Cayley Splitting for Second-Order Langevin Stochastic Partial Differential Equations

TL;DR

This paper introduces accurate, ergodic numerical methods for semilinear second-order Langevin SPDEs, leveraging Krein's theory to ensure stability in highly oscillatory Hamiltonian problems, with applications to Hamiltonian Monte Carlo.

Contribution

It presents novel geometric numerical methods for infinite-dimensional Hamiltonian SPDEs, utilizing Krein's theory to guarantee stability without preconditioning.

Findings

Methods are accurate and ergodic for second-order Langevin SPDEs.

Stable symplectic splitting schemes are achieved for highly oscillatory problems.

Applicable to Hamiltonian Monte Carlo on Hilbert spaces without preconditioning.

Abstract

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.