Probabilistic sensitivity methods for correlated normal variables

A methodology for computing probabilistic sensitivities with respect to the means, standard deviations, and correlation coefficients of correlated normal variables is presented. This approach is an extension of the 'Score Function' method for use with sampling methods and arbitrary limit states. The method is formulated in terms of an expected value operator such that the existing samples from a probabilistic analysis can be used to obtain the partial derivatives of the probability-of-failure or response moments as a function of the input distribution's means, standard deviations and correlation coefficients with typically negligible cost. The key ingredient is the development of multidimensional kernel functions for correlated normal variables which are herein derived. The method is straightforward to implement in any sampling-based method given the kernel functions. No restrictions exist on the form of the limit state nor are transformations required. The method indicates that sensitivities with respect to the correlation coefficient may be nonzero even if the correlation coefficient itself is zero. Numerical comparisons with analytical and numerical derivatives are presented.