Non-reversible sampling schemes on submanifolds

TL;DR

This paper introduces and analyzes a non-reversible sampling scheme on submanifolds, demonstrating improved variance reduction over reversible methods through theoretical proofs and numerical experiments.

Contribution

It presents a novel non-reversible scheme for sampling on submanifolds, with proven consistency, error estimates, and superior asymptotic variance performance.

Findings

Scheme outperforms reversible methods in variance reduction

Proven consistency and error bounds for the scheme

Demonstrated effectiveness on a test example

Abstract

Calculating averages with respect to probability measures on submanifolds is often necessary in various application areas such as molecular dynamics, computational statistical mechanics and Bayesian statistics. In recent years, various numerical schemes have been proposed in the literature to study this problem based on appropriate reversible constrained stochastic dynamics. In this paper we present and analyse a non-reversible generalisation of the projection-based scheme developed by one of the authors [ESAIM: M2AN, 54 (2020), pp. 391-430]. This scheme consists of two steps - starting from a state on the submanifold, we first update the state using a non-reversible stochastic differential equation which takes the state away from the submanifold, and in the second step we project the state back onto the manifold using the long-time limit of an ordinary differential equation. We prove the consistency of this numerical scheme and provide quantitative error estimates for estimators based on finite-time running averages. Furthermore, we present theoretical analysis which shows that this scheme outperforms its reversible counterpart in terms of asymptotic variance. We demonstrate our findings on an illustrative test example.