  Second derivatives of mathematical models for real-world phenomena are
fundamental ingredients of a wide range of numerical simulation methods
including parameter sensitivity analysis, uncertainty quantification, nonlinear
optimization and model calibration. The evaluation of such Hessians often
dominates the overall computational effort. Various combinatorial optimization
problems can be formulated based on the highly desirable exploitation of the
associativity of the chain rule of differential calculus. The fundamental
Hessian Accumulation problem aiming to minimize the number of floating-point
operations required for the computation of a Hessian turns out to be
NP-complete. The restriction to suitable subspaces of the exponential search
space proposed in this paper ensures computational tractability while yielding
improvements by factors of ten and higher over standard approaches based on
second-order tangent and adjoint algorithmic differentiation.
  Motivated by second-order parameter sensitivity analysis of surrogate
numerical models obtained through training and pruning of deep neural networks
this paper focusses on bracketing of dense Hessian chain products with the aim
of minimizing the total number of floating-point operations to be performed.
The results from a given dynamic programming algorithm for optimized bracketing
of the underlying dense Jacobian chain product are used to reduce the
computational cost of the corresponding Hessian. Minimal additional algorithmic
effort is required.
