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Center for Nonlinear Studies |
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 (Physarum Polycephalum). We conjecture that the long-time solution of the proposed model approaches the solution of the PDE base Monge-Kantorovich Optimal Transport equations. This new OT formulation assumes that the transport density and potential are time dependent and satisfy the elliptic PDE. The classical constraint on the norm of the gradient is then replaced by an ODE describing transient dynamics of the transport density. The conjecture is supported by yet largely incomplete analytical results and several numerical experiments performed on a number of published test cases. One of the most important advantages of the proposed formulation is that its numerical solution is very efficient and well-defined using simple discretization schemes. Moreover, simple modifications of the proposed model yield dynamic versions of the branched and congested transport problems. We shall discuss our numerical approaches, which are based on either a standard linear (P1) Galerkin method or a pseudspectral scheme combined with a P0 approximation of the diffusion coefficient, for the discretization of the elliptic PDE and Euler time stepping and Picard iteration to solve the resulting nonlinear differential-algebraic equation. Preliminary numerical simulations are used to show that the proposed formulation is efficient in finding solutions also of congested and branched transport tests. Although imitations arise when trying to solve highly discontinuous problems, we present experimental convergence results showing robustness of the scheme for a wide range of sample problems. Finally, we will examine models and related numerical results of diverse applications ranging from slime-mold dynamics to geomorphological problems, and discuss current and future progress.
Host: Vitaliy Gyrya