In nuclear design it is important to know the effect of uncertainties in key parameters on reaction rates. This gives the sensitivity of reaction rates to small changes (or perturbations) to these key parameters. For complex systems like nuclear reactors, it is often necessary to resort to Monte Carlo methods for this kind of sensitivity study. The uncertainty in a reaction rate, due to small changes in a system parameter, can be defined as the derivative of the reaction rate w.r.t. the system parameter. Correlated simulation estimates the change due to a given variation of parameter(s), whereas a differential Monte Carlo simulation estimates the change due to arbitrary (but small) variations in the system parameters. Hence, the derivatives are characteristic of the sensitivity of the reaction rate to variations in the parameters. In the case of several parameters, multivariate Taylor series applies and the partial derivatives of the reaction rate w.r.t. the various parameters are estimated.
For eigenvalue perturbation calculations, the scores of first and second order derivatives of the unperturbed system are stored in separate matrices. After the fission matrix of the unperturbed system is completed, the fission matrix of the perturbed system is calculated by a Taylor series approximation for the perturbation of the system.
An unbiased procedure of estimating the first derivative, for reactivity changes due to small variations in system parameters, was first proposed by Mikhailov [Mik67] and independently by Miller [Mil67] and Takahashi [Tak70]. A constructive derivation of a multiparameter second-order derivative estimation procedure was shown by Hall [Hal80, Hal82].
Both the correlated sampling and derivative operator sampling have the advantage that they require little additional computing effort to calculate the perturbation effects. No calculation for importance parameters are required, as is the case for the importance function approach, which is described next.