Use of Polynomial Interpolation to Reduce Bias and Uncertainty of Free Energy Estimates via Thermodynamic Integration

This report presents several approaches aimed at reducing the bias and uncertainty of the free energy difference estimates obtained using the thermodynamic integration simulation strategy. The central idea is to utilize interpolation schemes, rather than the often-used trapezoidal rule quadrature, to fit the thermodynamic integration data, and obtain a more accurate and precise estimate of the free energy difference. Two systems with analytical solutions were chosen as test cases to verify the accuracy and precision of these interpolation approaches. The specific interpolation techniques utilized in the current study are Lagrange polynomials, Newton polynomials, and cubic spline. For the systems considered here, Lagrange and Newton interpolation polynomials provide the most accurate estimates of free energy differences, but often with slightly larger uncertainty compared to quadrature. Cubic spline interpolation delivers slightly biased estimates but with much smaller uncertainty compared to quadrature. To further improve the overall accuracy of free energy estimates obtained from polynomial interpolation, we also investigate the use of non-equidistant λ values (Chebyshev nodes) for thermodynamic integration simulations. Results clearly demonstrate that the use of non-equidistant λ values significantly reduce the variance, and improves the overall accuracy of the free energy estimates compared to that of equidistant λ values. The numerical stability of all these interpolation schemes is also discussed. We conclude that Lagrange and Newton interpolation polynomials perform best if the number of data points is less than a dozen. For analysis with a dozen or more data points, spline interpolation should be used as it is less susceptible to numerical instability. Free energy estimates can be further improved if non-equidistant λ values are utilized. To allow researchers to immediately utilize these methods, free software and documentation is provided via http://www.phys.uidaho.edu/ytreberg/software.