Typical applications are data assimilation, model tuning, and sensitivity analysis.
Both data assimilation and model tuning derive a set of control variables that achieves an optimal degree of consistency between simulated and observed quantities. To analyze the uncertainties in the inferred optimal values of the control variables, second order derivates of the scalar valued cost function are of interest. Since, usually, the number of control variables is large, evaluation of the full second order derivative, i.e. the Hessian matrix, is prohibitively expensive. However, Hessian vector products are relatively cheap and provide a module to evaluate certain properties of the Hessian matrix. For example the best constrained directions are the leading eigenvectors of the Hessian matrix and can be determined iteratively by Lanczos type algorithms.