Numerical integrators for confined Langevin dynamics

TL;DR

This paper develops and analyzes numerical integrators for underdamped Langevin dynamics with boundary reflections, achieving second-order accuracy in stochastic settings through composition schemes and demonstrating their effectiveness.

Contribution

It introduces novel composition-based integrators for confined Langevin dynamics that attain second-order weak accuracy with minimal gradient evaluations.

Findings

Second-order weak approximation achieved with a single gradient evaluation.

Numerical methods outperform first-order schemes in model problems.

Analysis confirms the stochastic integrators' efficiency and accuracy.

Abstract

We derive and analyze numerical methods for underdamped (kinetic) Langevin dynamics in a domain with elastic reflection at the boundary. First-order approximations are based on an Euler-type scheme incorporating collision-handling at the boundary. To achieve second order, composition schemes are derived based on decomposition of the generator into collisional drift, impulse, and stochastic momentum evolution. In a deterministic setting, this approach would typically lead to first-order approximation, even in symmetric compositions, but we find that the stochastic method can provide second-order weak approximation with a single gradient evaluation, both at finite times and in the ergodic limit. We provide analysis of this observation, as well as numerical demonstration, and we compare and contrast the performance of different variants of the integration method using model problems.