  We study the connections between Physarum Dynamics and Dynamic Monge
Kantorovich (DMK) Optimal Transport algorithms for the solution of Basis
Pursuit problems. We show the equivalence between these two models and unveil
their dynamic character by showing existence and uniqueness of the solution for
all times and constructing a Lyapunov functional with negative Lie-derivative
that drives the large-time convergence. We propose a discretization of the
equation by means of a combination of implicit time-stepping and Newton method
yielding an efficient and robust method for the solution of general basis
pursuit problems. Several numerical experiments run on literature benchmark
problems are used to show the accuracy, efficiency, and robustness of the
proposed method.
