This research concerns the algorithmic study of Hessian approximation in the context of multilevel nonlinear optimization problems. Methods using finite-difference approximations or secant equations are here considered. We present new methods for the Hessian approximation and update, and numerically compare their performance with existing methods. A software implementing an efficient method for Hessian approximation in this context has been developed, and is documented in this manuscript. Moreover an application of these developments to the modelling of snake-skin pigmentation patterns is presented.