An efficient unconditionally stable method for Dirichlet partitions in arbitrary domains

TL;DR

This paper introduces a simple, efficient, and unconditionally stable diffusion-based method for computing Dirichlet k-partitions in arbitrary domains, demonstrating high accuracy and significant speed improvements over previous methods.

Contribution

The paper presents a novel relaxation approach with a diffusion generated algorithm that is easy to implement, insensitive to initial guesses, and applicable to any domain without special discretization.

Findings

Method is simple, involving convolution, thresholding, and projection steps.

Algorithm is unconditionally stable with proven energy decay.

Achieves hundreds of times faster computation compared to previous methods.

Abstract

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 paper, 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.