Rigorous analysis of relative performance of IP-transformed Langevin splitting schemes

Develop a rigorous analysis that explains the relative performance of the splitting schemes proposed for the invariant measure-preserving transformed underdamped Langevin dynamics, specifically analyzing accuracy and stability to clarify why compositions such as hat-A hat-B hat-O hat-B hat-A, tilde-A tilde-B tilde-O tilde-B tilde-A, and the invariant-preserving transformed stochastic Verlet position method perform better than schemes such as hat-O hat-B hat-A hat-B hat-O and tilde-O tilde-B tilde-A tilde-B tilde-O.

Background

The paper introduces invariant measure-preserving (IP) transformed SDEs for overdamped and underdamped Langevin dynamics using a state-dependent monitor function g(x) and a correction term to preserve the Gibbs distribution. For the underdamped case, the authors construct multiple splitting schemes by distributing the correction term between the force (B) and Ornstein–Uhlenbeck (O) components and using an implicit midpoint method for the position update (A), yielding variants denoted with hats and tildes.

In numerical experiments on a modified harmonic potential, some IP-transformed splitting schemes (e.g., hat-A hat-B hat-O hat-B hat-A, tilde-A tilde-B tilde-O tilde-B tilde-A, and an IP-transformed stochastic Verlet position method) show better accuracy than others (e.g., hat-O hat-B hat-A hat-B hat-O and tilde-O tilde-B tilde-A tilde-B tilde-O). The authors identify the need for a rigorous theoretical explanation of these observed performance differences.