Eigenvector method for umbrella sampling enables error analysis

TL;DR

This paper introduces an eigenvector-based mathematical framework for umbrella sampling that enables rigorous error analysis, improves understanding of error scaling, and guides adaptive simulation strategies for more efficient sampling.

Contribution

It presents a novel eigenproblem formulation for combining umbrella sampling data, facilitating error estimation and analysis, and proposes an adaptive approach to accelerate convergence.

Findings

Eigenvector method provides a computationally inexpensive error estimator.

The estimator emphasizes low free energy pathways in alanine dipeptide simulations.

Error scales predictably with the number of sampling windows.

Abstract

Umbrella sampling efficiently yields equilibrium averages that depend on exploring rare states of a model by biasing simulations to windows of coordinate values and then combining the resulting data with physical weighting. Here, we introduce a mathematical framework that casts the step of combining the data as an eigenproblem. The advantage to this approach is that it facilitates error analysis. We discuss how the error scales with the number of windows. Then, we derive a central limit theorem for averages that are obtained from umbrella sampling. The central limit theorem suggests an estimator of the error contributions from individual windows, and we develop a simple and computationally inexpensive procedure for implementing it. We demonstrate this estimator for simulations of the alanine dipeptide and show that it emphasizes low free energy pathways between stable states in comparison to existing approaches for assessing error contributions. We discuss the possibility of using the estimator and, more generally, the eigenvector method for umbrella sampling to guide adaptation of the simulation parameters to accelerate convergence.