Analysis of multiscale integrators for multiple attractors and irreversible Langevin samplers

TL;DR

This paper analyzes multiscale integrator schemes for stiff stochastic differential equations with multiple attractors, demonstrating their stability and convergence, and explores their application in improving Langevin samplers for faster equilibrium convergence.

Contribution

It provides a rigorous convergence analysis of multiscale integrators for SDEs with multiple attractors and applies these findings to enhance Langevin sampling methods.

Findings

Multiscale integrators stabilize numerical solutions for stiff SDEs.

Convergence of the integrators to diffusions on graphs is established.

Numerical experiments show improved sampling efficiency with irreversibility.

Abstract

We study multiscale integrator numerical schemes for a class of stiff stochastic differential equations (SDEs). We consider multiscale SDEs with potentially multiple attractors that behave as diffusions on graphs as the stiffness parameter goes to its limit. Classical numerical discretization schemes, such as the Euler-Maruyama scheme, become unstable as the stiffness parameter converges to its limit and appropriate multiscale integrators can correct for this. We rigorously establish the convergence of the numerical method to the related diffusion on graph, identifying the appropriate choice of discretization parameters. Theoretical results are supplemented by numerical studies on the problem of the recently developing area of introducing irreversibility in Langevin samplers in order to accelerate convergence to equilibrium.