Cayley Splitting for Second-Order Langevin Stochastic Partial Differential Equations
This work addresses the need for stable and ergodic numerical integration of stochastic PDEs and infinite-dimensional Hamiltonian systems, which is crucial for Bayesian inference and molecular dynamics.
The paper develops accurate and ergodic numerical methods for semilinear second-order Langevin SPDEs, also providing geometric methods for infinite-dimensional Hamiltonian systems, enabling Hamiltonian Monte Carlo on Hilbert spaces without preconditioning. The methods leverage Krein's theory for stability of symplectic splitting schemes.
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.