Most approximation methods in high dimensions exploit smoothness of the function being approximated. These methods provide poor convergence results for non-smooth functions with kinks. For example, such kinks can arise in the uncertainty quantification of quantities of interest for gas networks. This is due to the regulation of the gas flow, pressure, or temperature. But, one can exploit that for each sample in the parameter space it is known if a regulator was active or not, which can be obtained from the result of the corresponding numerical solution. This information can be exploited in a stochastic collocation method. We approximate the function separately on each smooth region by polynomial interpolation and obtain an approximation to the kink. Note that we do not need information about the exact location of kinks, but only an indicator assigning each sample point to its smooth region. We obtain a global order of convergence of $(p+1)/d$, where $p$ is the degree of the employed polynomials and $d$ the dimension of the parameter space.

中文翻译：

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带扭结的分段平滑函数的单纯形随机搭配

大多数高维逼近方法都利用了被逼近函数的平滑性。这些方法为具有扭结的非平滑函数提供了较差的收敛结果。例如，这种扭结可能出现在天然气网络感兴趣的数量的不确定性量化中。这是由于气体流量、压力或温度的调节。但是，可以利用参数空间中的每个样本知道调节器是否处于活动状态，这可以从相应的数值解的结果中获得。可以在随机搭配方法中利用此信息。我们通过多项式插值在每个平滑区域上分别近似函数，并获得扭结的近似值。请注意，我们不需要有关扭结的确切位置的信息，但只有一个指标将每个样本点分配到其平滑区域。我们获得了 $(p+1)/d$ 的全局收敛阶数，其中 $p$ 是所采用的多项式的次数，而 $d$ 是参数空间的维度。