论文标题
关于内核截断方法的最佳零盖
On optimal zero-padding of kernel truncation method
论文作者
论文摘要
内核截断方法（KTM）是一种通常使用的算法，用于计算卷积类型的非局部潜在$φ（x）=（u \ astρ）（x）（x），〜x \ in {\ mathbb r^d} $，在卷积kernel $ u（x）$中可能是sighto kernel $ u（x）$，或者可能是singulin和fare of is/field and和/field and and and和field and and and and and and and and（快速付费。在KTM中，为了捕获内核截断带来的傅立叶积分的振荡，需要对密度进行零盖，这意味着通过二元性在傅立叶空间中具有更大的物理计算域和更细的网格。经验四倍零盖[Vico等人J. Comput。物理。 （2016）]给记忆要求造成了沉重的负担，尤其是对于更高的维度问题。在本文中，我们得出了最佳的零填充因子，即$ \ sqrt {d}+1 $，首次与严格的证明在一起。内存成本大大降低到很小的部分，即$（\ frac {\ sqrt {d} +1} {4} {4}）^d $，是原始四倍算法中所需的内容。 For example, in the precomputation step, a double-precision computation on a $256^3$ grid requires a minimum $3.4$ Gb memory with the optimal threefold zero-padding, while the fourfold algorithm requires around $8$ Gb where the reduction factor is $\frac{37}{64}\approx \frac{3}{5}$.然后，我们介绍了$ d $尺寸中电势和密度的错误估计。接下来，我们重新投资于各向异性密度的最佳零填充因子。最后，提供了广泛的数值结果，以确认各向异性密度的准确性，效率，最佳的零填充因子，以及某些应用于不同类型的非局部潜力的应用，包括1D/2D/3D POISSON，2D COULOMB，QUASI-2D/3D DIPOLE-DIPOLE-DIPOLE-DIPOLE ITACTION互动和3D QuadRup Quid QuadRup Quarlup Quarlup Quallup Quarlup all purel＆3D quad QuadRup oltimal。
The kernel truncation method (KTM) is a commonly-used algorithm to compute the convolution-type nonlocal potential $Φ(x)=(U\ast ρ)(x), ~x \in {\mathbb R^d}$, where the convolution kernel $U(x)$ might be singular at the origin and/or far-field and the density $ρ(x)$ is smooth and fast-decaying. In KTM, in order to capture the Fourier integrand's oscillations that is 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 empirical fourfold zero-padding [ Vico et al J. Comput. Phys. (2016) ] puts a heavy burden on memory requirement especially for higher dimension problems. In this paper, we derive the optimal zero-padding factor, that is, $\sqrt{d}+1$, for the first time together with a rigorous proof. The memory cost is greatly reduced to a small fraction, i.e., $(\frac{\sqrt{d}+1}{4})^d$, of what is needed in the original fourfold algorithm. For example, in the precomputation step, a double-precision computation on a $256^3$ grid requires a minimum $3.4$ Gb memory with the optimal threefold zero-padding, while the fourfold algorithm requires around $8$ Gb where the reduction factor is $\frac{37}{64}\approx \frac{3}{5}$. Then, we present the error estimates of the potential and density in $d$ dimension. Next, we re-investigate the optimal zero-padding factor for the anisotropic density. 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 types of nonlocal potential, including the 1D/2D/3D Poisson, 2D Coulomb, quasi-2D/3D Dipole-Dipole Interaction and 3D quadrupolar potential.
