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

Adaptive multi-stage integrators for optimal energy conservation in molecular simulations

We introduce a new Adaptive Integration Approach (AIA) to be used in a wide range of molecular simulations. Given a simulation problem and a step size, the method automatically chooses the optimal scheme out of an available family of numerical integrators. Although we focus on two-stage splitting integrators, the idea may be used with more general families. In each instance, the system-specific integrating scheme identified by our approach is optimal in the sense that it provides the best conservation of energy for harmonic forces. The AIA method has been implemented in the BCAM-modified GROMACS software package. Numerical tests in molecular dynamics and hybrid Monte Carlo simulations of constrained and unconstrained physical systems show that the method successfully realises the fail-safe strategy. In all experiments, and for each of the criteria employed, the AIA is at least as good as, and often significantly outperforms the standard Verlet scheme, as well as fixed parameter, optimized two-stage integrators. In particular, the sampling efficiency found in simulations using the AIA is up to 5 times better than the one achieved with other tested schemes

Palindromic 3-stage splitting integrators, a roadmap

The implementation of multi-stage splitting integrators is essentially the same as the implementation of the familiar Strang/Verlet method. Therefore multi-stage formulas may be easily incorporated into software that now uses the Strang/Verlet integrator. We study in detail the two-parameter family of palindromic, three-stage splitting formulas and identify choices of parameters that may outperform the Strang/Verlet method. One of these choices leads to a method of effective order four suitable to integrate in time some partial differential equations. Other choices may be seen as perturbations of the Strang method that increase efficiency in molecular dynamics simulations and in Hybrid Monte Carlo sampling.Comment: 20 pages, 8 figures, 2 table

Adapting Hybrid Monte Carlo methods for solving complex problems in life and materials sciences

Efficient sampling is the key to success of molecular simulation of complex physical systems. Still, a unique recipe for achieving this goal is unavailable. Hybrid Monte Carlo (HMC) is a promising sampling tool offering a smart, free of discretization errors, propagation in phase space, rigorous temperature control, and flexibility. However, its inability to provide dynamical information and its weakness in simulations of reasonably large systems do not allow HMC to become a sampler of choice in molecular simulation of complex systems. In this thesis, we show that performance of HMC can be dramatically improved by introducing in the method the splitting numerical integrators and importance sampling. We propose a novel splitting integration scheme called Adaptive Integration Approach or AIA, which leads to very promising improvements in accuracy and sampling in HMC simulations. Given a simulation problem and a time step, AIA automatically chooses the optimal scheme out of the family of two-stage splitting integrators. A system-specific integrator identified by our approach is optimal in the sense that it provides the best conservation of energy for harmonic forces. The role of importance sampling on the performance of HMC is studied through the modified Hamiltonian Monte Carlo (MHMC) methods, sampling with respect to a modified or shadow Hamiltonian. The particular attention is paid to Generalized Shadow Hybrid Monte Carlo (GSHMC), introduced by Akhmatskaya and Reich in 2008. To improve the performance of MHMC in general and GSHMC in particular, we develop and test the new multi-stage splitting integrators, specially formulated for sampling with respect to modified Hamiltonians. The novel adaptive two-stage integration approach or MAIA, specifically derived for MHMC is presented. We also discuss in detail the adaptation of GSHMC to the NPT ensemble and provide the thorough analysis of its performance. Moreover, for the first time, we formulate GSHMC in the grand canonical ensemble. A general framework, useful for an extension of other Hybrid Monte Carlo methods to the grand canonical ensemble, is also provided. The software development is another fundamental part of the present work. The algorithms presented in this thesis are implemented in MultiHMC-GROMACS, an in-house version of the popular software package GROMACS. We explain the details of such implementation and give useful recommendations and hints for implementation of the new algorithms in other software packages. In summary, in this thesis, we propose the new numerical algorithms that are capable of improving the accuracy and sampling efficiency of molecular simulations with Hybrid Monte Carlo methods. We show that equipping the Hybrid Monte Carlo algorithm with extra features makes it even a “smarter” sampler and, no doubts, a strong competitor to the well-established molecular simulation techniques such as molecular dynamics (MD) and Monte Carlo. The up to 60 times increase in sampling efficiency of GSHMC over MD, due to the new algorithms in simulations of selected systems, supports such a belief.MTM2013-46553-C3-1-