An $hp$-adaptive multi-element stochastic collocation method for surrogate modeling with information re-use

Abstract

This paper introduces an $hp$-adaptive multi-element stochastic collocation method, which additionally allows to re-use existing model evaluations during either $h$- or $p$-refinement. The collocation method is based on weighted Leja nodes. After $h$-refinement, local interpolations are stabilized by adding and sorting Leja nodes on each newly created sub-element in a hierarchical manner. For $p$-refinement, the local polynomial approximations are based on total-degree or dimension-adaptive bases. The method is applied in the context of forward and inverse uncertainty quantification to handle non-smooth or strongly localised response surfaces. The performance of the proposed method is assessed in several test cases, also in comparison to competing methods.