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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References

SHOWING 1-10 OF 48 REFERENCES

Calculation of free energy through successive umbrella sampling.

An implementation of umbrella sampling in which the pertinent range of states is subdivided into small windows that are sampled consecutively and linked together is considered, which is comparable to a multicanonical simulation with a very good weight function.

Optimal estimators and asymptotic variances for nonequilibrium path-ensemble averages.

A general minimal-variance estimator is derived that can combine nonequ equilibrium trajectory data sampled from multiple path-ensembles to estimate arbitrary functions of nonequilibrium expectations and develop asymptotic variance estimates pertaining to Jarzynski's equality for free energies and the Hummer-Szabo expressions for the potential of mean force.

Statistically optimal analysis of samples from multiple equilibrium states.

A new estimator for computing free energy differences and thermodynamic expectations as well as their uncertainties from samples obtained from multiple equilibrium states via either simulation or experiment is presented, which has significant advantages over multiple histogram reweighting methods for combining data from multiple states.

xTRAM: Estimating equilibrium expectations from time-correlated simulation data at multiple thermodynamic states

The expanded TRAM (xTRAM) estimator is formulated, shown to be asymptotically unbiased and a generalization of MBAR, and a random-swapping simulation protocol is introduced that can be used with xTRAM, gaining orders-of-magnitude advantages over simulation protocols that require the constraint of sampling from a global equilibrium.

On a Likelihood Approach for Monte Carlo Integration

The use of estimating equations has been a common approach for constructing Monte Carlo estimators. Recently, Kong et al. proposed a formulation of Monte Carlo integration as a statistical model,

Self-Learning Adaptive Umbrella Sampling Method for the Determination of Free Energy Landscapes in Multiple Dimensions.

This work presents an efficient automatized umbrella sampling strategy for calculating multidimensional potential of mean force and demonstrates that a significant smaller number of umbrella windows needs to be employed to characterize the free energy landscape over the most relevant regions without any loss in accuracy.

Free energies from dynamic weighted histogram analysis using unbiased Markov state model.

The dynamic histogram analysis method (DHAM) is developed, which finds that DHAM gives accurate free energies even in cases where WHAM fails, and may also prove useful in the construction of Markov state models from biased simulations in phase-space regions with otherwise low population.

Using Multistate Reweighting to Rapidly and Efficiently Explore Molecular Simulation Parameters Space for Nonbonded Interactions.

Free energy estimates of three molecular transformations in a benchmark molecular set as well as the enthalpy of vaporization of TIP3P are examined to demonstrate the power of this multistate reweighting approach for measuring changes in free energy differences or other estimators with respect to simulation or model parameters with very high precision and/or very low computational effort.

THE weighted histogram analysis method for free‐energy calculations on biomolecules. I. The method

The Weighted Histogram Analysis Method (WHAM), an extension of Ferrenberg and Swendsen's Multiple Histogram Technique, has been applied for the first time on complex biomolecular Hamiltonians. The