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.

翻译：本文提出了一种$hp$-自适应多单元随机配置方法，该方法允许在$h$-或$p$-细化过程中复用已有的模型评估结果。该配置方法基于加权Leja节点。在$h$-细化后，通过在每个新创建的子单元上以分层方式添加和排序Leja节点，实现局部插值的稳定性。对于$p$-细化，局部多项式近似基于总阶数或维度自适应基函数。该方法被应用于前向和逆不确定性量化中，以处理非光滑或高度局部化的响应面。通过多个算例对所提方法的性能进行了评估，并与竞争方法进行了比较。