arXiv subjects
Publications and source records attributed to G. Wolansky.
We prove the convergence of a particle method for the approximation of diffusive gradient flows in one dimension. This method relies on the discretisation of the energy via non-overlapping balls centred at the particles and preserves the gradient flow structure at the particle level. The strategy of the proof is based on an abstract result for the convergence of curves of maximal slope in metric spaces.
We study the numerical behaviour of a particle method for gradient flows involving linear and nonlinear diffusion. This method relies on the discretisation of the energy via non-overlapping balls centred at the particles. The resulting scheme preserves the gradient flow structure at the particle level, and enables us to obtain a gradient descent formulation after time discretisation. We give several simulations to illustrate the validity of this method, as well as a detailed study of one-dimensional aggregation-diffusion equations.
The extended principle of minimal action is described in the presence of prescribed source and sink points. Under the assumption of zero net flux, it leads to an optimal Monge-Kantorovich transport problem of metric type. We concentrate on action corresponding to a mecahnical Lagrangian. The optimal solution turns out to be a measure supprted on a graph composed of geodesic arcs connecting pairs of sources and sinks.
The definition of a mobilized system and its efficiency are introduced. The existence of an optimal (maximally efficient) system is proved by an application of Young measures and compensated compactness.