Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Iterative Thresholding Methods for Longest Minimal Length Partitions

Shilong Hu, Hao Liu, Dong Wang

TL;DR

This work addresses the longest minimal length partitions problem under volume constraints by proposing two iterative schemes that couple short-time heat-flow approximations with threshold dynamics and auction dynamics. The objective is approximated via indicator functions and Gaussian convolution, yielding a constrained max–min formulation solved by alternating updates of the total region and the multiphase partition, with two variants differing in stability and computational effort. Numerical experiments in 2D and 3D consistently support the conjecture that the disc maximizes the longest minimal length partitions in 2D and the ball does so in 3D across various partition counts and volume proportions. The methods provide a practical framework to study shape optimization under partition constraints and offer avenues for rigorous convergence analysis and level-set extensions.

Abstract

In this paper, we introduce two iterative methods for longest minimal length partition problem, which asks whether the disc (ball) is the set maximizing the total perimeter of the shortest partition that divides the total region into sub-regions with given volume proportions, under a volume constraint. The objective functional is approximated by a short-time heat flow using indicator functions of regions and Gaussian convolution. The problem is then represented as a constrained max-min optimization problem. Auction dynamics is used to find the shortest partition in a fixed region, and threshold dynamics is used to update the region. Numerical experiments in two-dimensional and three-dimensional cases are shown with different numbers of partitions, unequal volume proportions, and different initial shapes. The results of both methods are consistent with the conjecture that the disc in two dimensions and the ball in three dimensions are the solution of the longest minimal length partition problem.

Iterative Thresholding Methods for Longest Minimal Length Partitions

TL;DR

This work addresses the longest minimal length partitions problem under volume constraints by proposing two iterative schemes that couple short-time heat-flow approximations with threshold dynamics and auction dynamics. The objective is approximated via indicator functions and Gaussian convolution, yielding a constrained max–min formulation solved by alternating updates of the total region and the multiphase partition, with two variants differing in stability and computational effort. Numerical experiments in 2D and 3D consistently support the conjecture that the disc maximizes the longest minimal length partitions in 2D and the ball does so in 3D across various partition counts and volume proportions. The methods provide a practical framework to study shape optimization under partition constraints and offer avenues for rigorous convergence analysis and level-set extensions.

Abstract

In this paper, we introduce two iterative methods for longest minimal length partition problem, which asks whether the disc (ball) is the set maximizing the total perimeter of the shortest partition that divides the total region into sub-regions with given volume proportions, under a volume constraint. The objective functional is approximated by a short-time heat flow using indicator functions of regions and Gaussian convolution. The problem is then represented as a constrained max-min optimization problem. Auction dynamics is used to find the shortest partition in a fixed region, and threshold dynamics is used to update the region. Numerical experiments in two-dimensional and three-dimensional cases are shown with different numbers of partitions, unequal volume proportions, and different initial shapes. The results of both methods are consistent with the conjecture that the disc in two dimensions and the ball in three dimensions are the solution of the longest minimal length partition problem.
Paper Structure (17 sections, 1 theorem, 49 equations, 12 figures, 2 tables, 4 algorithms)

This paper contains 17 sections, 1 theorem, 49 equations, 12 figures, 2 tables, 4 algorithms.

Table of Contents

  1. Introduction
  2. Approximations of the objective functional
  3. Derivation of the methods
  4. The first method
  5. Update of
  6. Update of 1
  7. The second method
  8. Update of
  9. Numerical experiments
  10. Numerical experiments in two dimensions
  11. Two partitions
  12. Three partitions
  13. Sensitivity to initial shape
  14. Six and nine partitions
  15. Numerical experiments in three dimensions
  16. ...and 2 more sections

Key Result

Lemma 3.1

The problems update Omega: 1 and update Omega: 1.2 have the same maximizer, i.e.,

Figures (12)

  • Figure 1: The shortest partitions of different $\Omega$ in Pólya's question. The partitions are computed numerically by auction dynamics in Section \ref{['sec:method']}.
  • Figure 2: The initial condition is a disc. After one iteration by \ref{['update total']}, the total region changes instead of remaining round.
  • Figure 3: Two uniform partitions in two dimensions. The initial condition is set to be a five-petal flower. The total regions and their shortest partitions of both methods in the initial, $5th$, $10th$, and final iterations are plotted.
  • Figure 4: Two uniform partitions in two dimensions. The approximate objective functionals $\Tilde{E}_{\tau}(u_{\Omega}^{k},(u_i^k)_{i=1}^{n})$ of two methods versus iteration times. The horizontal axis represents the number of iterations. The vertical axis represents the functional value.
  • Figure 5: Three partitions in two dimensions. The initial condition is set to be a five-petal flower. The total regions and their shortest partitions of both methods in the final iterations are plotted for different values of $\mathbf{c}$ of the three partition case.
  • ...and 7 more figures

Theorems & Definitions (6)

  • Lemma 3.1
  • proof
  • Remark 3.2
  • Remark 3.3
  • Remark 3.4
  • Remark 3.5