Refinement Criteria for Simplex Stochastic Collocation with Local Extremum Diminishing Robustness

The simplex stochastic collocation method is developed for adaptive uncertainty quantification in computational problems with random inputs. The effectiveness of adaptive formulations is determined to a large extent by both the $h$- and $p$-refinement measures and the stopping criterion. An improved stopping criterion and different $h$-refinement measures are derived from an error estimate based on the root mean square of the hierarchical surpluses. A $p$-criterion for the polynomial degree $p$ is established that achieves, in principle, a constant order of convergence with dimensionality. The robustness of the adaptive method is guaranteed by the local extremum diminishing concept extended to probability space using the introduced local extremum conserving limiter for $p$. Numerical results show both that the stopping criterion is reliable owing to the accurate and conservative error estimate, and that the refinement measures outperform uniform refinement.