讲座题目：Fast convolution-type nonlocal potential solvers in Nonlinear Schrödinger equation and Lightning simulation
主 讲 人：张勇 博士后
张勇博士于清华大学获得博士学位，先后在奥地利维也纳大学的Wolfgang pauli 研究所，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撑的薛定谔基金，2018年入选国家“青年千人”计划，将到天津大学应用数学中心参加工作。张勇博士的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，已在SIAM Journal on Applied Mathemathics, Journal of Computational Physics（计算物理）, Computer Physics Communications（计算机物理通讯）， Mathematics of Computation （科学计算）等期刊发表SCI 论文20余篇。
Convolution-type potential are common and important in many science and engineeringfields. Effcient and accurate evaluation of such nonlocal potentials are essential in practical simulations. In this talk, I will focus on those arising from quantum physics/chemistry and lightning-shield protection, including Coulomb, dipolar and Yukawa potential that are generated by isotropic and anisotropic smooth and fast-decaying density, as well as convolutions defined on a one-dimensional adaptive finite difference grid. The convolution kernel is usually singular or discontinuous at the origin and/or at the far field, and density might be anisotropic, which together present great challenges for numerics in both accuracy and efficiency. The state-of-art fast algorithms include Wavelet based Method(WavM), kernel truncation method(KTM), Non Uniform-FFT based method(NUFFT) and Gaussian-Sum based method(GSM). Gaussian-sum/exponential-sum approximation and kernel truncation technique, combined with finite Fourier series and Taylor expansion, finally lead to a O(N log N) fast algorithm achieving spectral accuracy. Applications to NLSE, together with a useful recently-developed sum-of exponential algorithm are reviewed. Tree-algorithm for computing the one-dimensional convolutions in lighting-shield simulation is also covered as the last application.