Sparse collocation method for global sensitivity analysis and calculation of statistics of solutions in SPDEs

This paper explores advantages offered by the stochastic collocation method based on the Smolyak grids for the solution of differential equations with random inputs in the parameter space. We use sparse Smolyak grids and the Chebyshev polynomials to construct multidimensional basis and approximate decoupled stochastic differential equations via interpolation. Disjoint set of grid points and basis functions allow us to gain significant improvement to conventional Smolyak algorithm. Density function and statistical moments of the solution are obtained by means of quadrature rules if inputs are uncorrelated and uniformly distributed. Otherwise, the Monte Carlo analysis can run inexpensively using obtained sparse approximation. An adaptive technique to sample from a multivariate density function using sparse grid is proposed to reduce the number of required sampling points. Global sensitivity analysis is viewed as an extension of the sparse interpolant construction and is performed by means of the Sobol' variance-based or the Kullback-Leibler entropy methods identifying the degree of contribution from the individual inputs as well as the cross terms. Copyright (c) 2016 John Wiley & Sons, Ltd.