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:本文引入了一种可调整的多要素混合定位法,该方法还允许在以美元或美元为单位的精炼期间重新使用现有的模型评价。合用法以加权Leja节点为基础。在以美元为单位的精炼后,通过对每个新创建的子元素按等级方式添加和排序Leja节点,稳定了地方的内插。对于以美元为单位的精炼,当地多面近似值以总度或尺寸适应基数为基础。该方法用于前方和反向不确定性量化,以处理非光学或强定位反应表面。在几个测试案例中,也与相互竞争的方法相比,对拟议方法的性能进行了评估。