Quantifying Configuration-Sampling Error in Langevin Simulations of Complex Molecular Systems

• Main Authors: Josh Fass, David A. Sivak, Gavin E. Crooks, Kyle A. Beauchamp, Benedict Leimkuhler, John D. Chodera
• Format: Article
• Language: English
• Published: MDPI AG 2018-04-01
• Series: Entropy
• Subjects: Langevin dynamics; Langevin integrators; KL divergence; nonequilibrium free energy; molecular dynamics integrators; integrator error; sampling error; BAOAB; velocity verlet with velocity randomization (VVVR); Bussi-Parrinello; shadow work
• Online Access: http://www.mdpi.com/1099-4300/20/5/318

While Langevin integrators are popular in the study of equilibrium properties of complex systems, it is challenging to estimate the timestep-induced discretization error: the degree to which the sampled phase-space or configuration-space probability density departs from the desired target density due to the use of a finite integration timestep. Sivak et al., introduced a convenient approach to approximating a natural measure of error between the sampled density and the target equilibrium density, the Kullback-Leibler (KL) divergence, in phase space, but did not specifically address the issue of configuration-space properties, which are much more commonly of interest in molecular simulations. Here, we introduce a variant of this near-equilibrium estimator capable of measuring the error in the configuration-space marginal density, validating it against a complex but exact nested Monte Carlo estimator to show that it reproduces the KL divergence with high fidelity. To illustrate its utility, we employ this new near-equilibrium estimator to assess a claim that a recently proposed Langevin integrator introduces extremely small configuration-space density errors up to the stability limit at no extra computational expense. Finally, we show how this approach to quantifying sampling bias can be applied to a wide variety of stochastic integrators by following a straightforward procedure to compute the appropriate shadow work, and describe how it can be extended to quantify the error in arbitrary marginal or conditional distributions of interest.