Our aim is to consider a variational approach for different evolution problems, parabolic and hyperbolic in nature, arising in geometry,
physics, and complex systems. We borrow ideas from nonsmooth convex optimization and optimal
transport, and develop suitable novel efficient numerical schemes for a constructive approximation of their solutions.
Geometric evolution problems typically develop singularities in finite time. We aim to prove global existence for certain hyperbolic flows by minimizing suitable energy functionals, and to develop accordingly efficient numerical schemes preserving dynamical features (e.g.
symplectic schemes), adapting level set techniques (e.g. BMO-type schemes), time discretization (e.g. Morse semiflow) in combination with
primal-dual methods, splitting methods and exponential integrators. The robustness of the schemes and their validation along special solutions will enable us to perform simulations giving significant information of the evolution also after the formation of singularities in relevant situations completely unexplored so far, such as topology changes in the evolution, and in the case of networks of curves and surfaces.
Second, we consider the control of multi-agent systems whose microscopic dynamics is given by a differential inclusion with nonlocal terms. Thispoint of view can be used also to model uncertainty in classical deterministic control systems. A natural approach inspired to optimal
transportation consists in minimizing the value function in a metric measure space endowed with the Wasserstein distance. We aim to address
relevant issues concerning the existence and numerical approximation of optimal trajectories, to provide necessary conditions, and to develop suitablenotions of metric differentiability in order to derive regularity estimates for the value function, characterizing it as a
viscosity solution of a Hamilton-Jacobi-Bellman equation in the space of measures.
