Using Perturbed Underdamped Langevin Dynamics to Efficiently Sample from Probability Distributions

Area of Science:

• Computational Statistics
• Markov Chain Monte Carlo Methods

Background:

• Standard underdamped Langevin dynamics are widely used for sampling.
• Improving the efficiency and reducing the variance of these samplers is crucial for practical applications.

Purpose of the Study:

• To introduce and analyze novel Langevin samplers based on perturbations of standard underdamped Langevin dynamics.
• To demonstrate that these perturbed samplers can achieve improved properties, particularly reduced asymptotic variance.

Main Methods:

• Developing perturbed Langevin dynamics that preserve the invariant measure of the original dynamics.
• Conducting theoretical analysis for Gaussian target distributions.
• Performing numerical experiments with non-Gaussian target measures to validate theoretical findings.

Main Results:

• The proposed perturbed Langevin samplers exhibit reduced asymptotic variance compared to standard methods.
• Theoretical analysis confirms the effectiveness for Gaussian distributions.
• Numerical experiments validate performance on non-Gaussian distributions.

Conclusions:

• The novel perturbed Langevin samplers offer enhanced efficiency for statistical sampling.
• These methods provide a valuable alternative for MCMC applications requiring variance reduction.