Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

Measure-based approach to mesoscopic modeling of optimal transportation networks

Jan Haskovec, Peter Markowich, Simone Portaro

TL;DR

This work derives the gradient flow of the constrained energy functional with respect to the Fisher-Rao (or Hellinger-Kakutani) metric, which gives a reaction-type PDE system and construct equilibrium measures of the resulting PDE system.

Abstract

We propose a mesoscopic modeling framework for optimal transportation networks with biological applications. The network is described in terms of a joint probability measure on the phase space of tensor-valued conductivity and position in physical space. The energy expenditure of the network is given by a functional consisting of a pumping (kinetic) and metabolic power-law term, constrained by a Poisson equation accounting for local mass conservation. We establish convexity and lower semicontinuity of the functional on approriate sets. We then derive its gradient flow with respect to the 2-Wasserstein topology on the space of probability measures, which leads to a transport equation, coupled to the Poisson equation. To lessen the mathematical complexity of the problem, we derive a reduced Wasserstein gradient flow, taken with respect to the tensor-valued conductivity variable only. We then construct equilibrium measures of the resulting PDE system. Finally, we derive the gradient flow of the constrained energy functional with respect to the Fisher-Rao (or Hellinger-Kakutani) metric, which gives a reaction-type PDE. We calculate its equilibrium states, represented by measures concentrated on a hypersurface in the phase space.

Measure-based approach to mesoscopic modeling of optimal transportation networks

TL;DR

This work derives the gradient flow of the constrained energy functional with respect to the Fisher-Rao (or Hellinger-Kakutani) metric, which gives a reaction-type PDE system and construct equilibrium measures of the resulting PDE system.

Abstract

We propose a mesoscopic modeling framework for optimal transportation networks with biological applications. The network is described in terms of a joint probability measure on the phase space of tensor-valued conductivity and position in physical space. The energy expenditure of the network is given by a functional consisting of a pumping (kinetic) and metabolic power-law term, constrained by a Poisson equation accounting for local mass conservation. We establish convexity and lower semicontinuity of the functional on approriate sets. We then derive its gradient flow with respect to the 2-Wasserstein topology on the space of probability measures, which leads to a transport equation, coupled to the Poisson equation. To lessen the mathematical complexity of the problem, we derive a reduced Wasserstein gradient flow, taken with respect to the tensor-valued conductivity variable only. We then construct equilibrium measures of the resulting PDE system. Finally, we derive the gradient flow of the constrained energy functional with respect to the Fisher-Rao (or Hellinger-Kakutani) metric, which gives a reaction-type PDE. We calculate its equilibrium states, represented by measures concentrated on a hypersurface in the phase space.
Paper Structure (16 sections, 19 theorems, 182 equations)

This paper contains 16 sections, 19 theorems, 182 equations.

Table of Contents

  1. Introduction
  2. Discrete model
  3. Mesoscopic model and Wasserstein gradient flow
  4. Gradient flow in the $2$-Wasserstein topology
  5. Mathematical properties of the energy functional
  6. Wasserstein gradient flows
  7. Reduced Wasserstein gradient flow
  8. Scalar model.
  9. Stationary solutions
  10. Reduced Wasserstein model
  11. Wasserstein model.
  12. Monokinetic model
  13. Scalar model
  14. Semi-discrete model
  15. Fisher-Rao gradient flow
  16. ...and 1 more sections

Key Result

Lemma 1

Let $S \in L^2(\Omega)$, $\mathbb{P}[\mu] \in L^1 (\Omega)$. Then the Poisson equation eq:poisson1-eq:bc_poisson admits a unique weak solution $p \in H$ satisfying where $C_{\Omega} > 0$ is the Poincarè constant.

Theorems & Definitions (39)

  • Lemma 1
  • proof
  • Proposition 1
  • proof
  • Proposition 2
  • proof
  • Proposition 3
  • proof
  • Theorem 1
  • proof
  • ...and 29 more