Seminars on Numerical Algorithms, Analyses, and Applications：Efficient methods for partition problems

：王东（香港中文大学（深圳））
：2023-03-03 10:30
：海韵园数理大楼天元会议室686

报告人：王东（香港中文大学（深圳））

时  间：2023年3月3日上午10:30-12:00

地  点：海韵园数理大楼天元会议室686

内容摘要:

A Dirichlet k-partition of a domain is a collection of k pairwise disjoint open subsets such that the sum of their first Laplace--Dirichlet eigenvalues is minimal. In this talk, we propose a new relaxation of the problem by introducing auxiliary indicator functions of domains and develop a simple and efficient diffusion generated method to compute Dirichlet k-partitions for arbitrary domains. The method only alternates three steps: 1. convolution, 2. thresholding, and 3. projection. The method is simple, easy to implement, insensitive to initial guesses and can be effectively applied to arbitrary domains without any special discretization. At each iteration, the computational complexity is linear in the discretization of the computational domain. Moreover, we theoretically prove the energy decaying property of the method. Experiments are performed to show the accuracy of approximation, efficiency and unconditional stability of the algorithm. We apply the proposed algorithms on both 2- and 3-dimensional flat tori, triangle, square, pentagon, hexagon, disk, three-fold star, five-fold star, cube, ball, and tetrahedron domains to compute Dirichlet k-partitions for different k to show the effectiveness of the proposed method. Compared to previous work with reported computational time, the proposed method achieves hundreds of times acceleration.

个人简介：

Dong is an Assistant Professor in Mathematics at the Chinese University of Hong Kong, Shenzhen from Aug. 2020. Dong has broad research interests in computational mathematics, especially in modelling and simulation of multiphase problems, image processing, threshold dynamics method, and optimization and control, and has published papers in Math. Comp., SISC, JCP and Multiscale Modeling & Simulation et al. Dong received a Ph.D degree in Applied Mathematics in June 2017 at the Hong Kong University of Science and Technology and a Bachelor degree in Mathematics in 2013 at Sichuan University. Before moving to CUHKSZ, Dong was an Assistant Professor Lecturer in the Department of Mathematics at the University of Utah between July 2017 and June 2020.

 

联系人：陈黄鑫