  This report presents the application of polynomial regression for estimating
free energy differences using thermodynamic integration. We employ linear
regression to construct a polynomial that optimally fits the thermodynamic
integration data, and thus reduces the bias and uncertainty of the resulting
free energy estimate. Two test systems with analytical solutions were used to
verify the accuracy and precision of the approach. Our results suggest that
regression with a high degree of polynomials give the most accurate free energy
difference estimates, but often with a slightly larger variance, compared to
commonly used quadrature techniques. High degrees of polynomials possess the
flexibility to closely fit the thermodynamic integration data but are often
sensitive to small changes in data points. To further improve overall accuracy
and reduce uncertainty, we also examine the use of Chebyshev nodes to guide the
selection of non-equidistant lambda values for the thermodynamic integration
scheme. We conclude that polynomial regression with non-equidistant lambda
values delivers the most accurate and precise free energy estimates for
thermodynamic integration data. Software and documentation is available at
http://www.phys.uidaho.edu/ytreberg/software
