Title: Multi-stage splitting integrators for sampling with modified Hamiltonian Monte Carlo methods,
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.