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

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 Leja nodes on each newly created sub-element in a hierarchical manner. A dimension-adaptive algorithm is employed for local $p$-refinement. The suggested multi-element stochastic collocation method is employed in the context of forward and inverse uncertainty quantification to handle non-smooth or strongly localised response surfaces. Comparisons against existing multi-element collocation schemes and other competing methods in five different test cases verify the advantages of the suggested method in terms of accuracy versus computational cost.