Biologically inspired formulation of optimal transportation problems. An unexpected branching source

Enrico Facca, Franco Cardin, Mario Putti

Dipartimento di Matematica, Universita’ di Padova

Monday 17th October 2016

10:30:00

AULA CONSIGLIO VII Piano DIPARTIMENTO DI MATEMATICA, POLITECNICO DI MILANO

We have recently developed an approach, based on an extension of a model proposed by Tero et al (2007) for the simulation of the dynamics of a slime mold (Physarium Polycephalum). We conjecture that this model is an original formulation of the PDE-based OT problem. This new formulation assumes that the potential and the diffusion coefficient (the latter yielding the transportation plan) are time dependent. The classical constraint on the norm of the gradient is then replaced by an ODE describing transient dynamics of the diffusion coefficient. Analytical results in the case of the Monge-Kantorovich problem, although yet largely incomplete, suggest that indeed the conjecture is true. This is supported by several numerical experiments showing that at large times the solution to this problem is equivalent to the solution of the classical Monge-Kantorovich PDE based OT. One of the most important advantages of the proposed formulation is that its numerical solution is very efficient and well-defined using simple numerical approaches. Moreover, this dynamical extension allows the reconstruction of the time-history of the process, thus enlarging the applicability of the OT model to wider sets of processes. The proposed model can also be easily adapted to branched and congested transport problems. Preliminary numerical simulations show that the proposed formulation is efficient in finding solutions also of congested transport and branched transport problems. While the interpretation via $p$-Laplacians allows a straight forward interpretation of the new formulation in the simulation of congested transport, only numerical evidence is currently available for applications to branched transport problems. Although obvious limitations are present in the numerical solution of highly discontinuous problems, we present some experimental convergence results showing robustness of the scheme for sufficiently regular solutions. Finally, we will present models and related numerical results of diverse applications of this formulation ranging from slime-mold dynamics to geomorphological applications, and discuss current and future progress. contact: christian.vergara@polimi.it