Multi-stage splitting integrators for sampling with modified Hamiltonian Monte Carlo methods
For practitioners using MHMC methods, this work provides easily implementable integrators that enhance accuracy and efficiency, though it is an incremental improvement over existing splitting techniques.
The paper introduces novel multi-stage splitting integrators for modified Hamiltonian Monte Carlo methods that significantly improve sampling performance, with numerical experiments showing substantial gains in both statistical and molecular dynamics problems.
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.