Variance-Based Global Sensitivity Analysis via Sparse-Grid Interpolation and Cubature (Communications in Computational Physics, 2011)
The stochastic collocation method using sparse grids has become a popularchoice for performing stochastic computations in high dimensional (random) parameterspace. In addition to providing highly accurate stochastic solutions, the sparse gridcollocation results naturally contain sensitivity information with respect to the inputrandom parameters. In this paper, we use the sparse grid interpolation and cubaturemethods of Smolyak together with combinatorial analysis to give a computationallyefficient method for computing the global sensitivity values of Sobol'. This method allowsfor approximation of all main effect and total effect values from evaluation of f ona single set of sparse grids. We discuss convergence of this method, apply it to severaltest cases and compare to existing methods. As a result which may be of independentinterest, we recover an explicit formula for evaluating a Lagrange basis interpolatingpolynomial associated with the Chebyshev extrema. This allows one to manipulatethe sparse grid collocation results in a highly efficient manner.