On Optimal Zero-Padding of Kernel Truncation Method
Xin Liu, Qinglin Tang, S. D. Zhang, Yong Zhang
Abstract
.The kernel truncation method (KTM) is a commonly used algorithm to compute the convolution-type nonlocal potential \(\Phi ({\textbf x})=(U\ast \rho )({\textbf x}),\,{\textbf x} \in{\mathbb R^d},\) where the convolution kernel \(U({\textbf x})\) might be singular at the origin and/or far-field and the density \(\rho ({\textbf x})\) is smooth and fast decaying. In KTM, in order to capture the Fourier integrand's oscillations that are brought by the kernel truncation, one needs to carry out a zero-padding of the density, which means a larger physical computation domain and a finer mesh in the Fourier space by duality. The zero-padding factor, \(\sqrt{d}+1\), was first given as empirical formula for the 2D/3D Coulomb potential in [M. R. Jarvis et al. Phys. Rev. B, 56 (1997), pp. 14972–14978; C. A. Rozzi et al. Phys. Rev. B, 73 (2006), 205119]. In this article, we first rederive the optimal zero-padding factor in a rigorous way for arbitrary space dimension and different nonlocal potentials. Then, we present the error estimates of the density and potential. Next, we investigate the anisotropic density case, and provide details of tensor acceleration [F. Vico, L. Greengard, and M. Ferrando, J. Comput. Phys. 323 (2016), pp. 191–203], which is an important performance improvement. Finally, extensive numerical results are provided to confirm the accuracy, efficiency, optimal zero-padding factor for the anisotropic density, together with some applications to different nonlocal potentials, including the one-dimensional/two-dimensional (2D)/three-dimensional (3D) Poisson, 2D Coulomb, quasi-2D/3D dipole-dipole Interaction, and 3D quadrupolar potential.Keywordsconvolution-type nonlocal potentialkernel truncation methodoptimal zero-paddingerror estimatesanisotropic densityMSC codes65D3268Q25