Modelling Stable Backward Diffusion and Repulsive Swarms with Convex Energies and Range Constraints

Backward diffusion and purely repulsive swarm dynamics are generally feared as ill-posed, highly unstable processes. On the other hand, it is well-known that minimising strictly convex energy functionals by gradient descent creates well-posed, stable evolutions. We prove a result that appears counterintuitive at first glance: We derive a class of one-dimensional backward evolutions from the minimisation of strictly convex energies. Moreover, we stabilise these inverse evolutions by imposing range constraints. This allows us to establish a comprehensive theory for the time-continuous evolution, and to prove a stability condition for an explicit time discretisation. Prototypical experiments confirm this stability and demonstrate that our model is useful for global contrast enhancement in digital greyscale images and for modelling purely repulsive swarm dynamics.