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A Mollification Approach to Ramified Transport and Tree Shape Optimization

Alberto Bressan, Giacomo Vecchiato, Ludmil Zikatanov

Abstract

The paper analyzes a mollification algorithm, for the numerical computation of optimal irrigation patterns. This provides a regularization of the standard irrigation cost functional, in a Lagrangian framework. Lower semicontinuity and Gamma-convergence results are proved. The technique is then applied to some numerical optimization problems, related to the optimal shape of tree roots and branches.

A Mollification Approach to Ramified Transport and Tree Shape Optimization

Abstract

The paper analyzes a mollification algorithm, for the numerical computation of optimal irrigation patterns. This provides a regularization of the standard irrigation cost functional, in a Lagrangian framework. Lower semicontinuity and Gamma-convergence results are proved. The technique is then applied to some numerical optimization problems, related to the optimal shape of tree roots and branches.

Paper Structure

This paper contains 8 sections, 9 theorems, 92 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Review of optimal irrigation patterns
  3. A mollified Lagrangian approach
  4. $\Gamma$-limit of the mollified irrigation costs
  5. An alternative mollification procedure
  6. Numerical simulations of optimal irrigation patterns
  7. Optimal shapes of tree roots and branches
  8. Numerical simulations of optimal tree shapes

Key Result

Proposition 3.1

(lower semicontinuity). Let $(x_n)_{n\geq 1}$ be a sequence of points in ${\mathbb R}^d$ and let $( \chi_n)_{n\geq 1}$ a sequence of irrigation plans for a measure $\mu$, such that for some $x \in {\mathbb R}^d$ and $\chi \in {\cal IP}(\mu)$. In addition, assume that the corresponding stopping times satisfy for some constant $C > 0$ and for all $n \ge 1$. Then the mollified multiplicities mmi sa

Figures (5)

  • Figure 1: An example showing that, when the mollified multiplicity \ref{['mm1']} is adopted, the mollified irrigation cost is not lower semicontinuous. In the limit as $n\to\infty$, the paths $\gamma_{2,n}$ are replaced by the shorter path $\gamma_2$. Hence the transportation cost along $\gamma_2$ decreases. However, along $\gamma_1$ the mollified multiplicity decreases and hence the transportation cost is larger.
  • Figure 2: Different stages of the minimization of (\ref{['mic1']}), applied to the irrigation of 25 equal masses located along an arc of circumference. Here $\alpha= 0.4$. The left image represents the initial configuration. The remaining three figures represent the local minimizers obtained by gradient descent, where the parameter in the mollifier takes the values $\varepsilon = 0.25, 0.1, 0.05$, respectively.
  • Figure 3: Different stages of the minimization of (\ref{['mic1']}), applied to the irrigation of 29 equal masses located along an arc of circumference. Here $\alpha= 0.9$. The left image represents the initial configuration. The remaining three figures represent the local minimizers obtained by gradient descent, taking $\varepsilon= 0.1, 0.05, 0.01$, respectively.
  • Figure 4: A simulation with 11 branches. Here $\alpha=0.4$, $c_1=0.4$, $c_2 = 1.4$. The left figure shows the initial configuration. The other three figures show different stages of the minimization algorithm, where the mollification parameter takes the values $\varepsilon=0.5$, $\varepsilon= 0.1$, $\varepsilon=0.03$.
  • Figure 5: A simulation with 15 branches. Here $\alpha=0.5$, $c_1=0.5$, $c_2 = 1.5$. The left figure shows the initial configuration. The other three figures show different stages of the minimization algorithm, where the mollification parameter takes the values $\varepsilon=0.8$, $\varepsilon= 0.1$, $\varepsilon=0.01$.

Theorems & Definitions (15)

  • Definition 2.1
  • Definition 2.2
  • Remark 2.3
  • Remark 2.4
  • Definition 2.5
  • Proposition 3.1
  • Proposition 3.2
  • Theorem 3.3
  • Proposition 4.1
  • Proposition 4.2
  • ...and 5 more