Analysis of multiscale integrators for multiple attractors and irreversible Langevin samplers
For researchers in computational statistics and molecular dynamics, this provides a rigorous foundation for using multiscale integrators in irreversible Langevin sampling, though the results are incremental.
The paper analyzes multiscale integrators for stiff SDEs with multiple attractors, proving convergence to a diffusion on a graph. Numerical studies on irreversible Langevin samplers show that the method accelerates convergence to equilibrium.
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.