Different numerical estimators for main effect global sensitivity indices

The variance-based method of global sensitivity indices based on Sobol sensitivity indices became very popular among practitioners due to its easiness of interpretation. For complex practical problems computation of Sobol indices generally requires a large number of function evaluations to achieve reasonable convergence. Four different direct formulas for computing Sobol main effect sensitivity indices are compared on a set of test problems for which there are analytical results. These formulas are based on high-dimensional integrals which are evaluated using MC and QMC techniques. Direct formulas are also compared with a different approach based on the so-called double loop reordering formula. It is found that the double loop reordering (DLR) approach shows a superior performance among all methods both for models with independent and dependent variables.