Ricerc@Sapienza

In this work, we consider the discretization of some nonlinear Fokker-Planck-Kolmogorov equations. The scheme we propose preserves the nonnegativity of the solution, conserves the mass, and, as the discretization parameters tend to zero, has limit measure-valued trajectories which are shown to solve the equation. The main assumptions to obtain a convergence result are that the coefficients are continuous and satisfy a suitable linear growth property with respect to the space variable. In particular, we obtain a new proof of existence of solutions for such equations.

In this survey, we present some numerical schemes for the approximation of level set models related to image processing and segmentation, focusing in particular on finite difference and semi-Lagrangian methods. The schemes reviewed here show various interesting features: in particular, they are explicit, may allow for large time steps, and handle degenerate problems and nonsmooth solutions without adding too much numerical diffusion. Several examples will illustrate their use in the image processing problems under consideration