Information metrics for long-time errors in splitting schemes for stochastic dynamics and parallel KMC

TL;DR

This paper introduces an information-theoretic framework to analyze and improve long-time accuracy of splitting schemes in stochastic dynamics, especially for Parallel KMC algorithms, by using path-space relative entropy rate.

Contribution

It develops a novel information-theoretic approach to control long-time errors and guide scheme selection in stochastic numerical methods, extending beyond finite-time error estimates.

Findings

Path-space RER controls long-time error in splitting schemes.

A new information criterion for scheme comparison and design.

Improved scheme selection for Parallel KMC with better accuracy and efficiency.

Abstract

We propose an information-theoretic approach to analyze the long-time behavior of numerical splitting schemes for stochastic dynamics, focusing primarily on Parallel Kinetic Monte Carlo (KMC) algorithms.Established methods for numerical operator splittings provide error estimates in finite-time regimes, in terms of the order of the local error and the associated commutator. Path-space information-theoretic tools such as the relative entropy rate (RER) allow us to control long-time error through commutator calculations. Furthermore, they give rise to an a posteriori representation of the error which can thus be tracked in the course of a simulation. Another outcome of our analysis is the derivation of a path-space information criterion for comparison (and possibly design) of numerical schemes, in analogy to classical information criteria for model selection and discrimination. In the context of Parallel KMC, our analysis allows us to select schemes with improved numerical error and more efficient processor communication. We expect that such a path-space information perspective on numerical methods will be broadly applicable in stochastic dynamics, both for the finite and the long-time regime.