Third- and Fourth-Order Adjoint Sensitivity and Uncertainty Analysis of the PERP Benchmark

Abstract

The sensitivity analyses performed in previous chapters indicated that the largest first- and second-order sensitivities of the PERP leakage response were with respect to the group-averaged total microscopic cross sections, which were thus singled out to be the most influential parameters underlying the PERP benchmark. It is therefore of interest to investigate the magnitudes of the third-order sensitivities of the PERP’s leakage response to the total microscopic cross sections, which will be performed in this chapter by applying the 3rd-CASAM-L. It will be shown that some of the third-order sensitivities are even larger than the second-order ones, which consequently has motivated the investigation of the fourth-order sensitivities that correspond to the largest of the third-order ones. The computation of the fourth-order sensitivities is performed by applying the principles underlying the 4th-CASAM-L, and the results thus obtained indicated that several of the fourth-order sensitivities are even larger than the largest third-order ones. The impact of the sensitivities of various orders on the moments (expected value, standard deviation, and skewness) of the leakage response’s distribution in the parameter space is also investigated in this chapter. It will be shown that if the parameters’ standard deviations are very small (<1%), the first-order sensitivities have the largest impact on these response distribution moments. However, if the parameters’ standard deviations are larger than ca. 5%, the impact of the third- and fourth-order sensitivities far outweigh the impacts of the first- and second-order ones. Also, the relative importance of the higher-order sensitivities increases with increasing parameter standard deviations. The third- and fourth-order sensitivities also cause the leakage response distribution to be skewed toward the positive direction from its expected value. Finally, it will also be shown in this chapter that using finite-differences becomes computationally unfeasible for obtaining higher-order sensitivities because of the “curse of dimensionality.” Furthermore, finding the optimal step size, which would minimize the error between the finite difference result and the exact result, is practically impossible to achieve unless one knows beforehand the exact result—which is possible only by using the adjoint sensitivity analysis methodology.