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 finite‐difference methods and updating schemes based on the secant equation are presented and compared numerically inside the multilevel trust‐region algorithm proposed by Gratton et al. (2008).