Aggregation Methods for Computing Steady-States in Statistical Physics

TL;DR

This paper presents a new proof of convergence for the IAD multigrid method in Markov chain steady-state computation, offering insights into its efficiency and potential applications in statistical physics.

Contribution

It provides a rigorous convergence proof and convergence rate estimate for IAD, and explores its relevance to complex statistical physics methods for steady-state calculations.

Findings

Proves local convergence of IAD for Markov chains.

Provides a convergence rate estimate for IAD.

Discusses potential for efficient steady-state computation in statistical physics.

Abstract

We give a new proof of local convergence of a multigrid method called iterative aggregation/disaggregation (IAD) for computing steady-states of Markov chains. Our proof leads naturally to a precise and interpretable estimate of the asymptotic rate of convergence. We study IAD as a model of more complex methods from statistical physics for computing nonequilibrium steady-states, such as the nonequilibrium umbrella sampling method of Warmflash, et al. We explain why it may be possible to use methods like IAD to efficiently calculate steady-states of models in statistical physics and how to choose parameters to optimize efficiency.