# Rigorous continuum limit for the discrete network formation problem

@article{Haskovec2018RigorousCL, title={Rigorous continuum limit for the discrete network formation problem}, author={Jan Haskovec and Lisa Maria Kreusser and Peter A. Markowich}, journal={Communications in Partial Differential Equations}, year={2018}, volume={44}, pages={1159 - 1185} }

Abstract Motivated by recent papers describing the formation of biological transport networks we study a discrete model proposed by Hu and Cai consisting of an energy consumption function constrained by a linear system on a graph. For the spatially two-dimensional rectangular setting we prove the rigorous continuum limit of the constrained energy functional as the number of nodes of the underlying graph tends to infinity and the edge lengths shrink to zero uniformly. The proof is based on… Expand

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#### References

SHOWING 1-10 OF 26 REFERENCES

ODE and PDE based modeling of biological transportation networks.

- Mathematics, Physics
- 2018

We study the global existence of solutions of a discrete (ODE based) model on a graph describing the formation of biological transportation networks, introduced by Hu and Cai. We propose an… Expand

Mathematical Analysis of a PDE System for Biological Network Formation

- Mathematics, Physics
- 2014

Motivated by recent physics papers describing rules for natural network formation, we study an elliptic-parabolic system of partial differential equations proposed by Hu and Cai [13, 15]. The model… Expand

Continuum Modeling of Biological Network Formation

- Mathematics
- 2017

We present an overview of recent analytical and numerical results for the elliptic–parabolic system of partial differential equations proposed by Hu and Cai, which models the formation of biological… Expand

A mesoscopic model of biological transportation networks.

- Computer Science, Mathematics
- 2018

A mesoscopic model for natural network formation processes is introduced, acting as a bridge between the discrete and continuous network approach proposed by Hu and Cai, and it is proved that if the metabolic energy consumption term is concave with respect to the conductivities, the optimal network structure is a tree. Expand

Notes on a PDE System for Biological Network Formation

- Mathematics
- 2015

We present new analytical and numerical results for the elliptic-parabolic system of partial differential equations proposed by Hu and Cai [8, 10], which models the formation of biological transport… Expand

Biological transportation networks: Modeling and simulation

- Mathematics
- 2016

We present a model for biological network formation originally introduced by Cai and Hu [Adaptation and optimization of biological transport networks, Phys. Rev. Lett. 111 (2013) 138701]. The… Expand

Adaptation and optimization of biological transport networks.

- Computer Science, Medicine
- Physical review letters
- 2013

The adaptation dynamics minimizes the global energy consumption to produce optimal networks, which may possess hierarchical loop structures in the presence of strong fluctuations in flow distribution, and shows that there may exist a new phase transition as there is a critical open probability of sinks. Expand

Chapter 2 A handbook of Г-convergence

- Mathematics
- 2006

Publisher Summary This chapter discusses the main properties of Γ -convergence, in particular those that are useful in the actual computation of Γ -limits. For some classes of functional, a common… Expand

Handbook of graph theory

- Computer Science, Mathematics
- Discrete mathematics and its applications
- 2003

Introduction to Graphs Fundamentals of Graph Theory, Jonathan L. Gross and Jay Yellen Families of Graphs and Digraphs, Lowell W. Wilson Graph Representation Computer Representation of graphs, Alfred V. Aho Graph Isomorphism, Brendan D. McKay The Reconstruction Problem, Josef Lauri Recursively Constructed Graphs, Richard B. Tovey Structural Graph Theory. Expand

An Introduction to-convergence

- Mathematics
- 1993

1. The direct method in the calculus of variations.- 2. Minimum problems for integral functionals.- 3. Relaxation.- 4. ?-convergence and K-convergence.- 5. Comparison with pointwise convergence.- 6.… Expand