Concepts de base
This paper derives and analyzes efficient numerical methods for weak approximation of confined Langevin dynamics, including first-order Euler-type schemes and second-order splitting schemes. The methods demonstrate strong performance in sampling from stationary distributions with compact support.
Résumé
The paper focuses on the efficient numerical approximation of confined Langevin dynamics (CLD), which describe the motion of particles in a bounded domain with elastic collisions at the boundary. The authors derive and analyze two classes of numerical methods:
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First-order weak methods:
- [PAc] scheme: Performs an Euler-type update in momentum, followed by an elastic collision step.
- [AcP] scheme: Performs an elastic collision step, followed by an Euler-type update in momentum.
- These schemes are shown to have first-order weak convergence for CLD over finite time intervals, as well as for approximating ergodic limits of the ergodic confined Langevin dynamics (ECLD).
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Second-order weak splitting schemes:
- [OBAcBO], [BAcOAcB], [OAcBAcO] schemes: These schemes decompose the generator of the Markov process into components representing collisional drift, impulse, and stochastic momentum evolution, and then apply a symmetric splitting approach.
- Surprisingly, these schemes demonstrate second-order weak convergence for ECLD, despite the fact that the deterministic counterparts would typically only achieve first-order accuracy.
- The authors provide theoretical justification for this counterintuitive result, especially for the case of a half-space domain.
The paper also discusses the relationship between the confined Langevin dynamics and the reflected gradient SDE (overdamped Langevin dynamics with reflection), as well as comparisons with other known numerical integrators for CLD. The proposed methods are shown to be effective for sampling from stationary distributions with compact support, with applications in areas such as molecular dynamics, computational fluid dynamics, and optimization.