数学科学学院

计算与应用讨论班——Fast integral equation methods and the DMK framework

来源:数学科学学院 发布时间:2024-07-02   10

报告人:蒋世东,Flatiron Insitute


时间:2024年7月2日10:00-11:00    


地点:浙江大学紫金港校区基础交叉研究院(筹)101会议室


摘要:In this talk, we will first present an overview of fast integral equation methods (FIEMs). Fast algorithms, such as the fast multipole methods and their descendants, have made profound impacts on many areas of scientific computing. Boundary integral equations of the second kind are the natural formulation for boundary value problems of elliptic partial differential equations due to the reduction of dimensionality by one, automatic satisfaction of conditions at infinity for exterior and scattering problems, and well conditioning. The computational bottleneck that arises because the resulting linear system is dense due to the nonlocal and long-range nature of integral operators is largely removed by fast algorithms. FIEMs have been highly successful in solving a large class of problems in fluid mechanics, elasticity, acoustics, and electromagnetics.

Second, we introduce a new class of multilevel, adaptive, dual-space methods for computing fast convolutional transforms. The DMK (dual-space multilevel kernel-splitting) framework uses a hierarchy of grids, 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. Unlike earlier multilevel summation schemes, DMK exploits the fact that the interaction at each scale is diagonalized by a short Fourier transform, permitting the use of separation of variables without relying on the FFT. The DMK framework substantially simplifies the algorithmic structure of the fast multipole method (FMM) and unifies the FMM, Ewald summation, and multilevel summation, achieving speeds comparable to the FFT in work per grid point, even in a fully adaptive context.


报告人简介:Shidong holds a Ph.D. in Mathematics from New York University in 2001, an MSc in Physics from New York University in 1998 and a BSc in Applied Physics from Shanghai Jiao Tong University in 1994. From 2001 to 2004, he held a postdoctoral position in the Department of Computer Science at Yale University. From 2004 to 2021, he was affiliated with the Department of Mathematical Sciences at the New Jersey Institute of Technology. Since August 2021, he has been a Senior Research Scientist at the Center for Computational Mathematics at the Flatiron Institute, Simons Foundation. His research interests include applied and computational mathematics, fast algorithms, and integral equation methods.


联系人:赖俊(laijun6@zju.edu.cn)

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