Remarks about the numerical inversion of injective nonlinear maps

The problem of designing an algorithm which computes the left inverse of an injective immersion is studied. It is first shown how standard Newton and gradient descent algorithms fail to work when no assumption is made on the image set of the map to be inverted. In particular, solutions might converge to the wrong point or diverge in finite/infinite time even when the map is a (global) diffeomorphism. Inspired by recent results concerning the implementation of observers in given coordinates, we investigate possible means to modify the given map in order to guarantee completeness and convergence of those algorithms.