Architecture of optimal transport networks

This article presents a set of tools for the modeling of a spatial allocation problem in a large geographic market and gives examples of applications. In our settings, the market is described by a network that maps the cost of travel between each pair of adjacent locations. Two types of agents are located at the nodes of this network. The buyers choose the most competitive sellers depending on their prices and the cost to reach them. Their utility is assumed additive in both ...

Topology optimization (TopOpt) is a mathematical-driven design procedure to realize optimal material architectures. This procedure is often used to automate the design of devices involving flow through porous media, such as micro-fluidic devices. TopOpt offers material layouts that control the flow of fluids through porous materials, providing desired functionalities. Many prior studies in this application area have used Darcy equations for primal analysis and the minimum pow...

In this work, we present a novel tool for reconstructing networks from corrupted images. The reconstructed network is the result of a minimization problem that has a misfit term with respect to the observed data, and a physics-based regularizing term coming from the theory of optimal transport. Through a range of numerical tests, we demonstrate that our suggested approach can effectively rebuild the primary features of damaged networks, even when artifacts are present.

The branched transport problem, a popular recent variant of optimal transport, is a non-convex and non-smooth variational problem on Radon measures. The so-called urban planning problem, on the contrary, is a shape optimization problem that seeks the optimal geometry of a street or pipe network. We show that the branched transport problem with concave cost function is equivalent to a generalized version of the urban planning problem. Apart from unifying these two different mo...

While many large infrastructure networks, such as power, water, and natural gas systems, have similar physical properties governing flows, these systems tend to have distinctly different sizes and topological structures. This paper seeks to understand how these different size-scales and topological features can emerge from relatively simple design principles. Specifically, we seek to describe the conditions under which it is optimal to build decentralized network infrastructu...

We consider a dissipative flow network that obeys the standard linear nodal flow conservation, and where flows on edges are driven by potential difference between adjacent nodes. We show that in the case when the flow is a monotonically increasing function of the potential difference, solution of the network flow equations is unique and can be equivalently recast as the solution of a strictly convex optimization problem. We also analyze the maximum throughput problem on such ...

The analysis and boundary optimal control of the nonlinear transport of gas on a network of pipelines is considered. The evolution of the gas distribution on a given pipe is modeled by an isothermal semilinear compressible Euler system in one space dimension. On the network, solutions satisfying (at nodes) the so called Kirchhoff flux continuity conditions are shown to exist in a neighborhood of an equilibrium state. The associated nonlinear optimization problem then aims at ...

From the vasculature of animals to the porous media making up batteries, transport by fluid flow within complex networks is crucial to service all cells or media with resources. Yet, living flow networks have a key advantage over porous media: they are adaptive and can self-organize their geometry to achieve a homogeneous perfusion throughout the network. Here, we show that, through erosion, artificial flow networks self-organize to a geometry where perfusion is more homogene...

We introduce a numerical method for extracting minimal geodesics along the group of volume preserving maps, equipped with the L2 metric, which as observed by Arnold solve Euler's equations of inviscid incompressible fluids. The method relies on the generalized polar decomposition of Brenier, numerically implemented through semi-discrete optimal transport. It is robust enough to extract non-classical, multi-valued solutions of Euler's equations, for which the flow dimension is...

Existing techniques for the cost optimization of water distribution networks either employ meta-heuristics, or try to develop problem-specific optimization techniques. Instead, we exploit recent advances in generic NLP solvers and explore a rich set of model refinement techniques. The networks that we study contain a single source and multiple demand nodes with residual pressure constraints. Indeterminism of flow values and flow direction in the network leads to non-linearity...