Comparing Markov Chain Samplers for Molecular Simulation

Abstract

Markov chain Monte Carlo sampling propagators, including numerical integrators for stochastic dynamics, are central to the calculation of thermodynamic quantities and determination of structure for molecular systems. Efficiency is paramount, and to a great extent, this is determined by the integrated autocorrelation time (IAcT). This quantity varies depending on the observable that is being estimated. It is suggested that it is the maximum of the IAcT over all observables that is the relevant metric. Reviewed here is a method for estimating this quantity. For reversible propagators (which are those that satisfy detailed balance), the maximum IAcT is determined by the spectral gap in the forward transfer operator, but for irreversible propagators, the maximum IAcT can be far less than or greater than what might be inferred from the spectral gap. This is consistent with recent theoretical results (not to mention past practical experience) suggesting that irreversible propagators generally perform better if not much better than reversible ones. Typical irreversible propagators have a parameter controlling the mix of ballistic and diffusive movement. To gain insight into the effect of the damping parameter for Langevin dynamics, its optimal value is obtained here for a multidimensional quadratic potential energy function.

Dynamical Sampling with Langevin Normalization Flows

“…Since the transformation function is invertible, the integral of KL(q|p) can be calculated precisely. However, for NFL, the transformation functions demonstrated in Equations (12) and (13) are only approximately invertible because of the usage of the gradient information of the target distribution. Since the precise value of KL(q|p) cannot be obtained through integration, Monte Carlo estimation is used to calculate KL(q|p).…”

“…where f −1 θ is the inverse transformation function which can be calculated through Equation (12) and Equation (13). Since the update of x 1:D is divided into two parts, the calculation of ln det…”

“…where y 1:d , t 1 and t 2 can be calculated through Equations (12) and (13). However, in the progress of optimizing Equation (17), we find that the KL divergence may not be strictly non-negative because of the Monte Carlo integral, so we introduce a new objective function to overcome this problem.…”

“…Recent MCMC methods can be divided into two aspects. One class of the MCMC methods is slice sampling [11] and the other one is dynamical sampling [12,13]. The main problem of the slice sampler is that when sampling from the distributions with high dimensions, solving the slice interval can be very difficult.…”

Molecular Dynamics vs. Stochastic Processes: Are We Heading Anywhere?

“…Given the growing importance of advanced statistical techniques and stochastic modeling to understand and develop a more solid basis for computational statistical mechanics and, in particular, MD simulation, we need a novel effort to overcome the traditional way of approaching this problem. The present Special Issue is a first attempt to collect some of these new theory-related research efforts in the framework of MD, specially addressing algorithms [ 1 , 2 , 3 , 4 ], theoretical methods [ 5 , 6 , 7 ] and rigorous mathematical formulations [ 8 , 9 ]. The issue also presents some further applications [ 10 , 11 ] of molecular dynamics which can be of use while widening the perspective.…”