The strong convergence of operator-splitting methods for the Langevin dynamics model

TL;DR

This paper investigates the strong convergence properties of operator-splitting methods for Langevin dynamics, proposing new algorithms with higher order accuracy verified through analysis and numerical experiments.

Contribution

The paper introduces a novel class of operator-splitting methods based on Kunita's solution, achieving strong orders up to 3 for Langevin dynamics.

Findings

Symmetric splitting methods only achieve strong order 1.

New methods based on Kunita's solution reach strong order 3.

Numerical results confirm theoretical convergence orders.

Abstract

We study the strong convergence of some operator-splitting methods for the Langevin dynamics model with additive noise. It will be shown that a direct splitting of deterministic and random terms, including the symmetric splitting methods, only offers strong convergence of order 1. To improve the order of strong convergence, a new class of operator-splitting methods based on Kunita's solution representation are proposed. We present stochastic algorithms with strong orders up to 3. Both mathematical analysis and numerical evidence are provided to verify the desired order of accuracy.