Kaiser-Bessel basis for particle-mesh interpolation

In this work, we introduce the Kaiser-Bessel interpolation basis for the particle-mesh interpolation in the fast Ewald method. A reliable a priori error estimate is developed to measure the accuracy of the force computation in correlated charge systems, and is shown to be effective in optimizing the shape parameter of the Kaiser-Bessel basis in terms of accuracy. By comparing the optimized Kaiser-Bessel basis with the traditional $B$-spline basis, we demonstrate that the former is more accurate than the latter in part of the working parameter space, say, a relatively small real-space cutoff, a relatively small reciprocal space mesh, and a relatively large truncation of basis. In some cases, the Kaiser-Bessel basis is found to be more than one order of magnitude more accurate.

Figure 1

The truncated Kaiser-Bessel basis with different shape parameters and the $B$-spline basis [plot (a)], and their Fourier transforms [plot (b)]. In the legend, the “KB” stands for Kaiser-Bessel. The truncation radius $C$ is taken to be 2. The number of mesh points is $K=32$. In the plots, all the basis and the Fourier transform pairs are rescaled by the basis value measured at 0 for an easier comparison.

Figure 2

The reciprocal error plotted against the shape parameter $h$ of the Kaiser-Bessel interpolation basis. The truncation radius of the interpolation basis is $C=2$. The number of mesh points on each direction is $K_{\alpha }=32$. The reciprocal force error is plotted by solid color lines for the splitting parameter $\beta =1.0{\mathrm{nm}}^{-1}$ (red), 2.0 ${\mathrm{nm}}^{-1}$ (green), 4.0 ${\mathrm{nm}}^{-1}$ (blue), and 7.0 ${\mathrm{nm}}^{-1}$ (pink). The dashed lines are the corresponding error estimates [given by Eq. (25)] of the solid lines with the same color. The black cross indicates the optimal shape parameter that minimizes the reciprocal force error for $\beta =1.0,1.5,2.0,3.0,4.0,5.0,6.0,7.0{\mathrm{nm}}^{-1}$. The orange square indicates the optimal shape parameter determined by minimizing the estimated error.

Figure 3

The actual and estimated reciprocal errors of the Kaiser-Bessel basis plotted against the splitting parameter $\beta$ in the homogeneous and uncorrelated charge system. The solid lines present the actual error, while the dashed lines present the estimated error. The actual reciprocal error of $B$-spline basis is added as a comparison using dotted lines. Different colors denote different combinations of the truncation radius and the mesh size, which are listed in the legend of the figure. The shape parameter is optimized by minimizing the estimated error.

Figure 4

The actual and estimated reciprocal errors of the Kaiser-Bessel basis plotted against the splitting parameter $\beta$ in the TIP3P water system. The solid lines present the actual error, while the dashed lines present the estimated error. Different colors denote different combinations of the truncation radius and the mesh size, which are listed in the legend of the figure. The shape parameter is optimized by minimizing the estimated error.

Figure 5

The relative error reduction due to the charge correlation in the TIP3P water system. The force is computed by the Kaiser-Bessel basis with the parameters listed in the legend of the figure. The shape parameter of the Kaiser-Bessel basis is optimized by minimizing the estimated error.

Figure 6

The comparison between the reciprocal force errors computed by the $B$-spline basis (dashed lines), the Bessel basis (dotted lines), and the Kaiser-Bessel basis (KB for short; solid lines) in the TIP3P water system. The error as a function of the splitting parameter $\beta$ is presented. In each plot, the red, green, and blue colors represent the errors computed with truncation radius $C=2$, 3, and 4, respectively. The plots (a)–(d) from top to bottom take the mesh size of $K_{\alpha }=32,64,96$, and 128, respectively. The crossover of the Kaiser-Bessel and the $B$-spline bases is indicated by the black arrow in the plots. The dashed, solid, and dotted gray lines represent the direct errors of cutoff radii 0.6, 0.9, and 1.4 nm, respectively.

Figure 7

The energy drift of a microcanonical MD simulation using the Kaiser-Bessel basis and the ik-differentiation force scheme. The lower plot (b) is a zoom-in of the upper plot (a) in the energy range $[-10,100]$ kJ/(mol nm). Different colors denote different combinations of the truncation radius and the mesh size, which are listed in the legend of the figure. The shape parameter is optimized by minimizing the estimated error.