Optimal zero-padding of kernel truncation method
报告专家：张勇教授 （天津大学）
报告时间：2024年1月4日：16:45-17:20
报告地点：西南中心516
报告摘要：The kernel truncation method (KTM) is a commonly-used algorithm to compute the convolution-type nonlocal potential, where the convolution kernel might be singular at the origin and/or far-field and the density 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 around 60%. Then, we present the error estimates of the potential and density in d space 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.
专家简介：张勇，2007年本科毕业于天津大学数学系，2012年在清华大学获得博士学位，曾先后在奥地利维也纳大学，法国雷恩一大和美国纽约大学克朗所从事博士后研究工作。2015年7月获得奥地利自然科学基金委支持的薛定谔基金，2018年入选国家高层次人才计划。研究兴趣主要是偏微分方程的数值计算和分析工作，尤其是快速算法的设计和应用。迄今发表论文30余篇，主要发表在包括SIAM Journal on Scientific Computing, SIAM journal on Applied Mathematics, Multiscale Modeling and Simulation, Mathematics of Computation, Journal of Computational Physics, Computer Physics Communication等计算数学顶尖杂志。
邀请人：唐庆粦
