Homogenisation of dynamical optimal transport on periodic graphs

Area of Science:

• Applied Mathematics
• Scientific Computing
• Optimization Theory

Background:

• Dynamical optimal transport is crucial for modeling various physical phenomena.
• Understanding large-scale behavior on discrete structures like graphs is computationally challenging.
• Homogenization techniques are essential for bridging discrete and continuous models.

Purpose of the Study:

• To analyze the large-scale behavior of dynamical optimal transport on periodic graphs.
• To derive a homogenization result connecting discrete and continuous optimal transport problems.
• To characterize the effective energy density using a cell formula.

Main Methods:

• Utilizing Gamma-convergence for action functionals on curves of measures.
• Proving a homogenization result for discrete optimal transport problems.
• Developing a cell formula based on finite-dimensional convex programming.

Main Results:

• A homogenization result is established, describing the effective behavior of discrete optimal transport.
• The effective energy density is explicitly expressed via a cell formula.
• The cell formula depends on the graph's local geometry and energy density.

Conclusions:

• The study provides a rigorous mathematical framework for understanding large-scale optimal transport on periodic graphs.
• The derived cell formula offers explicit computation of effective properties.
• The findings are applicable to discretizations like finite-volume methods for Wasserstein distance.