Optimal Micropatterns in 2D Transport Networks and Their Relation to Image Inpainting

Abstract

We consider two different variational models of transport networks, the so-called branched transport problem and the urban planning problem. Based on a novel relation to Mumford-Shah image inpainting and techniques developed in that field, we show for a two-dimensional situation that both highly non-convex network optimization tasks can be transformed into a convex variational problem, which may be very useful from analytical and numerical perspectives.As applications of the convex formulation, we use it to perform numerical simulations (to our knowledge this is the first numerical treatment of urban planning), and we prove the lower bound of an energy scaling law which helps better understand optimal networks and their minimal energies.

Asymptotic Self-Similarity of Minimizers and Local Bounds in a Model of Shape-Memory Alloys

“…Taking the limit j → ∞ the same holds with d 1 instead of d 1,j , and by continuity of u we obtain (9).…”

“…These vectorial generalizations have confirmed that the Kohn-Müller scalar model indeed captures the correct scaling of the energy. A number of other problems have then been addressed with similar tools, including magnetization patterns in uniaxial ferromagnets [14,15,37], diblock copolymers [1,13], flux tubes in type-I superconductors [16,20,22,28], wrinkling in thin elastic films [4,6,7,34], dislocation microstructures [18,19], transport network structures [8,9], and compliance minimization [42].…”

Inexact first-order primal–dual algorithms

“…The (inexact) proximum may be computed in a straightforward fashion using Dykstra's algorithm [29], which has e.g. been done in [11] or [1,17,Ex. 7.7] for Mumford-Shah-type segmentation problems.…”

On the well-posedness of branched transportation

“…Explicit optima are known only in few (mainly discrete) cases; for this reason, some effort has been put in developing numerical strategies to compute minimizers, for instance in term of phasefield approximations [37,12,4], in the spirit of numerical calibrations [35,3], or exploiting the convex nature of different formulations of some aspects of the problem (which is overall highly nonconvex) [30,31,6].…”