In this paper, we show that the approximation of high-dimensional functions, which are effectively low-dimensional, does not suffer from the curse of dimensionality. This is shown first in a general reproducing kernel Hilbert space set-up and then specifically for Sobolev and mixed-regularity Sobolev spaces. Finally, efficient estimates are derived for deciding whether a high-dimensional function is effectively low-dimensional by studying error bounds in weighted reproducing kernel Hilbert spaces. The results are applied to parametric partial differential equations, a typical problem from uncertainty quantification.

翻译：本文证明，对于本质上是低维的高维函数，其逼近过程不会遭受维度灾难。这一结论首先在一般的再生核希尔伯特空间框架下得到证明，随后特别针对Sobolev空间和混合正则性Sobolev空间进行了验证。最后，通过研究加权再生核希尔伯特空间中的误差界，推导出用于判定高维函数是否具有低维本质的有效估计方法。相关结果被应用于参数化偏微分方程——这是不确定性量化领域的典型问题。