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

报告人：张勇，天津大学应用数学中心

时间：2024年6月15日，10:30--11:30

地点：学术交流中心第三会议室

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 the kernel truncation method (KTM), NonUniform-FFT based method (NUFFT) and Gaussian-Sum based method (GauSum). 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，天津大学应用数学中心教授，博士生导师，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等计算数学顶尖杂志。

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报告题目：Convergence of some numerical methods for parabolic inverse Robin problems

报告人：蒋代军，华中师范大学数学学院

时间：2024年6月15日，9:30--10:30

地点：学术交流中心第三会议室

摘要：We study in this talk some numerical methods for solving the highly nonlinear and ill-posed inverse problem of identifying the Robin coefficients in  parabolic systems. We first apply the Levenberg-Marquardt method (LMM) to transform the Tikhonov regularized nonlinear non-convex minimizations into convex minimizations. And the quadratic convergence of the LMM is rigorously established for the nonlinear  parabolic inverse problems for the first time, under a simple novel adaptive strategy for selecting regularization parameters during the LM iteration. Then the domain decomposition methods (DDMs) are used to solve the convex minimizations. The methods are completely local and the local minimizers have explicit expressions within the subdomains. Numerical experiments are presented to show the accuracy and efficiency of the methods, in particular, the convergence seems nearly optimal in the sense that the iteration number of the methods is independent on the mesh size.

报告人简介： 蒋代军，华中师范大学数学学院副教授，博士生导师，2007年获得华中师范大学学士学位，2009年和2012年分别获得武汉大学硕士和博士学位。蒋代军博士的研究领域包括偏微分方程反问题、稀疏优化及控制，快速算法等，主持国家自然科学基金项目4项，在SIAM 系列、 Inverse Problems等刊物上发表学术论文近30篇。

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