Simplex Stochastic Collocation with Random Sampling and Extrapolation for Nonhypercube Probability Spaces

• Authors: Jeroen A. S. Witteveen;Gianluca Iaccarino
• Venue: SIAM Journal on Scientific Computing
• Year: 2012

Stochastic collocation (SC) methods for uncertainty quantification (UQ) in computational problems are usually limited to hypercube probability spaces due to the structured grid of their quadrature rules. Nonhypercube probability spaces with an irregular shape of the parameter domain do, however, occur in practical engineering problems. For example, production tolerances and other geometrical uncertainties can lead to correlated random inputs on nonhypercube domains. In this paper, a simplex stochastic collocation (SSC) method is introduced, as a multielement UQ method based on simplex elements, that can efficiently discretize nonhypercube probability spaces. It combines the Delaunay triangulation of randomized sampling at adaptive element refinements with polynomial extrapolation to the boundaries of the probability domain. The robustness of the extrapolation is quantified by the definition of the essentially extremum diminishing (EED) robustness principle. Numerical examples show that the resulting SSC-EED method achieves superlinear convergence and a linear increase of the initial number of samples with increasing dimensionality. These properties are demonstrated for uniform and nonuniform distributions, and correlated and uncorrelated parameters in problems with 15 dimensions and discontinuous responses.