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Part II: Talk Information

Biography: Shidong Jiang joined the Center for Computational Mathematics, Flatiron Institute, Simons Foundation in August 2021 as a Senior Research Scientist. Jiang was previously a Professor of Mathematical Sciences at the New Jersey Institute of Technology. His research lies in the field of numerical analysis and scientific computing with particular emphasis on fast numerical algorithms and integral equation methods for solving initial/boundary value problems for various partial differential equations. Shidong holds a Ph.D. in Mathematics from New York University, an MSc in Physics from New York University and a BSc in Applied Physics and Applied Mathematics from Shanghai Jiao Tong University.

Title: A dual-space multilevel kernel-split framework for discrete and continuous convolution

Abstract: We introduce a new class of multilevel, adaptive, dual-space methods for computing fast convolutional transforms. These methods can be applied to a broad class of kernels, from the Green’s functions for classical partial differential equations (PDEs) to power functions and radial basis functions such as those used in statistics and machine learning. The DMK (dual-space multilevel kernel-splitting) framework uses a hierarchy of grids, initialized by computing a smoothed interaction at the coarsest level, followed by a sequence of corrections at finer and finer scales until the problem is entirely local, at which point direct summation is applied.

The main novelty of DMK is that the interaction at each scale is diagonalized by a short Fourier transform, permitting the use of separation of variables, but without requiring the FFT for its linear complexity. It substantially simplifies the algorithmic structure of the fast multipole methods (FMMs), unifies the tree-based algorithms such as the FMM and FFT-based algorithms  such as the Ewald summation, and achieves speeds comparable to the FFT in work per gridpoint, even in a fully adaptive context. This is joint work with Leslie Greengard.

Biography: Zhenli Xu is a professor at the School of Mathematical Sciences and Institute of Natural Sciences, Shanghai Jiao Tong University. He received B.S. and Ph.D. degrees from University of Science and Technology of China, and was a postdoctoral fellow at University of North Carolina at Charlotte before joining the SJTU. He was selected in the Program of New Century Talents in University of Chinese Ministry of Education in 2010. Professor Xu works in a number of fields in applied mathematics, including fast Monte Carlo algorithms, computer simulations and continuum theory of biological and colloidal systems, and numerical PDEs, and published papers in core journals such as SIAM Review and other SIAM series journals, Physical Review Letters, Journal of Computational Physics, and Journal of Chemical Physics. His research projects has been financially supported by Chinese Ministry of Education, and Natural Science Foundation of China.

Title: Random Batch Ewald and Sum-of-Gaussians Methods for Molecular Dynamics Simulations

Abstract: The development of efficient methods for long-range systems plays important role in all-atom simulations of biomolecules and materials. We present a random-batch Ewald (RBE) and a random-batch sum-of-Gaussians (SOG) methods for molecular dynamics simulations of particle systems with long-range Coulomb interactions. These algorithms take advantage of the random minibatch strategy for the force calculation between particles, leading to an order N algorithm. By the Ewald or the SOG splitting of the Coulomb kernel, the random importance sampling is employed in the Fourier part such that the force variance can be reduced. Thus, it avoids the use of the FFT and greatly improves the scalability of the molecular simulations. We present the comparison between different approaches and the progress of software development. Numerical results are presented to show the attractive performance of the algorithms, including the superscalability for large-scale simulations.

Biography: Wenjun Ying , distinguished research fellow; bachelor of Tsinghua University, doctorate and post-doctorate of Duke University, and the tenure-track assistant professor of Michigan Technological University. He has put forward the computational method of spatial-temporal adaptation for the analogue simulation of heart transmission in electrocardiac wave, which has reached the internationally advanced level. The method can be applied in the hydromechanics (such as supersonic flight simulation), or other important science and engineering problems in mathematical biology. In the case of the simulation of electrocardiac wave transmission, he has proposed the full –implicit temporal integral method for the reaction-diffusion equation. And on the studies of the impact of biological cells on electrical field stimulation, he has proposed the method of hybrid finite element, which significantly improved the computation precision and efficiency. He proposed the nucleus-free boundary integral method as to partial differential equation of elliptic type, which has overcome several limitations of traditional boundary integral method. Researchers can use the nucleus-free boundary integral method without needing to know the analytic expression of integral kernel. The method has extended the boundary integral method to the partial differential equation of solvable coefficient and anisotropy. He has won the funding of NSF in 2009 for his studies on the nucleus-free boundary integral method.

Title:  A correction function-based kernel-free boundary integral method for elliptic PDEs with implicitly defined interfaces

Abstract: In this talk, I will present a new version of the kernel-free boundary integral(KFBI)method for elliptic PDEs with implicitly defined irregular boundaries and interfaces. The KFBI method evaluates boundary or volume integrals indirectly by solving equivalent but much simpler interface problems. A correction function is introduced for both evaluation of right hand side correction terms and interpolation of a non-smooth potential function. It allows the new method to avoid computation of high-order partial derivatives on interfaces or boundaries, greatly reducing the algorithm complexity and improving the efficiency, especially for fourth-order methods in three space dimensions. Challenging numerical examples including high-contrast coefficients, arbitrarily close interfaces and heterogeneous interface problems, will be reported to demonstrate the efficiency and accuracy of the method.

Biography: 张继伟，武汉大学数学与统计学院教授，博士生导师，2003和2006年在郑州大学获得学士和硕士学位，2009年在香港浸会大学获得博士学位。随后在南洋理工大学和纽约大学克朗所从事博士后研究，2014年5月在北京计算科学研究中心工作，2018年11月到武汉大学工作。主要研究领域包括偏微分方程和非局部模型的数值解法，以及神经科学的建模与计算。主要成果发表在SIAM Journal on Scientific Computing, SIAM Journal on Numerical Analysis, Mathematics of Computation，Journal of Computational Neuroscience等国际知名期刊上。

Title：三维非局部模型基于逼近球的有限元策略

Abstract: 三维非局部模型的有限元实现涉及高维积分、巨大的计算量和存储量，因此在算法设计、网格生成、数据结构优化、面向并行平台的算法架构等诸多方面都提出了挑战，本报告将简要介绍我们在高维有限元算法实现方面的一些思路和进展。

Biography:张勇，天津大学数学学院应用数学中心教授，2007年本科毕业于天津大学，2012年在清华大学获得博士学位。他先后在奥地利维也纳大学的Wolfgang Pauli 研究所、新加坡国立大学、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。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。主要成果发表在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上。

Title: A  Spectrally Accurate Numerical Method For Computing The Bogoliubov-De Gennes Excitations Of Dipolar Bose-Einstein Condensates

Abstract: In this paper, we propose an efficient and robust numerical method to study the ele- mentary excitation of dipolar Bose-Einstein condensates (BEC), which is governed by the Bogoliubov- de Gennes equations (BdGEs) with nonlocal dipole-dipole interaction, around the mean field ground state. Analytical properties of the BdGEs are investigated, which could serve as benchmarks for the numerical methods. To evaluate the nonlocal interactions accurately and efficiently, we propose a new Simple Fourier Spectral Convolution method (SFSC). Then, integrating SFSC with the stan- dard Fourier spectral method for spatial discretization and Implicitly Restarted Arnoldi Methods (IRAM) for the eigenvalue problem, we derive an efficient and spectrally accurate method, named as SFSC-IRAM method, for the BdGEs. Ample numerical tests are provided to illustrate the accuracy and efficiency. Finally, we apply the new method to study systematically the excitation spectrum and Bogoliubov amplitudes around the ground state with different parameters in different spatial dimensions.

Biography: Zecheng Gan is now an assistant professor of Advanced Materials of the Function Hub at HKUST(GZ), and an affiliate assistant professor in the Department of Mathematics, School of Science at HKUST(CWB). He obtained his PhD from Shanghai Jiao Tong University, under the supervision of Prof. Zhenli Xu (2010-2016), and worked as a postdoc assistant professor in mathematics at University of Michigan, Ann Arbor (2016-2019) and then postdoc associate at Courant institute, New York university until August 2021 before joining HKUST. 

Broadly, his research interest includes applied and computational mathematics, scientific computing, machine learning and data-driven methods, and their applications in interdisciplinary sciences. More specifically, his research focuses on the development of efficient, accurate and scalable numerical methods for the modeling and simulations of complex fluids at mesoscopic scales, with broad applications in soft condensed matter physics, materials science and biology.

Title: An O(N) simulation algorithm for charged particles under dielectric confinement

Abstract: We derive a fast convergent lattice summation formula for the simulation of charged particles under dielectric confinement, based on a novel “quasi-Ewald splitting” technique. We further achieve O(N) complexity by an optimal Gauss quadrature rule for the real space sum, and a “separation via sorting” technique combined with random batch importance sampling technique in reciprocal space. Specifically, for the challenging case of metamaterials confinement, charge/field oscillation is observed, which we show analytically due to the arising of a first-order pole in the quasi-2D Green's function, tailored quadrature scheme is further developed to efficient and accurately handle the corresponding Cauchy principle value problem. The influence of polarization, and field oscillation on collective behaviors of the confined particles will also be discussed.