Eigenvector method for umbrella sampling enables error analysis.

@article{Thiede2016EigenvectorMF,
  title={Eigenvector method for umbrella sampling enables error analysis.},
  author={Erik H. Thiede and Brian Van Koten and Jonathan Weare and Aaron R. Dinner},
  journal={The Journal of chemical physics},
  year={2016},
  volume={145 8},
  pages={
          084115
        }
}
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… 

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