2024/12/06 by Gladbach, Peter, Maas, Jan, Portinale, Lorenzo
#49J45 #49M25 #49Q22 #65K10 #74Q10 #Analysis of PDEs (math.AP) #FOS: Mathematics #Optimization and Control (math.OC)
paper · doi:10.48550/arxiv.2412.05217
This paper deals with the large-scale behaviour of nonlinear minimum-cost flow problems on random graphs. In such problems, a random nonlinear cost functional is minimised among all flows (discrete vector-fields) with a prescribed net flux through each vertex. On a stationary random graph embedded in ℝd, our main result asserts that these problems converge, in the large-scale limit, to a continuous minimisation problem where an effective cost functional is minimised among all vector fields with prescribed divergence. Our main result is formulated using Γ-convergence and applies to multi-species problems. The proof employs the blow-up technique by Fonseca and Müller in a discrete setting. One of the main challenges overcome is the construction of the homogenised energy density on random graphs without a periodic structure.