报告题目：Fast convolution-type nonlocal potential solvers in Nonlinear Schrödinger equation and Lightning simulation

报告人： 张勇 教授（天津大学数学学院）

报告时间：2021 年 01 月 21 日（周四）15：30          

报告地点：在线报告（ 腾讯会议 ID : 993 127 806）

报告摘要：

Convolution-type potential are common and important in many science and engineering fields. Efficient 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), Nonuniform-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

报告人简介：

张勇教授2007年本科毕业于天津大学，2012年在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持的薛定谔基金。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20余篇，主要发表在包括SIAM Journal on Scientific Computing, SIAM journal on Applied Mathematics, Journal of Computational Physics, Mathematics of Computation, Computer Physics Communication 等计算数学顶尖杂志。

参考文献：

[1] W. Bao, S. Jiang, Q. Tang and Y. Zhang, Computing the ground state and dynamics of the nonlinear Schrödinger equation with nonlocal interactions via the nonuniform FFT, J. Comput. Phys., 296 (2015), pp. 72–89.

[2] L. Exl, N. Mauser and Y. Zhang, Accurate and efficient computation of nonlocal potentials based on Gaussian-sum approximation, J. Comput. Phys., 327 (2016), pp. 629–642.

[3] X. Antoine, Q. Tang and Y. Zhang, On the ground states and dynamics of space fractional nonlinear Schrödinger/Gross-Pitaevskii equations with rotation term and nonlocal nonlinear interactions, J. Comput. Phys., 325 (2016), pp. 74–97.

[4] C. Zhuang, Y. Zhang, X.Zhou, R. Zeng and L. Liu, A fast tree algorithm for electrical field calculation in electrical discharge simulations, IEEE Transactions on Magnetics (2017).

[5] X. Antoine, Q. Tang, Y. Zhang A Preconditioned Conjugated Gradient method for computing ground states of rotating dipolar Bose-Einstein condensates via Kernel Truncation method for Dipole-Dipole Interaction evaluation, Communication in Computational Physics, 2017.

[6] L. Greengard, S. Jiang and Y. Zhang, A generic anisotropic kernel truncation method for convolution of free-space Green’s function, SIAM Journal on Scientific Computing, 2018.