Umbrella sampling efficiently yields equilibrium averages that depend onexploring rare states of a model by biasing simulations to windows ofcoordinate values and then combining the resulting data with physicalweighting. Here, we introduce a mathematical framework that casts the step ofcombining the data as an eigenproblem. The advantage to this approach is thatit facilitates error analysis. We discuss how the error scales with the numberof windows. Then, we derive a central limit theorem for averages that areobtained from umbrella sampling. The central limit theorem suggests anestimator of the error contributions from individual windows, and we develop asimple and computationally inexpensive procedure for implementing it. Wedemonstrate this estimator for simulations of the alanine dipeptide and showthat it emphasizes low free energy pathways between stable states in comparisonto existing approaches for assessing error contributions. We discuss thepossibility of using the estimator and, more generally, the eigenvector methodfor umbrella sampling to guide adaptation of the simulation parameters toaccelerate convergence.
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