论文标题
具有混合光滑度的功能的数值加权整合
Numerical weighted integration of functions having mixed smoothness
论文作者
论文摘要
我们研究了来自$ \ mathbb {r}^d $的加权积分的近似值，用于来自混合平滑度的加权Sobolev空间的integrands。我们证明，相对于来自这些空间的函数的$ n $集成节点，最佳四元素的收敛速率的上限和下限。在一维情况$（d = 1）$中，我们获得了最佳四足的合适收敛速率。对于$ d \ ge 2 $，上限是由稀疏网格二次构造在函数域中的台阶双曲线上的集成节点$ \ mathbb {r}^d $。
We investigate the approximation of weighted integrals over $\mathbb{R}^d$ for integrands from weighted Sobolev spaces of mixed smoothness. We prove upper and lower bounds of the convergence rate of optimal quadratures with respect to $n$ integration nodes for functions from these spaces. In the one-dimensional case $(d=1)$, we obtain the right convergence rate of optimal quadratures. For $d \ge 2$, the upper bound is performed by sparse-grid quadratures with integration nodes on step hyperbolic crosses in the function domain $\mathbb{R}^d$.
