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

Orthogonal parallel MCMC methods for sampling and optimization

Monte Carlo (MC) methods are widely used for Bayesian inference and optimization in statistics, signal processing and machine learning. A well-known class of MC methods are Markov Chain Monte Carlo (MCMC) algorithms. In order to foster better exploration of the state space, specially in high-dimensional applications, several schemes employing multiple parallel MCMC chains have been recently introduced. In this work, we describe a novel parallel interacting MCMC scheme, called {\it orthogonal MCMC} (O-MCMC), where a set of "vertical" parallel MCMC chains share information using some "horizontal" MCMC techniques working on the entire population of current states. More specifically, the vertical chains are led by random-walk proposals, whereas the horizontal MCMC techniques employ independent proposals, thus allowing an efficient combination of global exploration and local approximation. The interaction is contained in these horizontal iterations. Within the analysis of different implementations of O-MCMC, novel schemes in order to reduce the overall computational cost of parallel multiple try Metropolis (MTM) chains are also presented. Furthermore, a modified version of O-MCMC for optimization is provided by considering parallel simulated annealing (SA) algorithms. Numerical results show the advantages of the proposed sampling scheme in terms of efficiency in the estimation, as well as robustness in terms of independence with respect to initial values and the choice of the parameters

Efficient Gaussian Sampling for Solving Large-Scale Inverse Problems using MCMC Methods

The resolution of many large-scale inverse problems using MCMC methods requires a step of drawing samples from a high dimensional Gaussian distribution. While direct Gaussian sampling techniques, such as those based on Cholesky factorization, induce an excessive numerical complexity and memory requirement, sequential coordinate sampling methods present a low rate of convergence. Based on the reversible jump Markov chain framework, this paper proposes an efficient Gaussian sampling algorithm having a reduced computation cost and memory usage. The main feature of the algorithm is to perform an approximate resolution of a linear system with a truncation level adjusted using a self-tuning adaptive scheme allowing to achieve the minimal computation cost. The connection between this algorithm and some existing strategies is discussed and its efficiency is illustrated on a linear inverse problem of image resolution enhancement.Comment: 20 pages, 10 figures, under review for journal publicatio