Langevin dynamics with general kinetic energies

TL;DR

This paper investigates Langevin dynamics with non-standard kinetic energies to improve sampling efficiency, establishing convergence properties, developing stable numerical schemes, and applying them to optimize kinetic energy shapes and reduce metastability.

Contribution

It introduces new theoretical results on convergence with non-globally Lipschitz kinetic energies and develops stable, high-order numerical schemes with Metropolis corrections.

Findings

Proved exponential convergence to Boltzmann-Gibbs measure.

Developed stable, weak order two numerical schemes with Metropolis correction.

Demonstrated applications in optimizing kinetic energy and reducing metastability.

Abstract

We study Langevin dynamics with a kinetic energy different from the standard, quadratic one in order to accelerate the sampling of Boltzmann-Gibbs distributions. In particular, this kinetic energy can be non-globally Lipschitz, which raises issues for the stability of discretizations of the associated Langevin dynamics. We first prove the exponential convergence of the law of the continuous process to the Boltzmann-Gibbs measure by a hypocoercive approach, and characterize the asymptotic variance of empirical averages over trajectories. We next develop numerical schemes which are stable and of weak order two, by considering splitting strategies where the discretizations of the fluctuation/dissipation are corrected by a Metropolis procedure. We use the newly developped schemes for two applications: optimizing the shape of the kinetic energy for the so-called adaptively restrained Langevin dynamics (which considers perturbations of standard quadratic kinetic energies vanishing around the origin); and reducing the metastability of some toy models using non-globally Lipschitz kinetic energies.