Sensitivity analysis is the study of how uncertainty in the QoI can be apportioned to different sources of uncertainty in the input parameters. Unlike correlations where the effects of parameters are estimated based on the model evaluation data, the Sobol method looks at the entire input parameter distribution and decomposes the variance of the QoI into a sum of contributions from the input parameters and their interactions. The number of terms in the Sobol indices of the variance of the QoI with m input parameters grows as 2m. It is customary to compute only the m first-order effects (first-order Sobol indices) and the m total effects (total Sobol indices).

Given a scalar QoI defined as y = M(x1, x2, …, xm), the variance-based first-order Sobol index is defined as

where xi is the ith input parameter and x\i denotes all the other parameters but xi. The meaning of the inner expectation value is that the mean of y is taken over all possible values of x\i while fixing xi. The outer variance is then computed over all possible values of xi.

which measures the summation of the first-order effect and interactions with other parameters of the ith input parameter. The second term in the equation can be seen as the first-order effect of all parameters but the ith parameter. So, one minus the second term must give the contribution of all terms related to xi. Note that for a UQ study with multiple QoIs, the Sobol indices are computed for each QoI.

There are two types of methods to compute the Sobol indices. One is through postprocessing of a polynomial chaos expansion (PCE) model. Given the definition of the PCE, the contributions from each input parameter and their interactions are readily separable. Thus the Sobol indices can be computed solely based on the coefficients trained from the PCE. More details on the computation of Sobol indices based on the PCE model can be found in Ref. 6.

The other method of computing the Sobol indices is through Monte Carlo analysis. As it is computationally expensive to perform Monte Carlo simulation directly with model evaluation, it provides a Gaussian process as the surrogate model for this type of analysis. The algorithm follows the best practice of simultaneous computation of the first and total Sobol index described in Ref. 11.