[Paper Review] A diffusion generated method for computing Dirichlet partitions

Dong Wang, Braxton Osting|arXiv (Cornell University)|Feb 8, 2018

Graph theory and applications19 references3 citations

TL;DR

This paper introduces a diffusion-generated method for computing Dirichlet $k$-partitions on $d$-dimensional flat tori and spheres, using iterative diffusion, max selection, and normalization. The method efficiently computes optimal partitions, revealing hexagonal honeycombs in 2D, rhombic dodecahedral and Weaire-Phelan structures in 3D, and 24-cell honeycombs in 4D, with the first published results for 4D flat tori.

ABSTRACT

A Dirichlet $k$-partition of a closed $d$-dimensional surface is a collection of $k$ pairwise disjoint open subsets such that the sum of their first Laplace-Beltrami-Dirichlet eigenvalues is minimal. In this paper, we develop a simple and efficient diffusion generated method to compute Dirichlet $k$-partitions for $d$-dimensional flat tori and spheres. For the $2d$ flat torus, for most values of $k=3$-9,11,12,15,16, and 20, we obtain hexagonal honeycombs. For the $3d$ flat torus and $k=2,4,8,16$, we obtain the rhombic dodecahedral honeycomb, the Weaire-Phelan honeycomb, and Kelvin's tessellation by truncated octahedra. For the $4d$ flat torus, for $k=4$, we obtain a constant extension of the rhombic dodecahedral honeycomb along the fourth direction and for $k=8$, we obtain a 24-cell honeycomb. For the $2d$ sphere, we also compute Dirichlet partitions for $k=3$-7,9,10,12,14,20. Our computational results agree with previous studies when a comparison is available. As far as we are aware, these are the first published results for Dirichlet partitions of the $4d$ flat torus.

Motivation & Objective

• To develop an efficient numerical method for computing Dirichlet $k$-partitions on compact manifolds such as flat tori and spheres.
• To address the lack of computational studies on Dirichlet partitions in higher dimensions, particularly in 4D.
• To investigate the geometric structure of optimal partitions for various $k$ values and manifold types.
• To validate the method against known results and explore new configurations in higher dimensions.
• To provide the first published computational results for Dirichlet partitions on 4D flat tori.

Proposed method

• The method iteratively evolves $k$ functions via the heat equation on the domain for a fixed time $\tau$.
• At each spatial point, the function with the maximum value is retained while others are set to zero, enforcing a partitioning structure.
• Each function is then renormalized to maintain unit $L^2$-norm, preserving the energy minimization framework.
• The algorithm uses Fast Fourier Transform (FFT) for flat tori and Spherical Harmonic Transform (SHT) for the sphere to accelerate computations.
• The method is grounded in a mapping reformulation of the Dirichlet partition problem, where the partition corresponds to the non-zero level sets of a vector-valued ground state.
• The process is repeated until convergence to a stationary configuration that minimizes the total first Dirichlet eigenvalue sum.

Experimental results

Research questions

• RQ1 What geometric structures emerge as optimal Dirichlet $k$-partitions on 2D and 3D flat tori for various $k$ values?
• RQ2 Can the diffusion-generated method reliably compute Dirichlet partitions on higher-dimensional manifolds such as the 4D flat torus?
• RQ3 How do the computed partitions compare to known minimal surface or tiling structures like the Weaire-Phelan or 24-cell honeycombs?
• RQ4 Does the method recover previously known results for the 2D sphere, and can it extend to new $k$ values?
• RQ5 What is the asymptotic behavior of optimal partitions as $k \to \infty$ in higher dimensions, based on numerical evidence?

Key findings

• For the 2D flat torus with $k=3$ to $9$, $11$, $12$, $15$, $16$, and $20$, the method computes hexagonal honeycomb partitions, consistent with known asymptotic optimality.
• For the 3D flat torus with $k=2$, $4$, $8$, and $16$, the method produces the rhombic dodecahedral honeycomb, Weaire-Phelan structure, and Kelvin’s truncated octahedron tessellation, respectively.
• For the 4D flat torus, $k=4$ yields a constant extension of the rhombic dodecahedral honeycomb along the fourth dimension, and $k=8$ yields a 24-cell honeycomb, representing the first published results for this setting.
• On the 2D sphere, the method successfully computes partitions for $k=3$ to $7$, $9$, $10$, $12$, $14$, and $20$, with results consistent with prior studies.
• The computed energy values $\tilde{E}$ decrease with increasing $k$, and for $k=20$ on the sphere, $\tilde{E} = 12.20$, indicating improved energy efficiency.
• The method achieves convergence with reasonable CPU times, e.g., $9803$ seconds for $k=8$ on the 4D tesseract with $64^4$ grid points.