核心概念
Invariant measure-preserving transformed dynamics for efficient recovery of long-time behavior in Langevin dynamics.
要約
The article discusses the design of adaptive stepsize algorithms for Langevin dynamics, focusing on rescaling time to sample from an invariant measure efficiently. It covers overdamped and underdamped Langevin dynamics, emphasizing the incorporation of a correction term to preserve the invariant measure. The study includes model systems and Bayesian sampling problems with steep priors.
- Introduction to Underdamped Langevin dynamics.
- Designing an efficient monitor function for existence of solution.
- Continuous overdamped transformed dynamics with Gibbs distribution.
- Comparison of numerical integrators for Gibbs distribution simulation.
- Computational results and benefits of the method in one and two dimensions.
- Conclusions.
Key Insights:
- Importance of adaptive stepsize discretization in efficient sampling.
- Incorporation of correction terms for invariant measure preservation.
- Designing monitor functions for maintaining solution existence.
統計
"Given an appropriate monitor function which characterizes the numerical difficulty of the problem as a function of the state of the system..."
"When the trajectories of the system undergo sudden changes, or exhibit highly oscillatory modes, the stepsize must be small enough..."