Numerical Integrators for the Hybrid Monte Carlo Method

We construct numerical integrators for Hamiltonian problems that may advantageously replace the standard Verlet time-stepper within Hybrid Monte Carlo and related simulations. Past attempts have often aimed at boosting the order of accuracy of the integrator and/or reducing the size of its error constants; order and error constant are relevant concepts in the limit of vanishing step-length. We propose an alternative methodology based on the performance of the integrator when sampling from Gaussian distributions with not necessarily small step-lengths. We construct new splitting formulae that require two, three, or four force evaluations per time-step. Limited, proof-of-concept numerical experiments suggest that the new integrators may provide an improvement on the efficiency of the standard Verlet method, especially in problems with high dimensionality.This author's work was supported by project MTM2010-18246-C03-02 from Ministerio de Ciencia e Innovacion, Spain.Blanes Zamora, S.; Casas, F.; Sanz-Serna, JM. (2014). Numerical Integrators for the Hybrid Monte Carlo Method. SIAM Journal on Scientific Computing. 36(4):A1556-A1580. https://doi.org/10.1137/130932740A1556A158036