An efficient unconditionally stable method for computing Dirichlet partitions in arbitrary domains
数学专题报告
报告题目(Title)：An efficient unconditionally stable method for computing Dirichlet partitions in arbitrary domains
报告人(Speaker)：王东（香港中文大学（深圳））
地点(Place)：腾讯会议ID：768-477-524
时间(Time)：2022年10月14日（周五），15:00-16:00
邀请人(Inviter)：蔡勇勇
报告摘要
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
王东，香港中文大学（深圳）理工学院助理教授。2013年本科毕业于四川大学，2017年博士毕业于香港科技大学，2017-2020年在美国犹他大学从事博士后研究。他的主要研究兴趣包括计算流体力学、计算材料科学、图像处理、优化、及机器学习，并在SIAM Journal on Scientific Computing, SIAM Journal on Applied Mathematics，Mathematics of Computation, Journal of Computational Physics等国际著名期刊发表学术论文20余篇。
