Multi-stage splitting integrators for sampling with modified Hamiltonian Monte Carlo methods

TL;DR

This paper introduces new multi-stage splitting integrators for Modified Hamiltonian Monte Carlo (MHMC) methods, improving accuracy and efficiency in sampling for statistical and molecular dynamics applications.

Contribution

The paper develops novel splitting integrators for MHMC, optimizing their performance by minimizing energy errors and analyzing their modified Hamiltonians.

Findings

Significant improvement in sampling performance with new integrators

Enhanced accuracy in statistical and molecular dynamics problems

Ease of implementation due to splitting algorithm family

Abstract

Modified Hamiltonian Monte Carlo (MHMC) methods combine the ideas behind two popular sampling approaches: Hamiltonian Monte Carlo (HMC) and importance sampling. As in the HMC case, the bulk of the computational cost of MHMC algorithms lies in the numerical integration of a Hamiltonian system of differential equations. We suggest novel integrators designed to enhance accuracy and sampling performance of MHMC methods. The novel integrators belong to families of splitting algorithms and are therefore easily implemented. We identify optimal integrators within the families by minimizing the energy error or the average energy error. We derive and discuss in detail the modified Hamiltonians of the new integrators, as the evaluation of those Hamiltonians is key to the efficiency of the overall algorithms. Numerical experiments show that the use of the new integrators may improve very significantly the sampling performance of MHMC methods, in both statistical and molecular dynamics problems.