An Efficient Reduced-order Method for Hessian Matrix Construction

When nonlinear behavior must be considered in sensitivity analysis studies, one needs to approximate higher order derivatives of the response of interest with respect to all input data. This paper presents an application of a general reduced order method to constructing higher order derivatives of response of interest with respect to all input data. In particular, we apply the method to constructing all second order derivatives which are often compactly combined in a matrix denoted by the Hessian matrix. Compared to the state-of-the-art methods for Hessian constructions including notably rank-update formulas, the presented Subspace Reduced Order Method requires only r evaluations of the first order derivatives as opposed to n, where r is the rank of the Hessian matrix, and n is the number of input data. Moreover, the new method is less sensitive to numerical errors as it does not rely on rank-update formulas. Finally, in contract to a class of rank-update methods which requires serial evaluation of derivatives, our method evaluates the r derivatives in parallel, thereby rendering it more suitable for coarse-grain parallelization. The paper derives the new methods and applies it to a criticality problem employing the SCALE 6.1 (TSUNAMI-2D) code system.