荔园数学讲坛: An efficient unconditionally stable method for computing Dirichlet partitions in arbitrary domains

主讲人：香港中文大学（深圳） 王东 助理教授

时间：2023年3月1日10:30–11:30

地点：H312

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.

主讲人简介: 

Dong Wang is an Assistant Professor in the School of Science and Engineering at the Chinese University of Hong Kong (Shenzhen). He has broad interests in analytical and computational methods for problems in applied mathematics, especially in computational fluid dynamics, computational material science, image processing, optimization and machine learning. After receiving his Bachelor degree in Mathematics from Sichuan University in 2013, Dong earned his Ph.D. in Applied Mathematics at the Hong Kong University of Science and Technology in 2017. Before moving to CUHK(SZ), he was an Assistant Professor Lecturer in the Department of Mathematics at the University of Utah.