  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.
