Measure-based approach to mesoscopic modeling of optimal transportation networks
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arXiv
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| Format: | Preprint |
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2024
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| author | Haskovec, Jan Markowich, Peter Portaro, Simone |
| author_facet | Haskovec, Jan Markowich, Peter Portaro, Simone |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2401_07922 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Measure-based approach to mesoscopic modeling of optimal transportation networks Haskovec, Jan Markowich, Peter Portaro, Simone Analysis of PDEs 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. |
| title | Measure-based approach to mesoscopic modeling of optimal transportation networks |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2401.07922 |