics.
A key tool in our approach is Krein's theory on strong stability of symplectic
maps, which gives us sufficient conditions for stability of symplectic
splitting schemes in highly oscillatory Hamiltonian problems.
