Adaptive Splitting Integrators for Enhancing Sampling Efficiency of Modified Hamiltonian Monte Carlo Methods in Molecular Simulation.

The modified Hamiltonian Monte Carlo (MHMC) methods, i.e., importance sampling methods that use modified Hamiltonians within a Hybrid Monte Carlo (HMC) framework, often outperform in sampling efficiency standard techniques such as molecular dynamics (MD) and HMC. The performance of MHMC may be enhanced further through the rational choice of the simulation parameters and by replacing the standard Verlet integrator with more sophisticated splitting algorithms. Unfortunately, it is not easy to identify the appropriate values of the parameters that appear in those algorithms. We propose a technique, that we call MAIA (Modified Adaptive Integration Approach), which, for a given simulation system and a given time step, automatically selects the optimal integrator within a useful family of two-stage splitting formulas. Extended MAIA (or e-MAIA) is an enhanced version of MAIA, which additionally supplies a value of the method-specific parameter that, for the problem under consideration, keeps the momentum acceptance rate at a user-desired level. The MAIA and e-MAIA algorithms have been implemented, with no computational overhead during simulations, in MultiHMC-GROMACS, a modified version of the popular software package GROMACS. Tests performed on well-known molecular models demonstrate the superiority of the suggested approaches over a range of integrators (both standard and recently developed), as well as their capacity to improve the sampling efficiency of GSHMC, a noticeable method for molecular simulation in the MHMC family. GSHMC combined with e-MAIA shows a remarkably good performance when compared to MD and HMC coupled with the appropriate adaptive integrators.

Compressible generalized hybrid Monte Carlo.

One of the most demanding calculations is to generate random samples from a specified probability distribution (usually with an unknown normalizing prefactor) in a high-dimensional configuration space. One often has to resort to using a Markov chain Monte Carlo method, which converges only in the limit to the prescribed distribution. Such methods typically inch through configuration space step by step, with acceptance of a step based on a Metropolis(-Hastings) criterion. An acceptance rate of 100% is possible in principle by embedding configuration space in a higher dimensional phase space and using ordinary differential equations. In practice, numerical integrators must be used, lowering the acceptance rate. This is the essence of hybrid Monte Carlo methods. Presented is a general framework for constructing such methods under relaxed conditions: the only geometric property needed is (weakened) reversibility; volume preservation is not needed. The possibilities are illustrated by deriving a couple of explicit hybrid Monte Carlo methods, one based on barrier-lowering variable-metric dynamics and another based on isokinetic dynamics.