Approximating Hessians in unconstrained optimization arising from discretized problems

We consider Hessian approximation schemes for large-scale unconstrained optimization in the context of discretized problems. The considered Hessians typically present a nontrivial sparsity and partial separability structure. This allows iterative quasi-Newton methods to solve them despite of their size. Structured finitedifference methods and updating schemes based on the secant equation are presented and compared numerically inside the multilevel trust-region algorithm proposed by Gratton, Mouffe, Toint, and Weber-Mendoça (IMA J. Numer. Anal. 28(4), 2008).