Abstract: In this talk I will report on some joint activities with Jens Lang (TU Darmstadt) and Jan Verwer (CWI) regarding efficiency and reliability questions for finite difference approximations of parabolic problems. First, systems of ODEs are considered. Existing popular codes focus on efficiency by adaptively optimizing time grids in accordance to local error control. The reliability question, that is, how large are the global errors, has received much less attention. We have implemented classical global error estimation based on the first variational equation, and global error control, for which we have used the property of tolerance proportionality. We have found, using the Runge-Kutta-Rosenbrock method ROS3P as example integrator, that the classical approach is remarkably reliable. For finite difference approximations of parabolic PDEs, the ODE approach is combined with estimates for the spatial truncation errors based on Richardson extrapolation. Numerical examples are used to illustrate the reliability of the estimation and control strategies.

Abstract: In this conference we first recall the definition of sub-laplacian and mean curvature operators in a Lie group with a subriemannian metric. Then we focus on the minimal surfaces, which are graphs of a functions in the Heisenberg group, and prove that they are C^∞ in a suitable intrinsic sense. If the dimension of the space is grater than 3, this mean that the solutions are smooth in the classical sense. If the dimension of the space is 3, minimal graphs are foliated in legendrian curves.
Application to visual perception will be presented.